9355.0-60
EPA 540-R-96-003
PB96-963302
January 1996
DOCUMENTING GROUND-WATER MODELING
AT SITES CONTAMINATED
WITH RADIOACTIVE SUBSTANCES
A Cooperative Effort By
Office of Radiation and Indoor Air
Office of Solid Waste and Emergency Response
U. S. Environmental Protection Agency
Washington, D.C. 20460
Office of Environmental Restoration
U.S. Department of Energy
Washington, D.C. 20585
Office of Nuclear Material Safety and Safeguards
Nuclear Regulatory Commission
Washington, D.C. 20555
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FOREWORD
This report is the product of the Interagency Environmental Pathway Modeling Working Group. The
working group includes representatives of the U.S. Environmental Protection Agency' s Office of
Radiation and Indoor Air and Office of Solid Waste and Emergency Response, the U.S. Department of
Energy' s Office of Environmental Restoration, and the U. S. Nuclear Regulatory Commission' s Office
of Nuclear Material Safety and Safeguards. The purpose of the Working Group is to promote the
appropriate and consistent use of mathematical models in the remediation and restoration process at
sites containing-er contaminated with^adioactive and/or mixed waste materials. This report
demonstrates a thorough approach to documenting model applications in a consistent manner and is
intended to assist technical staff responsible for identifying and implementing flow and transport models
in support of cleanup decisions at radioactive and hazardous waste sites. It is hoped that adoption of
the tenets in the report will enhance the understanding between modelers and their managers of what
may be expected in model documentation; facilitate the peer-review process by ensuring that modeling
documentation is complete; ensure the institutional memory is preserved; and institute greater
consistency among modeling reports.
This document is one of several the working group is developing to bring a uniform approach to
solving environmental modeling problems common to all federal agencies. The interagency working
group has also prepared the following reports:
Computer Models Used to Support Cleanup Decision-Making at Hazardous and
Radioactive Waste Sites, EPA 402-R-93-005, March 1993.
Environmental Characteristics of EPA, NRC, and DOE Sites Contaminated with
Radioactive Substances, EPA 402-R-93-011, March 1993.
Environmental Pathway Models-Ground Water Modeling in Support of Remedial
Decision-Making at Sites Contaminated with Radioactive Material, EPA 402-R-93-009,
March 1993.
A Technical Guide to Ground Water Model Selection at Sites Contaminated with
Radioactive Substances, EPA 402-R-94-012, June 1994.
The Project Officers of the Interagency Working Group (Beverly MaEPA, Paul BeanDOE, Sam
NalluswamiNRC) acknowledge the cooperation and insight of many staffers in preparing this
document from organizations including EPA Regions 2, 3, 4, 5, 6, and 8; EPA Office of Emergency
and Remedial Response; EPA Office of Underground Storage Tanks; EPA Robert S. Kerr
Environmental Research Center; EPA Office of Radiation Programs/Las Vegas; EPA National Air and
Radiation Environmental Laboratory; EPA Office of Radiation and Indoor Air Criteria and Standards
Division; DOE Office of Environmental Restoration; and NRC Office of Material Safety and
Safeguards, who graciously agreed to provide review and comment. We also thank their managers
who permitted them the time to provide us with valuable input.
This report was prepared under EPA Contract 68D20155, Mr. David Back, Project Officer, Sanford
Cohen & Associates.
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CONTENTS
Page
Foreword i
Executive Summary ES-1
1. Introduction 1-1
1.1 Purpose of This Manual 1-1
1.2 How to Use This Manual 1-1
1.3 Key Terms 1-2
1.4 Conceptual Approach 1-5
1.5 Standardization in Ground-Water Modeling 1-6
2. Modeling Objectives and Data Requirements 2-1
2.1 Scoping Phase 2-1
2.2 Site Characterization Phase 2-3
2.3 Data Requirements 2-4
2.4 Remedial Design and Implementation Phase 2-15
3. Conceptual Model Development 3-1
3.1 Preliminary Conceptual Model 3-1
3.2 Evolution of the Conceptual Model 3-5
3.3 Remedial Design and Implementation 3-13
4. Model Application 4-1
4.1 Scoping Calculations 4-2
4.1.1 Release Analysis - Ground Water 4-2
4.1.2 Fate Analysis - Ground Water 4-3
4.1.3 Analytical Methods for Aquifer Flow and Transport 4-3
4.1.4 Uncertainty Analysis 4-3
4.2 Site Characterization Modeling 4-6
4.2.1 Code Selection 4-7
4.2.2 Model Construction 4-11
4.2.2.1 Layering and Gridding 4-12
4.2.2.2 Definition of Boundary and Initial Conditions 4-13
4.2.2.3 Specification of Time Steps 4-16
4.2.2.4 Specifying Parameter Values in the Model 4-16
4.2.3 Calibration of the Model 4-17
4.2.4 Uncertainty and Sensitivity Analyses 4-19
4.3 Predictive Simulations 4-21
4.4 Baseline Risk Assessment 4-21
4.5 Exposure Estimation-Ground Water 4-22
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CONTENTS (Continued)
Page
5. Report Presentation Guidelines 5-1
Modeling Objectives and Data Requirements 5-1
Conceptual Model Development 5-2
Model Application 5-3
6. References 6-1
Appendix A Fate and Transport of Radionuclides A-l
Appendix B Scoping Analysis Procedures B-l
Appendix C Default Parameter Values C-l
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TABLES
No. Page
ES-1 Major Steps in Modeling Evaluation Procedures ES-5
1-1 General Modeling Approach as a Function of Project Phase 1-7
2-1 Data Requirements 2-5
4-1 Bounding Analyses: Maximum Extent of Contamination 4-5
4-2 Bounding Analyses: Maximum Concentration of Contamination 4-6
4-3 Boundary Conditions of Ground-Water Flow Equations 4-14
4-4 Boundary Conditions of Solute Transport Equations 4-16
5-1 Major Steps in Modeling Evaluation Procedures 5-9
FIGURES
No. Page
ES-1 Exposure Pathways ES-1
3-1 Typical Conceptual Model(s) in the Scoping Phase 3-2
3-2 Representative Conceptual Model of the Unsaturated Zone 3-5
3-3 Representative Conceptual Model of the Saturated Zone 3-5
4-1 Modes in Which Ground Water May Become Contaminated 4-1
4-2 Three-Dimensional View of Model Grid 4-12
4-3 Cross-sectional View of Model Grid 4-13
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EXECUTIVE SUMMARY
This joint EPA/DOE/NRC program is concerned
with the selection and use of mathematical models
that simulate environmental behavior and the
impacts of radionuclides via all potential
pathways of exposure, including the air, surface
water, ground water, and terrestrial pathways.
Figure ES-1 gives an overview of these various
exposure pathways.
Though the joint program is concerned with all
pathways, this report focuses on ground-water
pathways. Ground-water pathways were selected
for first consideration for several reasons. At
many sites currently regulated by EPA and NRC
or managed by DOE, the principal concern is the
existence of, or potential for, contamination of the
underlying aquifers. Compared to the
contamination of air, surface water, and terrestrial
pathways, ground-water contamination is more
difficult to sample and monitor, resulting in
greater dependence on models to predict the
locations and levels of environmental
contamination.
The types of models used to simulate the behavior
of radionuclides in ground water are generally
more complex than models for surface water and
atmospheric pathway transport. The additional
complexity is necessary to address the complexity
and diversity of settings associated with different
sites. The methods used to model ground water
are not as standardized as the methods for surface
water and air dispersion modeling, and there is
considerably less guidance on appropriate
methods for such modeling. The information
presented in this report is consistent with recent
standards on ground-water modelingpublished by
the American Society for Testing and Materials
(ASTM). ASTM is a private organization that
publishes consensus standards for a variety of
fields, including ground-water modeling. The
ASTM Subcommittee D 18.21 on Ground-Water
and Vadose Zone Investigations has approved
seven new standards related to ground-water
modeling. These standards have been written in
the form of guides (not rigid standards) and
include the following publications:
Pl,l«
Uptake
fa&fftw
«*
V,,Mf
S*l«.G«i«i
!K3f
3XT^
Figure ES-1. Exposure pathways
ES-1
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D-5447 Standard Guide for Application of a
Ground-Water Flow Model to a Site-
Specific Problem
D-5490 Standard Guide for Comparing Ground-
Water Flow Model Simulations to Site-
Specific Information
D-5609 Standard Guide for Defining Boundary
Conditions in Ground-Water Modeling
D-5610 Standard Guide for Defining Initial
Conditions in Ground-Water Modeling
D-5611 Standard Guide for Conducting a
Sensitivity Analysis for a Ground-Water
Flow Model Application
D-5718 Standard Guide for Documenting a
Ground-Water Flow Model Application
D-5719 Standard Guide for Simulating
Subsurface Air Flow using Ground-
Water Flow Modeling Techniques
This report describes a process by which ground-
water flow and transport modeling can be
systematically reviewed during each phase of the
remedial process. The phases include the initial
scoping phase, the detailed characterization of the
site, and the selection and implementation of
remedial alternatives.
The proper application of the selected model(s) is
as important, if not more important, than its
selection. No matter how well a model is suited
to a particular application, it could give erroneous
and highly misleading results if used improperly
or with incomplete or incorrect input data.
Conversely, even a model with very limited
capabilities, or a model used at a site which has
not been well characterized, can give very useful
results if applied properly with a full appreciation
of the limitations of both the model and the input
data. This report describes the methods for
applying ground-water flow and transport models
to sites contaminated with radioactive materials.
The model application process is described in
terms of the objectives, data availability, and
various site characteristics and processes requiring
modeling.
A review by EPA of 20 site-specific modeling
studies (LEE95) cited modeling mistakes in all
aspects of the modeling process including: (1)
misunderstanding of the selected model, (2)
improper application of boundary conditions and
or initial conditions, (3) misconceptualization, (4)
improper or unjustifiable estimation of input data,
(5) lackof orimproper calibration/verification, (6)
omission of or insufficient sensitivity/uncertainty
analysis, and (7) misinterpretation of simulation
results. Any of these errors can lead to the use of
faulty assumptions as the basis for remedial and
risk decisions.
This model review guide is designed to provide, at
a minimum, a means to determine whether proper
modeling protocol has been followed. In some
cases, the guide provides sufficient information to
ensure that common modeling pitfalls are avoided.
For example, one of the errors indicated by
LEE95 was that, in at least one of the
investigations, ground-water extraction well(s)
had been placed too close to the model boundary,
which resulted in an underestimation of the
ground-water capture zones predicted for these
wells. The section of this guide dedicated to
model construction discusses the correct
placement of wells relative to model boundaries
and provides a simple means for determining if
the well has been placed too close to the
boundary.
However, the goal of this review guide is not to
detail exactly how ground-water modeling is
performed. Instead, the intention is to provide a
means to ensure that all modeling reports are
properly documented and provide sufficient detail
to allow a comprehensive peer review.
A checklist containing the major review steps is
presented. With this checklist, the analyst for a
specific project can identify potential problem
areas in applying and documenting the model
activities. The major steps in evaluating the
model are listed in Table ES-1. The first step is to
identify the objectives of the modeling study. Do
ES-2
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the objectives correspond to the project's
objectives?
The second step is to examine in detail the site
characterization data provided by the modeler.
Are there sufficient data to characterize the site?
Are there sufficient data to match its history?
Were aquifer tests and tracer tests performed? If
so, how were they analyzed? Does the
distribution of wells give a sufficient vertical as
well as horizontal picture? Are the wells deep
enough to delineate the greatest depths to which
contaminants are expected to migrate? Do the
data provide information on the soil profile as
well as the water levels?
The next step is to review the conceptual
approach used by the modeler to represent the
ground-water flow and contaminant transport
processes occurring at the site. Here, the modeler
should attempt to identify and list the key
assumptions used in developing the conceptual
ground-water flow and transport models. The
justifications for the individual assumptions
should be carefully examined, in conjunction with
a general review of field information or data on
site characterization provided by the modeler.
The key objective of the examination shouldbe to
determine if the modeler's conceptual approach is
consistent with the field data and the objectives.
Specific questions that should be addressed
include:
Can a steady flow system be assumed, or
must transient flow conditions be considered?
What transport processes are important?
Which of these processes are not considered
in the conceptual model?
What are the features of site characterization
that support or repudiate the conceptual
modeling assumptions?
Can two-dimensional horizontal flow be
assumed or must three-dimensional flow
conditions be considered?
Assuming the conceptual approach is appropriate,
the reviewer should then examine the
methodology selected by the modeler to solve the
flow and transport problems. The objective at this
stage is to identify the particular analytical or
numerical model used by the modeler and
determine if it can reliably predict solutions to the
ground-water flow and transport problems
identified during the conceptual stage. Specific
questions that should be addressed include:
Can the model treat all of the important
components or features identified in the
conceptual model?
Does the model provide the type of results
that are necessary to satisfy the objectives set
forth at the beginning of the study?
If a series of flow and transport models are
selected, how do they fit together?
Is the computer code well-documented and
has it received thorough testing?
If a numerical model is used in obtaining the
solution, the following question pertaining to
spatial and temporal approximations also should
be asked:
Are the grids and time increments selected
for the flow and transport simulations
sufficiently refined to give results of
acceptable accuracy?
Are the grids free of numerical
instability caused by rapid changes in
grid spacing or time step size?
Next, the critical input parameters and boundaries
of the model shouldbe identified and the rationale
for selecting the parameter values and boundary
conditions assessed. Are the parameter values
based on site-specific data or on previous studies?
What data support the selected boundary
conditions? It is the modeler's responsibility to
insure that this result is consistent with field
evidence. Specific questions may include:
ES-3
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Are the boundary conditions consistent with
the conceptual model and with natural
hydrologic boundaries? (Watch for arbitrary
boundaries, such as geographic boundaries)
Will the selected boundaries influence model
predictive simulations because they are too
close to the area of interest?
Are the parameter values consistent with the
conceptual model and within the range of
reported or measured values?
Are the parameter values assigned in a
patchwork pattern? (A common problem is
that parameters are adjusted on a block-by-
block basis to achieve a good calibration
without regard for geologic evidence)
If the model is calibrated to the field data, the
comparison of the observed and simulated results
should be examined. History matching or model
calibration refines estimates of hydrologic
parameters and boundary conditions by comparing
the model results with observed data. Estimates
of parameters are changed to improve the
comparison. It is important to constrain the
changes so that physically realistic parameters are
specified; this generally requires an experienced
ground-water hydrologist. The history matching
procedure can be done either by a trial and error
or by automatic regression. No matter which
approach is selected, sensitivity analysis will be
part of the matching phase. If the model is not
calibrated, clear justification should be provided.
Specific areas to evaluate in model calibration
include:
reasonable range in order to achieve the
calibration goals?
Does the model report discuss the rationale
for selection of the final calibrated model
parameters?
In the final phase of the study, the future behavior
of the system is predicted. Generally, this is the
shortest part of a study. Predictions are based on
the results using the best estimate of the system's
parameters obtained by history matching.
Because the set of parameters is not unique, it is
important to assess the uncertainty in the
predicted results, which is usually accomplished
by using a sensitivity analysis. The model's
predictive results and the sensitivity analysis
should be examined to determine if sufficient
conservatism has been made in the simulation.
Any numerical error that may have been
introduced as a result of inappropriate solution
techniques or poor choice of grid spacing and time
increment should be assessed carefully.
Finally, the validity of the modeler's conclusions
should be reviewed. Do these conclusions satisfy
the original objectives? The modeler should trace
back each conclusion to ensure that the conclusion
is valid and follows from supporting
documentation. Is there sufficient information to
allow the modeling study to be reproduced?
It is the reviewer's responsibility to review data
and modeling results. It is critical that the
reviewer has sufficient experience to interpret
data and assess the conceptualization as well as to
evaluate the results.
Have calibration criteria been established and
have these criteria been met by the
calibration?
Are calibration errors (differences between
measured and computed
heads/concentrations) spatially biased? (e.g.,
too high in one area of the model and too low
in another)
Were model parameters varied beyond a
ES-4
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Table ES-1. Major Steps in Modeling Evaluation Procedures
MODELING AND EVALUATION CRITERIA
APPRAISAL
Yes
No
Comments
CHAPTER 2
OBJECTIVES AND DATA REQUIREMENTS
Are the purpose and scope outlined?
Are the objectives consistent with decision-making needs?
Are the objectives satisfactory?
Are a site description and waste disposal history provided?
Are the data requirements for the proposed modeling outlined?
Are the sources of data adequately presented?
Are data uncertainties discussed?
Is the probable sensitivity of the future modeling results presented for the
data?
Are the potential data limitations and weaknesses provided?
Are the plans to resolve data limitations discussed?
CHAPTER 3
CONCEPTUAL MODEL DEVELOPMENT
Is the physical framework discussed in detail?
Both regional and local?
Is the hydro geologic framework described in detail?
Both regional and local?
Is the nature of the contaminant source term described?
Are the hydraulic boundaries described in detail?
Are data base deficiencies clearly identified and modeling implications
discussed?
Is the conceptual model consistent with the field data?
Are the uncertainties inherent in the conceptual model discussed?
Are the simplifying assumptions outlined?
Are the assumptions justified?
Are the natural boundaries or the aquifer system described?
Are the following figures and/or tables included:
Map showing location of study area.
Geologic map and cross sections indicating the areal and vertical extent of
the system.
In some instances tabular representation of the data may be appropriate.
ES-5
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Table ES-1 (Continued)
MODELING AND EVALUATION CRITERIA
Topographic map with the surface water bodies.
Contour maps showing the tops and/or bottoms of the aquifers and
confining units.
Isopach maps of hydro stratigraphic units.
Maps showing extent and thicknesses of stream and lake sediments.
Maps indicating discrete features (e.g., faults), if present.
Maps and cross sections showing the unsaturated zone properties (e.g.,
thickness, Ksat).
Potentiometric surface maps of aquifer(s) and hydraulic boundaries.
Maps and cross sections showing storage properties of the aquifers and
confining units.1
Maps and cross sections showing hydraulic conductivity of the aquifers,
confining units and stream and lake sediments.
Maps and hydrographs of water-budget information.
Maps and cross sections indicating transport parameters (e.g., Kj).1
Areal and cross sectional isoconcentration maps of primary contaminants
in soil and ground water.
Time-series graphs of contaminant concentrations.
Relevant source-term inventory information.
APPRAISAL
Yes
No
Comments
CHAPTER 4
MODEL APPLICATION
Section
(4.1) SCOPING ANALYSIS
Are scoping analyses performed?
Do scoping results lead to proposed modeling approach?
(4.2) SITE CHARACTERIZATION MODELING
(4.2.1) Code Selection
Is the rationale for the selection clearly presented for proposed code(s)?
Are the general features of the code(s) presented?
Are the assumptions and limitations of the code(s) presented and compared
to the conceptual model?
Is the basis for regulatory acceptance presented?
Is the source documentation for the code included?
Is an executable version of the code included?
ES-6
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Table ES-1 (Continued)
MODELING AND EVALUATION CRITERIA
Is the source code readily available for inspection?
Does the code have a history of use?
Is the code well documented?
Is the code adequately tested?
Are the hardware requirements compatible with those available?
(4.2.2) Model Construction
(4.2.2.1) Layering and Gridding:
Is the domain of the grid large enough so that the boundaries will not
interfere with the results?
Do the nodes fall near pumping centers on existing and potential future wells
and along the boundaries?
Is the grid oriented along the principal axes of hydraulic conductivity?
Is the grid discretized at the scale appropriate for the problem?
Are areas of sharp contrasts (e.g., hydraulic conductivity, concentration,
gradient) more finely discretized?
Is the Peclet number less than 2?
Do adjacent elements vary in size by a distance less than a factor of 1.5?
Are strong vertical gradients within a single aquifer accommodated by
multiple planes or layers of nodes?
If matrix diffusion is important, are the confining units adequately
discretized in the relevant regions of the model?
Is the grid more finely spaced along the longitudinal direction of simulated
contaminant plumes?
Is the aspect ratio less than 100:1?
Are the following figures included:
Grid presented as an overlay of a map of the area to be modeled.
A vertical cross section(s) which displays the vertical layering of the
model grid.
(4.2.2.2) Boundary and Initial Conditions
Is justification provided for the selection of all boundary and initial
conditions?
Are model boundaries consistent with natural hydro logic features?
Are the boundary and initial conditions consistent with the conceptual
model?
APPRAISAL
Yes
No
Comments
ES-7
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Table ES-1 (Continued)
MODELING AND EVALUATION CRITERIA
Are the uncertainties associated with the boundaries and initial conditions
addressed?
Are the boundaries far enough away from any pumping/injection centered to
prevent "boundary effects"?
Are transient boundaries discussed?
Is the rationale given for simplifying the boundaries from the conceptual
model discussed?
Are the values for the assigned boundaries presented?
(4.2.2.3) Specification of Time Steps
Is the Courant criterion satisfied?
(4.2.2.4) Model Parameterization
Are data input requirements fully described?
Is the discussion of the data well founded with respect to Objectives and
Data Review Section?
Are the interpretation and extrapolation methods (e.g., Kriging) adequately
presented?
Do the figures and tables completely describe the data input with respect to
discrete components of the model?
Are the model parameters within the range of reported or measured values?
(4.2.3) MODEL CALIBRATION
Has calibration been attempted?
Is the rationale for model calibration approach presented?
Are the calibration procedures described in detail?
Are the calibration criteria presented?
Does the calibration satisfactorily meet specified criteria?
Is the rationale presented for selecting convergence criteria?
Are code convergences and numerical instabilities discussed?
Do the calibrated parameters fall within their expected ranges?
Are discrepancies explained?
Has the calibration been tested against actual field data?
Are the differences between steady-state and transient calibrations
presented?
Could other sets or parameters have calibrated the code just as well? Is this
discussed?
APPRAISAL
Yes
No
Comments
ES-5
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Table ES-1 (Continued)
MODELING AND EVALUATION CRITERIA
Are areal and cross-sectional representations of the final calibrated results
included for both hydraulic heads and radionuclide plume(s)?
Does calibration of the model take into account the inconsistency between
point measurements at wells and areal averages of model output?
Is the match between the calibration targets and final parameters shown
diagrammatically?
Were calibrating errors presented quantitatively through the use of
descriptive statistics?
If particle-tracking was performed, are these results shown?
Is the calibrated model consistent with the conceptual model?
Are any changes to the conceptual model discussed and justified?
Is non-uniform areal recharge applied? Is this approach justified?
Does the calibration prop erly account for vertical gradients?
Is the calibrated hydraulic conductivity field consistent with the geologic
logs and aquifer stress tests?
Are the convergence criteria appropriate?
Was a mass balance performed?
Is the water-balance error less than 1%?
Are the mass balance results for the calibrated model discussed?
Is the model's water balance consistent with known flows of rivers and levels
of lakes?
APPRAISAL
Yes
No
Comments
ES-9
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Table ES-1 (Continued)
MODELING AND EVALUATION CRITERIA
(4.2.4) SENSITIVITY ANALYSES
Was a sensitivity analysis performed?
Is the approach to the sensitivity analysis detailed?
Were all input parameters selected for investigation?
If not, was rationale presented for excluding parameters?
Was a sensitivity analysis performed on the boundary conditions?
Are the ranges of parameters appropriate?
Were sufficient simulations performed? Was justification provided?
Was the relevance of the sensitivity analysis results to the overall project
objectives discussed?
Are the results presented so that they are easy to interpret?
Were sensitivity analyses performed for both the calibration and the
predictive simulations?
APPRAISAL
Yes
No
Comments
ES-10
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CHAPTER 1 INTRODUCTION
1.1 Purpose of This Manual
A review by EPA of 20 site-specific modeling studies
(LEE 95) cited modeling mistakes in all aspects of the
modeling process including: (1) misunderstanding of
the selected model, (2) improper application of
boundary conditions and or initial conditions, (3)
misconceptualization, (4) improper or unjustifiable
estimation of input data, (5) lack of or improper
calibration/verification, (6) omission of or insufficient
sensitivity/uncertainty analysis, and (7)
misinterpretation of simulation results. All of these
errors could lead to the use of faulty assumptions as the
basis for remedial and risk decisions.
Chapter 2 Modeling Objectives and Data Review
The goal of Chapter 2 is to illustrate the connection
between the modeling objectives and data requirements
for each phase in the remedial process. Specifically, it
should allow the reviewer to ensure that the modeling
report identifies:
the data needed for modeling.
the origins of data.
how the data will be used to meet modeling
objectives.
Chapter 3 Conceptual Model Development
This manual is designed to provide, at a minimum, a
means to determine whether proper modeling protocol
has been followed. In some cases, the guide provides
sufficient information to ensure that common modeling
pitfalls are avoided. Specific goals of this manual
include:
Enhance understanding between managers
and modelers of what is expected in terms of
modeling documentation.
Facilitate the peerreview process by ensuring
that the modeling documentation is complete.
Ensure that institutional memory is created
and/or utilized.
Institute greaterconsistency among modeling
reports.
It is not the goal of this manual to detail exactly how
ground-water modeling is performed. Instead, it is
intended to provide a means to ensure that all modeling
reports can be properly documented and provide
sufficient detail to allow a comprehensive peer review.
Furthermore, this document is not intended to be used
as a sole reference for reviewing modeling application
studies. Rather it is intended to be used along with
other published general references (e.g., EPA87, 88a,
88b, 94a, 94b, 94c, 94d, ASTM93, 94, 95).
1.2 How to Use This Manual
This manual has been designed to provide a basic
understanding of modeling terminology, modeling
approaches, and documentation requirements to
facilitate the peer review process. The content's of its
five chapters and three appendices are outlined below.
This chapter is designed to ensure that the reviewer has
sufficient information to assess the adequacy of the
conceptual model presented in the modeling report.
Chapter 4 Model Application
This chapter discusses model application in each phase
of the remedial process (i.e., scoping, site
characterization, remedial selection and design).
Before designing the field investigation, it is advisable,
that at a minimum, a series of scoping calculations be
made to assess the potential importance of the ground-
water pathway. The scoping section in Chapter 4 is
closely linked to calculational methods presented in
Appendix B and is intended to provide a means for
making preliminary estimates of the rate of contaminant
migration and the expected down-gradient contaminant
concentrations using a calculator. General equations,
data requirements, and example problems are given.
An integral part of the discussion is a description of the
dominant physical and chemical processes that may
affect the fate and transport of radionuclides. A basic
understanding of these processes will give a general
appreciation of the complexity of the controlling
processes, and of the limitations inherent in the scoping
calculations.
Application of the model during the site
characterization and remedial phases generally is fairly
sophisticated and typically will be undertaken by
experienced modelers. Therefore, these sections
emphasize the overall modeling approach and methods
that can be used as simple reality checks on modeling
performed by others. Guidelines are given as to which
information should be requested to facilitate a peer
1-1
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Chapter 5 Report Presentation Guidelines
This chapter summarizes the previously presented
guidelines for the presentation of reports and provides
a step-by-step checklist to facilitate the review of the
modeling activities.
Appendix A - Fate and Transport of Radionuclides
This appendix consists of a discussion of ground-water
flow and physical and chemical transport processes that
affect the mobility of radionuclides in ground water.
Appendix B - Scoping Analysis Procedures
This appendix discusses in detail a series of screening
calculations that can be used to estimate radionuclide
transit times and concentrations.
Appendix C - Default Parameter Values
Typical values of parameters that are frequently used in
modeling studies are presented.
1.3 Key Terms
The key terms and concepts that are fundamental to
understanding this report are explained below.
Conceptual Model. The conceptual model of a site is a
flow diagram, sketch, and/or a description of a site and
its setting. The conceptual model describes the
subsurface physical system including the nature,
properties, and variability of the aquifer system (e.g.,
aquifers, confining units), and also the types of
contaminants or wastes at a site, where they are located,
and how they are being transported off site by runoff,
percolation into the ground and transport in ground
water, or suspension or volatilization into the air and
transport by the prevailing meteorological conditions.
The conceptual model also attempts to visualize the
direction and path followed by the contaminants, the
actual or potential locations of the receptors, and the
ways in which receptors maybe exposed, such as direct
contact with the source, ingestion of contaminated food
or water, or inhalation of airborne contaminants.
As information about a site accumulates, the conceptual
model is continually revised and refined, in order to
consolidate site and regional hydrogeologic and
hydro logic data into a set of assumptions and concepts
that can be evaluated quantitatively. More specifically,
the conceptual model identifies and describes important
aspects of the physical hydrogeologic system or
subsystem for a given purpose. At a minimum, the
system conceptualization should include: the geologic
and hydrologic framework, media type (e.g., fractured
or porous), the nature of relevant physical and chemical
processes, time dependence, dimensionality of the
system, initial and boundary conditions, hydraulic
properties, and sources and sinks (water budget). The
conceptualmodel shouldnot only be consistent with the
physical system but also must be internally consistent.
Each of the components typical of the hydrogeological
conceptual model is briefly discussed below.
Geologic framework. The geologic framework is the
distribution and configuration of the transmissive (e.g.,
sands and gravels) and nontransmissive (e.g., clay)
geologic units. Of primary interest are the thickness,
continuity, lithology, and geologic structure of those
units relevant to the study.
Hydrologic framework. The hydrologic framework in
the conceptualmodel includes the physical extent of the
flow system, hydrologic features that affect or control
the ground-water flow system, analysis of ground-water
flow directions, and media type. The conceptual model
must address the degree to which the system behaves as
a porous media. If the system is significantly fractured
or solution channeled, the conceptual model must
address these issues.
Hydraulic properties. The hydraulic properties include
the transmissive and storage characteristics of the
geologic units (aquifers) and properties of the fluids.
Specific examples of aquifer and fluid properties
include transmissivity, hydraulic conductivity,
storativity, fluid viscosity, and densities.
Sources and sinks. Sources and/or sinks of water
and/or gas affect the pattern and rate of flow and
therefore will affect the transport of radionuclides from
the source. The most common examples of sources and
sinks include pumping or injection wells, infiltration,
evapotranspiration, drains, and flow from surface water
bodies.
Boundary and initial conditions. Boundary conditions
are the conditions the modeler specifies, typically on
the perimeter of the model domain, as known or
estimated flux, head, or concentration values in order to
solve for the unknowns in the problem domain. These
values may be associated with either the ground-water
flow or the contaminant transport aspects of the
problem. Ground-water boundaries may be described
in terms of where water and/or radionuclides are
flowing into or out of the ground-water system. Many
different types of boundaries exist, including surface
water bodies, ground-water divides, rainfall, wells, and
geologic features such as faults and sharp contrasts in
lithology. These real-world boundaries must be
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translated into their mathematical counterparts which
include fixed-head or concentration, constant flux, or
head-dependent flux. The most common contaminant-
source type boundaries either specify the source
concentration or prescribe the mass flux of
contamination entering the system. Initial conditions
are defined as values of ground-water elevation,
pressure, flow volumes, or contaminant concentrations
which are initially assigned to interior areas of the
modeled regions at the start of the simulation.
Transport processes. Various mechanical and
geochemical processes affect the transport of
radionuclides by flow through either a porous matrix or
a fracture system in a porous matrix. The dominant
mechanical processes are advection, dispersion
(hydrodynamic dispersion, channeling) and diffusion.
The chemical processes potentially affecting
radionuclide transport include radioactive decay,
adsorption on mineral surfaces (both internal and
external to the crystal structure), speciation,
precipitation, colloidal transport, radiolysis,biofixation,
natural organic matter interactions, anion exclusion, and
complexation.
Spatial dimensionality. Ground-water flow and
contaminant transport are seldom constrained to one or
two dimensions. However, in some instances, it may be
appropriate to restrict the analysis to one or two
dimensions. The particular number of dimensions
should be chosen based on the modeling objectives and
the availability of field and/or laboratory data.
Temporal dimensionality. Either steady-state or
transient flow simulations can be performed. At steady-
state, it is assumed that the flow field remains constant
with time, whereas a transient system changes with
time. Steady-state simulations produce average or long-
term results and generally require that a true
equilibrium case be physically possible. Transient
analyses are typically performed when boundary
conditions vary through time or when study objectives
require answers at more than one time. It is also
possible to mix temporally dimensionality in a
modeling study. A common technique is to use a
steady-state flow model and a transient transport model.
A conceptual model describes the present condition of
the system. To predict future behavior, it is necessary
to develop a dynamic model, such as physical scale
models, analog models, or mathematical models.
Laboratory sand tanks are physical scale models that
simulate ground-water flow directly. The flow of
ground water can also be implied using electrical
analog models. Mathematical models, including
analytical and numerical methods, which are discussed
below, are more widely used because they are easier to
develop and manipulate.
Model application. The model application is the
process of choosing and applying the appropriate
software algorithm, or other analysis techniques,
capable of simulating the characteristics of thephysical
hydrogeologic system, as identified in the conceptual
model. To enhance understanding and facilitate
implementation of the model application criteria
developed in this report, the evolution of the computer
model is traced from the inception of the conceptual
model, its progression through to the mathematical
model, and finally to the development of the computer
code where computer instructions for performing the
operations that are specified in the mathematical model
are programmed.
Mathematical Model. A mathematical model is
essentially a mathematical representation of a process
or system conceptual model. For example, the
mathematical model for ground-water flow is derived
by applying principles of mass conservation (resulting
in the continuity equation) and conservation of
momentum (resulting in the equation of motion). The
generally applicable equation of motion in ground-
water flow is Darcy's Law for laminar flow, which
originated in the mid-nineteenth century as an empirical
relationship. Later, a mechanistic approach related this
equation to the basic laws of fluid dynamics. In order
to solve the flow equation, both initial and boundary
conditions are necessary.
Solution Methodology. Solution methodology refers to
the strategy and techniques used to solve a set of
mathematical equations. In ground-water modeling, the
equations are normally solved for head (water
elevations in the subsurface) and/or contaminant
concentrations.
Mathematical methods developed to solve the ground-
water flow and transport equations can be broadly
classified as either deterministic or stochastic.
Deterministic methods assume that a system or process
operates such that the occurrence of a given set of
events leads to a uniquely definable outcome, while
stochastic methods presuppose the outcome to be
uncertain and are structured to account for this
uncertainty.
Most stochastic methods are not completely stochastic
in that they often utilize a deterministic representation
of soil processes and derive their stochastic nature from
their representation of inputs and/or spatial variation of
soil characteristics and resulting chemical movement.
While the deterministic approach results in a specific
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value of a soil variable (e.g., solute concentration) at
pre-specified points in the domain, the stochastic
approach provides the probability (within a level of
confidence) of a specific value occurring at any point.
The development of stochastic methods for solving
ground-water flow is a relatively recent endeavor. It
occurred as a result of the growing awareness of the
importance of the intrinsic variability of the
hydro geologic environment and the fact that the
variability cannot be fully characterized. Stochastic
methods are still primarily research tools; however, as
computer speeds continue to increase, the use of
stochastic methods will spread from the research
community into mainstream management applications.
This discussion focuses primarily on deterministic
methods, due to their more widespread use.
Deterministic Methods. Deterministic methods may
either be broadly classified as either analytical or
numerical. Analytical methods usually involve
approximate or exact solutions to simplified forms of
the differential equations for water movement and
solute transport. Simple analytical methods are based
on the solution of applicable differential equations
which make a simplified idealization of the field and
give qualitative estimates of the extent of contaminant
transport. Such methods are simpler to use than
numerical methods and can generally be solved with the
aid of a calculator, although computers are also used.
Analytical methods are restricted to simplified
representations of the physical situations and generally
require only limited site-specific input data. They are
useful for screening sites and scoping the problem to
determine data needs or the applicability of more
detailed sophisticated methods.
Analytical methods are used in ground-water
investigations to solve many different kinds of
problems. For example, aquifer parameters (e.g.,
transmissivity, storativity) are obtained from aquifer
tests through the use of analytical methods, and ground-
water flow and contaminant transport rates can also be
estimated by analytical methods.
Analytical methods that solve ground-water flow and
contaminant transport equations in porous media are
comparatively easy to use. However, because the
governing equations are relatively simple, analytical
solutions are generally restricted either to radial flow
problems or to cases where velocity is uniform over the
area of interest. Except for some radial flow problems,
almost all available analytical solutions are developed
for systems having a uniform and steady flow. This
means that the magnitude and direction of the velocity
throughout the system are uniform with respect to time
and space, which requires the system to be
homogeneous and isotropic with respect to the
hydraulic conductivity.
Unfortunately, the equations of flow and continuity in
the form of partial differential equations do not lend
themselves easily to rigorous analytical solutions when
boundaries are complex. Therefore, if a realistic
expression for hydraulic head or concentration as a
function of space cannot be written from the governing
equations and boundary and initial conditions, then
analytical methods are generally abandoned, and more
sophisticated numerical methods are used to solve the
set of equations.
Numerical methods provide solutions to the differential
equations describing water movement and solute
transport using approximate methods such as finite
differences and finite elements. Numerical methods can
account for complex geometry and heterogenous media,
as well as for dispersion, diffusion, and chemical
retardation processes (e.g., sorption, precipitation,
radioactive decay, ion exchange, degradation). These
methods always require a digital computer, greater
quantities of data than analytical modeling, and an
experienced modeler.
A numerical model for ground-water flow consists of
the mathematical framework for the solution of the
material balance equations that govern laminar flow
through porous media. These mass balance equations
depend on physical constraints and constitutive
relationships. The constraints simply state conditions
that components of the mass balance equations must
satisfy, whereas the constitutive relationships describe
the dependence of parameters, in the mass balance
equations, on other physical processes. Furthermore,
the mass balance equations are composed of both
spatial and temporal terms, both of which require
discretization within the model domain. These terms
describe the head or concentration in space and time.
The numerical methods mentioned above (i.e., finite
element and finite difference) are used as discretization
methods for the spatial term, whereas finite-difference
methods are generally used to discretize the temporal
term.
The mass balance equations, physical constraints, and
constitutive relationships lead to a series of equations
that must be solved in space and time. The means by
which the equations are discretized, linearized (e.g.,
Newton-Raphson), organized (i.e., matrix construction),
and solved via either direct or iterative methods are all
part of the numerical model.
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Following the formulation of the numerical model, the
computer pro gram is developed. The program consists
of the assembly of numerical techniques, bookkeeping,
and control language that represents the model from
acceptance of input data and instructions to delivery of
output.
In summary, the conceptual model is an interpretation
or working description of the characteristics and
dynamics of a physical system. Model construction is
the process of transforming the conceptual model into
a simplified, mathematical description of the physical
system, coded in computer programming language,
together with a quantification of the simulated system
(in the form of boundary and initial conditions, system
and process parameters, and system stresses). An
intermediate step in the model transformation process
is the mathematical model which consists of two
aspects: a process equation and a solution technique to
solve the process equation. An analytical solution
solves a very simple process equation analytically by
hand calculations. An analytical model solves a more
complex, but still relatively simple, process equation
analytically with a computer program. A numerical
model solves a simple or complex process equation
numerically with a computer program. In the context of
this document, mathematical model refers to all three
solution techniques of a process equation. The
complexity of the process equation dictates the solution
technique required. The model formulation process
concludes with the coding of the mathematical model
into computer programming language for performing a
specified set of operations.
1.4 Conceptual Approach
One of the primary goals of mathematical modeling is
to synthesize the conceptual model into mathematical
expressions, which, in turn, are solved by selecting and
applying an appropriate computer code. This section
discusses how the different components of the
conceptual model, in conjunction with the modeling
objectives, influence the model selectionand ultimately
the model application.
The underlying premise of model application is that the
various aspects of the conceptual model may be
simulated in a variety of ways, but the selected
approach must remain consistent with the objectives.
That is, the physical system cannot be overly simplified
to meet ambitious objectives, and less demanding
objectives should not be addressed with highly
sophisticated modeling approaches.
Table 1-1 presents an overview of how the overall
approach to modeling a site differs as a function of the
stage of the remedial process. The most common code
application mistakes are applying codes that are more
sophisticated than are appropriate for the available data
or the level of the result desired, and the application of
a code that does not account for the flow and transport
processes that dominate the system. For example, a
question that often arises is: when should three-
dimensional codes be used as opposed to two-
dimensional or one-dimensional codes? Inclusion of
the third dimension requires substantially more data
than one- and two-dimensional codes. Similar
questions involving underlying assumptions need to be
considered in the selection of a modeling approach and
the physical processes which are to be addressed. If the
modeler is not practical, sophisticated approaches may
be used too early in the problem analysis. In other
instances, the complexity of the modeling is
commensurate with the qualifications of the modeler.
An inexperienced modeler may take an unacceptably
simplistic approach. One should begin with the
simplest code that will satisfy the objectives and
progress toward the more sophisticated codes until the
modeling objectives are achieved.
1.5 Standardization in Ground-Water Modeling
On October 26, 1993, the Office of Management and
Budget (OMB) issued a revised version of OMB
Circular A-119, "Federal Participation in the
Development and Use of Voluntary Standards." The
revised circular encourages greater agency use of
voluntary standards. It reaffirms the basic federal
policy that voluntary standards should be given
preference over nonmandatory government standards
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unless the use of such voluntary standards would
adversely affect performance or cost, reduce
competition, or have significant disadvantages. Even
before this circular was revised, the American Society
for Testing and Materials (ASTM), U.S. EPA, the
USGS, and the U.S. Navy entered into a cooperative
agreement in 1988 (the Navy joined in 1990) to
accelerate the development of voluntary consensus
standards by ASTM. The cooperative agreement funds
eleven task groups within ASTM's Subcommittee
D18.21 on Ground-Water and Vadose Zone
Investigations. Task Group 10 (D18.21.10) was formed
to develop standards on subsurface fluid-flow modeling
and has produced a total of seven standards to date,
with numerous standards in draft form. The
information presented in this report is consistent with
these recent standards on ground-water modeling
published by ASTM. These standards have been
written in the form of guides (not rigid standards) and
include the following publications:
D-5447 Standard Guide for Application of a Ground-
Water Flow Model to a Site-Specific Problem
D-5490 Standard Guide for Comparing Ground-Water
Flow Model Simulations to Site-Specific
Information
D-5609 Standard Guide for Defining Boundary
Conditions in Ground-Water Modeling
D-5610 Standard Guide for Defining Initial Conditions
in Ground-Water Modeling
D-5611 Standard Guide for Conducting a Sensitivity
Analysis for a Ground-Water Flow Model
Application
D-5718 Standard Guide for Documenting a Ground-
Water Flow Model Application
D-5719 Standard Guide for Simulating Sub surface Air
Flow using Ground-Water Flow Modeling
Techniques
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Table 1-1. General Modeling Approach as a Function of Project Phase
Attributes
Accuracy
Temporal
Representation of
Flow and Transport
Processes
Dimensionality
Boundary and Initial
Conditions
Assumptions
Regarding Flow and
Transport Processes
Lithology
Methodology
Data Requirements
Scoping
Conservative
Approximations
Steady-State Flow
and Transport
Assumptions
1 -Dimensional
Uncomplicated
Boundary and
Uniform Initial
Conditions
Simplified Flow
and Transport
Processes
Homogeneous/
Isotropic
Analytical
Limited
Characterization
Site-Specific
Approximations
Steady-State
Flow/Transient
Transport
Assumptions
1 ,2-Dimensional/
Quasi-3 -Dimensional
Nontransient
Boundary and
Nonuniform Initial
Conditions
Complex Flow and
Transport Processes
Heterogeneous/
Anisotropic
Semi-
Analytical/Numerical
Moderate
Remediation
Remedial Action
Specific
Transient Flow and
Transport
Assumptions
Fully 3 -Dimensional/
Quasi-3 -Dimensional
Transient Boundary
and
Nonuniform Initial
Conditions
Specialized Flow and
Transport Processes
Heterogeneous/
Anisotropic
Numerical
Extensive
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CHAPTER 2 -^MODELING OBJECTIVES AND DATA REQUIREMENTS
Successful ground-water modeling must begin with an
approach that is consistent not only with the site
characteristics but also with the modeling objectives,
which depend strongly on the stage of the remedial
process (i.e., scoping vs. site characterization vs. the
selection and implementation of a remedy).
The most common mistakes in applying models are in
using models that are more sophisticated than are
appropriate for the available data or the level of the
result desired, and in using a model that does not
accurately account for the flow and transport processes
that dominate the system. The simplest model that will
satisfy the objectives should be used first, progressing
toward more sophisticated ones as understanding of
the system improves and objectives change.
The remedial process is generally structured in a way
that is consistent with this philosophy (i.e., as the
investigation proceeds, additional data become
available to support more sophisticated ground-water
modeling). The data available in the early stage of
remediation may limit the modeling to one or two
dimensions. In certain cases, this may be sufficient
to support decision-making. If the modeling
objectives cannot be met in this manner, additional
data will be needed to support more complex models.
The selection of more complex models in the later
phases often depends on the results obtained with
simpler models during the early phases.
Generally, in the later phases of the investigation,
enough data have been collected to meet more
ambitious objectives through complex three-
dimensional modeling. The necessary degree of
sophistication of the modeling effort can be evaluated
in terms of both site-related issues and objectives, as
well as the qualities inherent in the computational
methods available for solving ground-water flow and
transport.
Modeling objectives for each stage of the remedial
investigation must be very specific and well-defined
early within each phase. All too often modeling is
carried out without a clear rationale to meet the
objectives, and only after modeling is completed are
the weaknesses in the approach discovered.
The objectives must consider the decisions that the
results are intended to support. The selected approach
should not be driven by the availability of data, but by
the modeling objectives which should be defined in
terms of what can be accomplished with the available
data; also, the objectives should be reviewed and
possibly revised during the modeling process.
Furthermore, ground-water modeling should not be
thought of as static or linear process, but rather one
that can be continuously adapted to reflect changes in
modeling objectives, data needs, and available data.
2.1 Scoping Phase
A large part of ground-water modeling in this early
phase is understanding the decisions that need to be
made and determining which of these, which can be
assisted by using specific calculations when the data
are limited and the controlling hydrogeologic
processes at the site are not completely understood.
In the scoping phase, the objectives generally focus on
establishing order-of-magnitude estimates of the extent
of contamination and the probable maximum
radionuclide concentrations at actual or potential
receptor locations. At most sites, the migration rates
and contaminant concentrations are influenced by
several parameters and flow and transport processes
that typically are not fully characterized in the early
phase of the investigation. The parameters include
recharge, hydraulic conductivity, effective porosity,
hydraulic gradient, distribution coefficients,
thicknesses of the aquifer and confining unit, and
source concentrations. During this early phase,
questions pertaining to flow and transport processes
typically are limited to general considerations, such as
whether flow and transport are controlled by porous
media or fractures, and whether the wastes are
undergoing transformations from one phase to another
(e.g., liquid to gas).
At this point in the remedial program, one of the most
useful analyses is to evaluate the potential effects of
the controlling parameters on flow and transport. One
objective of the early analyses is to assess the
relationships among the parameters. How do changes
in one parameter affect the others and the outcome of
the modeling exercise? A better under standing of such
interdependencies assists in properly focusing the site-
characterization activities and ensuring that they are
adequately scoped. Also, it is desirable to evaluate the
effects that various processes have on flow and
transport; however, this generally has to be deferred
until additional information is obtained during site
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characterization. Furthermore, some caution is
needed: if simplistic assumptions have been made in
the model, the results may not be valid (i.e.,
transferable) to a more refined model that incorporates
more realistic or complex boundary conditions, initial
conditions, or variations in parameters.
In general, the uncertainty associated with each of the
parameters is expressed by a probability distribution,
which yields a likely range of values for each
parameter. At this early phase in the modeling
process, it is important to use a modeling approach
where values for individual parameters can be selected
systematically from the probable range and easily
substituted into the governing mathematical equations
describe the dominant flow and transport processes at
the site. In this manner, the effects that a single
parameter, or a multitude of parameters, have on the
rate of movement and concentrations of contaminants
may be evaluated. This technique of substituting one
value for another from within a range of values is
called a sensitivity analysis. It is important to ensure
that the range of individual values and combinations of
parameters selected allow for a conservative analysis
of the flow and transport processes.
In many cases, the potential range of values of
important parameters is unknown or very large.
Consequently, the analyst has little alternative but to
evaluate the sensitivity of the results to a very broad
range of possible values for the parameters. Many of
these results will be unrealistic but cannot be ruled out
until reliable site data are obtained during site
characterization. These types of analyses are useful
because they help to direct the field work. However,
they also can be used incorrectly. For example,
individuals not familiar with the scoping process could
come reach grossly inappropriate conclusions about
the potential public health impacts of the site based on
these results. Accordingly, care must be taken to
assure that the results of scoping analyses are used to
support the decisions for which they were intended.
An alternative to a detailed sensitivity analysis is a
conservative bounding approach. In this less
demanding analysis, values are selected from the
range of parameters to provide the highest probability
that the results are conservative (i.e., that the
contaminant migration rates and concentrations would
not be underestimated). For example, high values of
hydraulic conductivity combined with low effective
porosities and low distribution coefficients would
maximize the predicted migration rates of the
contaminant, although the higher flow rates may dilute
the concentrations predicted to reach the receptors.
Even though efforts are made to ensure a conservative
analysis, natural as well as anthropogenic influences
may adversely affect the migration of radionuclides.
For instance, published distribution coefficients are
frequently determined at neutral pH values. However,
even values conservatively selected from the low
range could be too high if acidic wastes were also
discarded with the radioactive material. Burrowing
animals and construction activities also have been
responsible for moving radioactive wastes beyond the
boundaries predicted by ground-water flow and
transport models.
Other processes that could invalidate an otherwise
conservative analysis include facilitative transport and
discrete features, such as soil macropores. Facilitative
transport is a term used to describe the mechanism by
which radionuclides may couple with either naturally
occurring material or other contaminants and move at
much faster rates than would otherwise be predicted
by their respective distribution coefficients.
Furthermore, discrete features are rarely considered in
early analyses, even though it is well known that
some, such as soil macropores, can allow the
movement of contaminants on the order of meters per
year in the vadose zone. Such features can result in a
gross underestimate of the time of arrival and
concentration of contaminants downgradient.
Nonetheless, the lack of site-specific data will
generally preclude the mathematical modeling of
anomalous flow and transport processes during the
project' s scoping phase. Therefore, it' s possible that
what normally would be considered conservative
modeling results actually underestimate the velocities
and concentrations of the contaminant. This
possibility highlights the need to confirm the modeling
results with site-specific field data, even when a
conservative approach has been taken.
In the scoping phase, the data generally available have
been collected over relatively short intervals.
Therefore, modeling objectives would be limited to
those which could be met without a detailed
understanding of the temporal processes affecting flow
and transport. For example, a typical analysis that
may not require detailed knowledge of the temporal
nature of recharge, source release rates, and other
flow and transport mechanisms would be an estimate
of the distance that radionuclides have traveled since
the beginning of waste management activities. This
analysis would use yearly average values for the input
parameters, such as infiltration and source release
rates. However, without accommodating the
transience of these processes, predictions of peak
concentrations of contaminants arriving at
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downgradient receptors would be associated with a
high degree of uncertainty.
Site-specific information is often limited in the scoping
phase. Therefore, modeling during the early planning
phase of most remedial investigations generally is
designed to support relatively simple objectives that
can be easily linked with more ambitious goals
developed during later phases of the investigation.
The iterative process of data collection, analysis, and
decision-making itself dictates that the preliminary
objectives must evolve to meet the needs of the overall
program. That is, it would be unreasonable to
assume that initial modeling based upon limited data
would do little more than direct future activities.
2.2 Site Characterization
In the site characterization phase, the plans developed
during the scoping phase are carried out. The
collected field data are used to characterize more fully
the nature and extent of the contamination at the site,
to define environmental and demographic
characteristics, and to support assessments of the
actual or potential impacts. The results of the site
characterization are analyzed to determine compliance
with applicable regulations and to begin to define
strategies for remediation.
The site characterization phase typically provides the
first opportunity to gain a detailed understanding of the
overall behavior of the system. This improved
understanding leads to a refinement of the conceptual
model and, in turn, allows more ambitious objectives
to be entertained.
The primary reasons for ground-water modeling in the
site characterization phase of the remedial process are
to: (1) refine the existing conceptual model; (2)
optimize the effectiveness of the site characterization
program; (3) support the baseline risk assessment; and
(4) provide preliminary input into the remedial
approach. To accomplish these goals, it is generally
necessary to apply relatively complex ground-water
models to simulate flow and transport in the saturated
zone and, in some instances, the unsaturated zone.
A properly designed site characterization program will
expand the data base to address very specific, often
demanding objectives. Consequently, the simplified
modeling approaches undertaken in the scoping phase
give way to more sophisticated means of evaluating
the data, but also convey far more complications in
developing the proper approach.
In many instances, several different approaches to
modeling will be taken to accomplish these objectives.
For example, the output of analytical modeling of the
unsaturated zone, in the form of radionuclide
concentrations at the interface between the saturated
and unsaturated zone, may be used as input to
numerical models of the saturated zone. Regardless of
the phase of the remedial process, the simplest
modeling approach that meets the objectives should be
taken.
The site characterization program is the first time in
the investigation that detailed flow and transport
processes are identified and investigated. Before site
characterization, the investigator could only evaluate
the effects of various parameter values on flow and
transport. In the scoping phase, the modeling focused
on estimating the dominant parameters rather than on
the effects that more complex chemical and physical
flow mechanisms have on the fate and transport of
contaminants. Examples of these mechanisms include
fractures, time-dependence of physical and chemical
processes, phase transformations, and changes in the
geochemical environment.
It is important to gain an appreciation for the
governing geochemical processes, as they may have a
significant impact on the transport of radionuclides,
and can be simulated indirectly in the analysis by
assuming a specified retardation of the contaminant.
Direct means (computer codes) for simulating
geochemical processes are available; however, a
detailed discussion of these methods is beyond the
scope of this report.
As additional data are acquired during the site
characterization program and the system boundary
conditions and hydrogeology become better
understood, the modeling approach becomes more
involved. Without the data limitations that constrained
the choice of methods in the scoping phase, the
number of possible alternatives in the modeling
approach increases dramatically.
2.3 Data Requirements
At most sites, the parameters that influence migration
rates and concentration, flow, and transport processes
of the contaminant would not have been fully
characterized in the early phase of the investigation.
These parameters are the basis upon which the early
conceptual model is formulated, and include variables
such as recharge, hydraulic conductivity, effective
porosity, hydraulic gradient, distribution coefficients,
aquifer thicknesses, and source concentrations. As the
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site characterization proceeds additional data are
obtained to support more advanced modeling. The
following information is generally required to estimate
the concentrations of contaminants released, although
the precise data needs will depend on the modeling
objectives:
Release Concentration
1. Curies of radionuclide(s)
2. Water solubility of radionuclide(s) (optional)
3. Half-life of the radionuclide
4. Distribution coefficient(s) of radionuclide(s)
5. Saturated hydraulic conductivities of soil
6. Source dimensions
7. Soil bulk densities
8. Total porosities
9. Volumetric water content(s)
10. Infiltration rates
11. Soil-specific moisture-release curve
Volumetric Release Rate
Percolation rate (evapotranspiration,
precipitation, runoff)
Area of contributing source
Water solubility of radionuclide(s)
Hydraulic conductivities
Hydraulic gradient
Table 2-1 shows the data typically required for
ground-water modeling at each stage in the remedial
process. For convenience, the data have been grouped
into three general categories: the Physical
Framework, the Hydrogeologic Framework, and
Source Characteristics. Data in the physical
framework category define the geometry of the
system, including the thickness and areal extent of
each hydrostratigraphic unit. Hydrogeologic data
include information on the system's boundary
conditions as well as the properties of the aquifer.
Source characteristics pertain to the contaminated
zone, which is the below-ground region within which
radionuclides are present in above-background
concentrations. Sometimes referred to as the source
term, it serves as the starting point for all pathways.
The following is a brief discussion of the relevance of
the individual parameters to the overall conceptual
models.
To estimate the velocity of ground water through the
unsaturated zone, the following information is needed:
1. Average percolation or recharge rate
2. Average volumetric water content
To estimate the velocity of ground water through the
saturated zone, the following information is needed:
1. Hydraulic conductivities (vertical and
horizontal)
2. Hydraulic gradient
3. Effective porosities
To estimate migration of radionuclides through the
saturated or unsaturated zones, the following
information is needed:
1. Pore-water velocity
2. Concentration in the liquid phase (optional)
3. Dispersion coefficients in the x, y, and z
directions (optional)
4. Decay coefficients (half-life)
5. Retardation factors (bulk density, distribution
coefficient, effective porosity)
2-4
-------
Table 2-1. Data Requirements
TYPICAL SOURCES OF INFORMATION
Scoping Calculations
Site Characterization/
Remedial Design
PHYSICAL FRAMEWORK
Depth to Ground Water
Areal Extent and Thickness of Aquifer(s)
Areal Extent and Thickness of Confining
Unit(s)
Elevation of Unsaturated Zone Base
Elevation of Top and Bottom of
Aquifer(s) and Confining Units
Areal Extent and Thickness of Stream and
Lake Sediments
Location and Orientation of Discrete
Features
Topographic Maps
Literature Values
Literature Values
Literature Values
Literature Values
Literature Values
Literature Values
Well Logs/Water -Level Data
Monitoring Well Logs/Geophysics
Monitoring Well Logs/Geophysics
Soil Borings/Elevation Survey
Monitoring Well Logs/Elevation
Survey /Geophysics
Soil Boring
Surface Mapping/Aerial Photo
Interpretation
HYDROLOGIC FRAMEWORK
UNSATURATED ZONE
Hydrodynamic Dispersion
Infiltration Rate
Moisture-Release Curve Parameters
Saturated Hydraulic Conductivity (ksat)
Soil Water Content
Gradient
Total Porosity
Bulk Density
Distribution Coefficients (kj)
SATURATED ZONE
Gradient
Hydraulic Conductivity
Storage Properties
Effective Porosity
Literature Values
Literature Values
Table C-2
Table C-3
Table C-l
Unit Gradient Assumed
Table C-9
Literature Values
Table C-5
Topographic Map
Table C-4
Literature Values
Table C-10
Literature Values
Field Measurements
Laboratory Test
Laboratory Analyses
Laboratory Analyses
Unit Gradient Assumed
Laboratory Analyses/ Field
Analyses
Laboratory Analyses/Literature
Values
Laboratory Analyses/Literature
Water Level Measurements/
Elevation Surveys
Aquifer Stress Tests
Aquifer Stress Tests
Field Tracer Tests/Laboratory
Analyses/Literature
2-5
-------
Table 2-1 (Continued)
TYPICAL SOURCES OF INFORMATION
Scoping Calculations
Site Characterization/
Remedial Design
HYDROGEOLOGIC FRAMEWORK
Bulk Density
Dispersivity
Distribution Coefficients
Diffusion Coefficients
Receptor Location(s)
HYDRAULIC BOUNDARIES
Precipitation
Evapotranspiration/Runoff
Surface- Water/Ground- Water
Interactions
Ground- Water Pumping
Literature Values
Table C-ll
Table C-5
Literature Values
Literature Values
Literature Values
Literature Values
Literature Values
Assumed Rates
Laboratory Analyses/Literature
Values
Literature Values/Field Tracer
Tests
Laboratory Analyses/Literature
Values/Field Tracer Tests
Literature Values/Laboratory
Experiments
Field Survey
Field Measurements/Literature
Values
Independent Calculations/
Literature Values
Field Measurements
Field Measurements
SOURCE CHARACTERISTICS
Areal and Vertical Extent
Composition
Release Mechanism
Concentration
Radioactive Decay
Assumption Guidelines
Historical Site Activities
Calculations (Appendix C)
Assumption
Literature Values
Radiation Surveys/ Soil Borings/
Immunoassays
Chemical Analyses
Soil Borings/Ground- Water
Monitoring Data/Site History
Field Data
Literature Values
2-6
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Physical Framework
The physical framework of the system defines the
geometry of the system and includes the depth to
water, areal extent of hydrostratigraphic units and
fluvial deposits, and discrete features. The relevance
of each of these data to ground-water flow and
contaminant transport is discussed below.
Areal Extent and Thickness of Hydrogeologic Units.
In heterogeneous formations, hydraulic properties
change spatially. These properties may include the
aquifer and/or confining unit (i.e., aquitard) thickness.
The thickness of the units directly impacts the volume
of flow and therefore, mass transport through the
system. Furthermore, areas where the confining units
are thin or discontinuous would provide avenues for
radionuclides to move more freely among aquifers.
For these reasons, the determination of the areal
extent and thicknesses of the hydrogeologic units is
one of the primary objectives during the site
characterization program.
The thickness of the aquifer is generally not a required
parameter for the scoping calculations, although it
does set upper bounds on the maximum amount of
vertical mixing that could potentially occur as is
discussed in Appendix B.
Areal Extent and Thickness of Stream and Lake
Sediments. The physical properties (e.g., hydraulic
conductivity, sorption properties) of fluvial deposits
are typically different from the underlying aquifer.
Therefore, ground-water flow and radionuclide
transport into and out of the ground-water system may
be very sensitive to the degree of interconnection of
the aquifer system with surface-water bodies.
Location and Orientation of Discrete Features. The
presence of discrete features, such as fractures, faults
and macropores could have a significant effect on the
ground-water flow and radionuclide transport.
Modeling flow through the unsaturated zone is
generally based on the assumption that the soil is a
continuous unsaturated solid matrix that holds water
within the pores. Actual soil, however, has a number
of cracks, root holes, animal burrows, etc., where the
physical properties differ enormously from the
surrounding soil matrix. Under appropriate
conditions, these flow channels have the capacity to
carry immense amounts of water at velocities that
greatly exceed those in the surrounding matrix. At
present, there is no complete theory describing water
flow through these structural voids or macropores.
There is uncertainty regarding the significance of
subsurface voids in water flow, since, if large, they
should fill only when the surrounding soil matrix is
close to saturation. Nonetheless, studies have shown
that contaminants can migrate to substantial depths
with only a small amount of water input.
Ground-water flow and radionuclide transport in the
saturated zone may be strongly influenced by the
presence of fractures. When a radionuclide is
introduced into a fractured porous medium, it migrates
through the fracture openings by means of advection
as well as hydrodynamic dispersion. The radionuclide
may also diffuse slowly into the porous matrix.
Molecular diffusion dominates flow and transport
within the porous matrix because the fluid velocity in
the porous matrix is usually very small. Upon
introduction of the radionuclide into a fractured
aquifer, the radionuclide moves rapidly within the
fracture network. As time progresses, the zone of
contamination will diffuse farther into the porous
matrix. Since the porous matrix has a very large
capacity to store the contaminant, it plays a significant
role in retarding the advance of the concentration front
in the fractures. If the source of contamination is
discontinued and the aquifer is flushed by fresh water,
the contaminant mass in the fractures will be removed
relatively quickly, whereas the contaminant in the
porous matrix will be removed very slowly via
diffusion back to the fracture openings.
Hydrogeologic Framework
The hydrological data have been divided into those
associated with the unsaturated zone and saturated
zones. These data include information on aquifer
properties, hydrologic stresses, and hydraulic heads.
The relevance of each of these to ground-water flow
and contaminant transport is discussed below.
Unsaturated Zone
The unsaturated zone is the zone between the land
surface and the water table including the capillary
fringe. In the subsurface environment, contaminants
migrate through this partially saturated zone (i.e.,
unsaturated zone) prior to reaching the saturated
zone. In this zone, flow is usually assumed to be in
the vertical direction. The flow is generally one-
2-7
-------
dimensional: therefore, scoping calculations are also
performed in one dimension. Generally, water in this
zone is under less than atmospheric pressure, and
some of the voids may contain air or other gases at
atmospheric pressure. Beneath flooded areas or in
perched water bodies the water pressure locally may
be greater than atmospheric.
The volumetric flux of liquid moving under isothermal
and isosmotic conditions through a partially saturated,
natural hydrogeologic unit, regarded as an equivalent
porous-medium continuum system, is determined by
the spatial gradients of matrix and gravitational
potentials and by the hydraulic properties. This
functional dependence obeys "Darcy's Law" for
unsaturated liquid flow (Appendix B).
The means by which water transports radionuclides
through the unsaturated zone is a complex process.
The passage of water is dynamic and depends on
detailed variations of the hydraulic properties of the
soil through which the water passes. Water storage by
a soil profile is characterized by water content
distribution, which ultimately depends on the detailed
spatial variability of hydraulic properties. Infiltrating
water that exceeds the soil water-holding capacity will
contribute to the net recharge of the underlying
aquifer.
A rigorous analysis of the flow and transport processes
through the unsaturated zone is accompanied by
demanding data requirements. However, rarely in the
scoping phase of the investigation would detailed data
be available. Even during the site characterization
phase, these data are rarely available. A discussion of
the data that are required for both scoping and site
characterization modeling of the unsaturated zone are
presented below.
Infiltration Rate (Recharge). Water from a
precipitation event moves downward through the soil
under the influence of gravity and matric pressures.
Water is extracted from the unsaturated zone as
surface evaporation and as plant transpiration; together
these processes are termed evapotranspiration. The
rates of both extraction processes depend primarily on
available solar energy, surface winds, and plant and
soil type.
A number of simple methodologies are available to
estimate the fraction of precipitation that recharges
into the aquifer (i.e., precipitation minus
evapotranspiration and runoff). Recharge estimates
are described by DAS77, FEI75, andTHOSS, and 57.
In areas drained by perennial streams, recharge may
be estimated by base-flow separation methods.
A somewhat more sophisticated method is
incorporated in the Hydrologic Evaluation of Landfill
Performance (HELP) model (SCH83 and 84). HELP
is a quasi-two-dimensional model that computes the
daily water budget for a landfill represented as a series
of horizontal layers.
As a first, rough approximation, net recharge can be
estimated by subtracting pan evaporation from
precipitation rates, both of which are generally readily
available. This approach would overestimate net
recharge because runoff is assumed to be negligible.
Typically, higher infiltration rates result in greater
health risks, but there are exceptions to this guideline.
When radium is a concern, the risk associated with
direct exposure to radium is generally greater than the
radium risk associated with the ground-water pathway.
Therefore, higher infiltration rates tend to flush the
radium from the source and, in turn, reduce the direct
exposure.
Soil Type. The soil type (e.g., sand, silt, clay) may be
used to obtain qualitative estimates of saturated
hydraulic conductivities, porosities, and the moisture
release information. In conjunction with infiltration
rates, the soil type can be used to make preliminary
estimates of moisture content. The moisture content
is assumed to fluctuate between field capacity and
saturation with the effective hydraulic conductivity
based on an empirical equation (Appendix B). Typical
values for these parameters are presented in Appendix
C. Appendix B describes how these parameters are
used to solve radionuclide transport.
Hydraulic Gradient. After water infiltrates beneath
the ground surface, it generally travels vertically
downward under the influence of gravity and matric
(i.e., suction) forces until it reaches the water table.
The gravity and capillary forces make up the hydraulic
gradient. Under partially saturated conditions, liquid
water is bound to the solid within the pore and fracture
openings either by surface-tension (capillary) forces
or, at very low saturations, by physical or chemical
adsorption. The strength of the bonding force is
measured in terms of an equivalent negative pressure,
or pressure head, designated as the matric potential.
2-8
-------
Since the gravitational head gradient has the value of
unity, it follows that scoping calculations will not
require site-specific information pertaining to the
partially saturated zone hydraulic gradient. For more
complex site characterization modeling, however,
capillary pressure relationships are used to determine
hydraulic gradients.
Matric potential is a function of liquid-water
saturation. Typically, an analytic or graphical
representation of the functional relationship defines the
moisture-retention curve for the porous medium.
Moisture-retention curves for most media are not
unique; they display hysteresis in which the precise
relation between matric potential and saturation
depends on the wetting and drying history of the
medium.
Standard techniques, using mercury intrusion,
pressure-plate apparatus, thermocouple psychrometers,
and centrifuges are used to measure the moisture-
retention curves for small soil and rock samples.
Thickness. The thickness of the unsaturated zone or
depth to water affects the travel time of radionuclides
leached from the surface to the ground water.
Typically, very little dilution occurs in the unsaturated
zone and radionuclide concentrations will be
diminished only through radioactive decay and
volatization. The depth to ground water has no direct
influence on transport mechanisms other than to create
a relatively thin region known as the capillary fringe
which is above the water table and has a higher
moisture content, and therefore, a higher relative
hydraulic conductivity.
Depth to ground water can also influence the extent of
upward water flow occurring to a surface layer which
has been evaporating without water input for an
extended period of time. It has been shown both
theoretically and experimentally that finer textured
soils can move water and radionuclides upward from
much greater depths than can coarse textured soils.
If site data are unavailable, approximate depth to
water estimates may be made based on land surface
topography, the elevation of nearby surface water
bodies, and the tendency of the shallow water table to
mirror the land surface topography. Detailed
modeling, however, cannot be performed without a
good understanding of the vadose zone geometry.
Distribution Coefficient. A distribution or partitioning
coefficient (designated Kj), which describes the degree
of sorption, is used to calculate the partitioning of
species such as radionuclides between the ground
water and aquifer and, thereby, calculate the sorption
capacity or retardation. The standard convention for
recording concentration units for soil samples is to
express the concentration in mass of constituent per
dry mass of soil. Based on this convention, the
dissolved liquid and absorbed-solid concentrations can
be expressed as follows:
C=-
s_
in which
and
where:
C =
CSTOT-
S =
-
C
dissolved-liquid phase concentration,
expressed as mass per volume of liquid
(Ci/ml or g/ml)
total contaminant concentration, expresse
d in
weight
of dry
soil
(Ci/g or
g/g)
bulk density of the soil (g/cm3)
moisture content
equilibrium (partition of distribution)
coefficient (ml/g)
particulate concentration, expressed in weight
of dry soil (Ci/g or g/g)
This expression assumes that there is a direct, linear
relationship between the amount of a solute sorbed
onto soil, S, and the concentration of the solute, C.
Therefore, the adsorption isotherm of C as a function
of S will graph as a straight line. The assumptions
regarding this linear relationship are presented in
2-9
-------
Appendix A.
If the soil contaminant concentrations are presented on
a per unit volume basis (i.e., mass of contaminant per
total volume of sample), the dissolved liquid and
absorbed-solid concentrations can be expressed as
follows:
C=
S=
where:
CT = total contaminant concentration, expressed in
activity or mass per unit volume (Ci/cm3 or
g/cm3)
These relationships are used in Appendix B to estimate
the radionuclide release leaching into the subsurface
environment and migrating with the ground water.
In the literature, distribution coefficients measured
from adsorption conditions abound; however, the Kd
values depend not only on the soil's physical and
chemical properties but also on the chemical properties
of the ground water. Because of its dependence on
many site-specific properties, the value of the
distribution coefficient for a specific radionuclide in
soils can range over several orders of magnitude under
different conditions.
Of particular significance in the unsaturated zone is
that sorption, rather than being dependent upon
effective porosity as in the saturated zone, is a
function of the moisture content as described by the
following:
because moisture content is transient in space and
time, the retardation factor will also exhibit these
characteristics.
Unless data are available to the contrary, scoping
calculations will generally use the same distribution
coefficient for the unsaturated zone and for the source
term. Whereas the distribution coefficient in the
source term is used to predict leaching concentrations
and rates, the distribution applied to the unsaturated
zone dictates the rate at which the radionuclide will be
transported as described in Appendix B. Typical
distribution coefficients are presented as Table C-5 in
Appendix C. However, the modeling results are
typically very sensitive to the distribution coefficient,
and caution should always be used when applying non-
site-specific data.
Hydrodynamic Dispersion. Since soil water flux is
represented as a continuous quantity which is volume-
averaged over many pores, the individual travel paths
around soil grains are mathematically replaced by an
equivalent one-dimensional flow. When this one-
dimensional flow of water is multiplied by the
dissolved solute concentration, the resulting mass flux
does not take into account the additional spreading of
solute which occurs by three-dimensional mass flow at
the pore scale in the actual system. This apparent
solute spreading arising from the mass flux effects
which are obscured by mathematical volume averaging
is called hydrodynamic dispersion.
Saturated Hydraulic Conductivity. Saturated hydraulic
conductivity is the proportionality coefficient between
the saturated water flux and the hydraulic head
gradient. In cases where water is ponded on the soil
surface, either through irrigation, rainfall or natural
lakes or man-made storage ponds or lagoons,
hydraulic conductivity will have a dominant influence
on the amount of water infiltrating into the soil and,
therefore, will strongly affect mass flow and transport.
where:
Rj- = Retardation Factor
Kd = Distribution Coefficient
8 = Moisture Content
p = Bulk Density
This relationship indicates that retardation will
increase as moisture content decreases. Furthermore,
Hydraulic conductivity and permeability are often used
synonymously in ground-water modeling; however,
they have different meanings. Hydraulic conductivity
combines the properties of the aquifer and of the fluid,
while permeability is a property of only the aquifer
material. The two parameters are related by the
following equation:
2-10
-------
where:
K =
k
P
hydraulic conductivity (m/s)
permeability (m )
fluid (water) density (kg/m3)
acceleration due to gravity (m/s )
fluid (water) viscosity (Pa-s)
Most saturated ground-water flow models require
hydraulic conductivity as input, while multi-phase
models require permeability data.
Soil Water Content. Volumetric soil water content has
a significant influence on the flow and transport
mechanisms. The curies of radionuclide moved per
unit time from one point to the next is inversely
proportional to the distance between the two points.
The actual path length in soil followed by a
radionuclide is strongly affected by water content.
Therefore, water content increases, the cross-sectional
area for flow increases, and the path length decreases
as liquid replaces air in the medium. Since flux of
radionuclide is proportional to water flux multiplied by
the dissolved radionuclide concentration, it is not
directly affected by water content. However, because
increasing the water content of a given soil will result
in a higher mass flux, some correlation between
radionuclide movement and water content may be
found.
The soil water content can influence adsorption in two
ways: it can modify the solution pathway leading to
the adsorption sites and thus increase or decrease the
accessibility of the surface to the solute, and it may
also affect the physical-chemical properties of the
adsorbent by increasing or decreasing the hydrolysis of
the clay lattice. However, the influence of soil water
content on adsorption is slight until the soil is
extremely dry. In dry soil, the preferential coverage
of water molecules on the soil' s adsorbing surfaces is
removed, and solute adsorption increases dramatically.
Total Porosity. The porosity of a rock or soil is its
property of containing interstices or voids. This may
be expressed quantitatively as the ratio of the volume
of its interstices to its total volume.
Flow and transport are indirectly affected by porosity
since regions of low porosity are likely to have lower
permeability to transport water. Although no reliable
models exist to describe the relationships between
porosity and permeability, permeability of a given soil
type strongly decreases as porosity decreases because
the pore sizes contract. However, finer-textured soils
such as clays generally have a higher porosity and
lower permeability than sandy soils.
Porosity is an important parameter in computing
ground-water velocity in both the saturated and
unsaturated zones. Velocity is inversely proportional
to porosity. Another way that porosity affects
transport is that decreasing soil porosity increases the
density of mineral adsorption sites and thus causes
increased adsorption of radionuclides with a
corresponding decrease in solution concentration.
Bulk Density. The soil or dry density is the ratio of
the mass of the solid phase of soil (i.e, dry soil) to its
total volume (solid and pore volumes together).
The influence of increasing bulk density on adsorption
is to increase the density of adsorption sites per unit
volume which will directly increase adsorption
capacity. However, the correlation between
adsorption and bulk density for a group of soils will be
small because clay and organic soils tent to be found
at lower bulk density than coarser textured soils which
are low in organic matter. Thus, the effect of
increasing bulk density on adsorption refers to
compressing a given soil volume.
Saturated Zone
The saturated zone is that part of the earth' s crust
beneath the regional water table in which all voids,
large and small, are filled with water under pressure
greater than atmospheric. The saturated zone may
depart from the ideal in some respects. A rising water
table may cause entrapment of air in the upper part of
the zone of saturation. The shallowest aquifer
typically would be under unconfined conditions or a
water-table aquifer. The ground water flowing within
a water-table surface is in immediate contact with the
atmosphere and is directly recharged through the
overlying unsaturated zone. This water-table surface
is free to rise and fall within the aquifer in response to
varying amounts of recharge (e.g., rain). The water-
table aquifer generally follows land-surface
2-11
-------
topography and is frequently revealed in the form of
surface-water bodies such as lakes and rivers. This
connection between the ground water and the surface
water also creates potential surface-water pathways.
Scoping calculations generally focus on contaminant
releases to the shallowest aquifer, whereas site
characterization modeling would tend to include all
aquifers and aquitards in the hydrogeologic flow
system. The data required to describe ground-water
flow and radionuclide transport through the saturated
zone are presented below.
Potentiometric Surface Maps. The potentiometric
surface is a way of depicting the static head in an
aquifer. It is defined by the levels to which water will
rise in tightly cased wells. In cases where the head
varies appreciably with depth (i.e., upward or
downward gradients) in the aquifer, a potentiometric
surface is meaningful only if it describes the static
head along a particular specified surface or stratum in
that aquifer. More than one potentiometric surface is
then required to describe the distribution of head. The
water table is a particular potentiometric surface for
an unconfined aquifer.
With respect to ground-water flow and contaminant
transport, the potentiometric surfaces define the
hydraulic gradients, which in turn, are used to
calculate the direction and volume of flow through the
system, as well as the ground-water and contaminant
velocities.
Hydraulic Gradient. The hydraulic gradient will play
a significant role in estimating the velocity at which
the radionuclides are migrating (Appendix B). If site-
specific data are not available for the scoping
calculations, hydraulic gradients maybe approximated
from the land-surface topography. However, in the
site characterization modeling, water levels from at
least three wells screened in the aquifer are needed to
determine the direction and magnitude of the gradient.
Hydraulic Conductivity. The hydraulic conductivity of
a soil or rock is a measure of the soil's ability to
transmit water under a hydraulic gradient. The values
of hydraulic conductivity in soils and rocks vary
within a wide range of several orders of magnitude,
depending on the grain size, the structure of the soil
matrix, the type of soil fluid, and the relative amount
of saturation present in the soil or rock matrix.
Aquifer tests are often performed for the purpose of
determining field values of aquifer hydraulic
conductivities. Analyses of the field test data are
based upon analytical solutions for radial flow towards
wells under a variety of conditions. The analytical
methods used are very straightforward and generally
do not require the use of a computer.
Before site characterization, only the most general
assumptions can be made about the relative flow
properties of the aquifers. For example, as a rule of
thumb for sedimentary deposits, it is often assumed
that the hydraulic conductivity in the horizontal
direction is ten times greater than that in the vertical
direction. In the absence of site-specific values,
literature values may be used for the scoping
calculations (Table C-6, Appendix C). For site
characterization modeling, hydraulic conductivity
values should be site specific.
Storage Properties. The storativity of a saturated
confined aquifer can be defined as the volume of water
that an aquifer releases from storage per unit surface
area of aquifer per unit decline in the component of
hydraulic head normal to that surface.
The storage term for unconfined aquifers is known as
the specific yield. It is defined as the volume of water
that an unconfined aquifer releases from storage per
unit surface area of aquifer per unit decline in the
water table.
Storage properties are required only for transient
ground-water flow simulations. If flow is assumed to
be steady state, the storativity of the aquifer is
assumed to be zero: in flow is equal to outflow with
no change in storage.
Effective Porosity. The effective porosity is the ratio
of the volume of interconnected pore spaces available
for transport to the total system volume. It is used to
estimate the velocity at which ground water and
radionuclides travel through a porous medium
(Appendix B). The smaller the effective porosity the
higher the ground-water velocity and the more rapidly
the transport of radionuclide(s) or other solubles.
Total porosity is the ratio of the total pore volume to
the total system volume and includes dead pore space.
Therefore, it is important not to confuse effective
porosity with total porosity, as total porosity will
always be greater than effective porosity.
In natural porous systems, such as subsurface soil,
2-12
-------
where the flow of water is caused by capillary,
molecular, and gravitational forces, the effective
porosity can be approximated by the specific yield,
which is defined as the ratio of the volume of water
drained by gravity from a saturated sample of soil to
the total volume of soil.
The most accurate means of obtaining effective
porosity data is by conducting site-specific field tracer
tests. These tests, however, are time consuming and
may not significantly reduce the uncertainty associated
with the effective porosity. Since the greatest source
of uncertainty relative to transport is typically the
distribution coefficient, it is generally best to estimate
effective porosities from the literature (Table C-10).
An analysis can be performed to evaluate the
sensitivity on flow and transport results (Section
4.1.4).
Bulk Density. The bulk density of the soil or rock is
used to determine the retardation factor as derived in
Appendix B. The soil or dry density is the ratio of the
mass of the solid phase of soil (i.e, dry soil) to its total
volume (solid and pore volumes together). The dry
density of most soils varies within the range of 1.1 -
1.6 gr/cm . In sandy soils, dry density can be as high
as 1.6 gr/cm3, in clayey soils and aggregated loams,
it can be as low as 1.1 gr/cm3. Although laboratory
measurements may be made to obtain accurate bulk
density values, it is rarely worth the effort as the
potential range is relatively narrow, and the modeling
results are typically insensitive to bulk density.
Dispersion Coefficients. The equations of solute
transport that are solved in contaminant-transport
codes are derived assuming that the solute migration
is due to advection and hydrodynamic dispersion.
Hydrodynamic dispersion is caused by the tendency of
the solute to spread out from the path that it would be
expected to follow if transported only by advection.
This spreading of the contamination over an ever-
increasing area is called hydrodynamic dispersion and
has two components: mechanical dispersion and
diffusion. Hydrodynamic dispersion causes dilution of
the solute and occurs because of spatial variations in
ground-water flow velocities and mechanical mixing
during fluid advection. Molecular diffusion, the other
component of hydrodynamic dispersion, is due to the
thermal kinetic energy of solute molecules and also
contributes to the dispersion process. Thus, if
hydrodynamic dispersion is factored into the solute
transport processes, ground-water contamination will
cover a much larger region than in the case of pure
advection, with a corresponding reduction in the
maximum and average concentrations of the
contaminant. Typical dispersivity values, obtained
from tracer tests, are presented in Table C-ll.
Because hydrodynamic dispersion is the sum of
mechanical dispersion and diffusion, it is possible to
divide the hydrodynamic dispersion term into the two
components and have two separate terms in the
equation. Under most conditions of ground-water
flow, diffusion is insignificant and is frequently
neglected in many of the contaminant transport codes.
However, this artificial exclusion of the diffusion term
may create problems in certain instances (see Section
3.3 under the topic of matrix diffusion).
Representing dispersion adequately in computer codes
is difficult, because dispersion is related to spatial
scale and variations in aquifer properties that are
generally not explicitly simulated in the code (e.g.,
tortuosity). Furthermore, dispersion coefficients are
very difficult to measure in the field and have been
shown generally to increase with scale of observation.
These difficulties are generally addressed by using
dispersivity values from the published literature and
refining these estimates during the model calibration
process.
Distribution Coefficient. As with the unsaturated
zone, the distribution coefficient assigned to the
saturated zone will help determine the rate at which
the radionuclides migrate. This distribution coefficient
should be consistent with the rock or soil types that
make up the aquifer or water-bearing unit.
A detailed discussion of distribution coefficients is
presented in Appendix A. The use of literature values
for site characterization modeling is rarely defensible.
Furthermore, modeling results are typically very
sensitive to the magnitude of the
distribution coefficients. Therefore, site-specific
distribution coefficients should be obtained during the
site characterization program.
The two most common experimental techniques for the
determination of the distribution coefficient are the
batch and column methods. The batch method is used
to measure the distribution coefficient under saturated
equilibrium conditions. The column method is used to
obtain a more representative value as the soil has not
been altered (e.g., grinding, agitated) as much as in
2-13
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the batch experiments.
Precipitation/Irrigation. The characteristics of the
precipitation or rainfall events (i.e., intensity and
distribution) greatly affect the extent of radionuclide
transport. Precipitation will have a dominant
influence on flow because the rainfall rate is directly
related to the water flow rate in the soil. Therefore,
soils that receive intense, frequent rainfall will have
high water fluxes and hence high radionuclide activity
fluxes. Furthermore, extremely intense rainfall might
induce saturation which could result in a greatly
enhanced mass transport through soils of high
permeability. That is, the saturated hydraulic
conductivity may be reached. These same
characteristics may be observed in aquifers underlying
irrigated fields.
Evapotranspiration. Evapotranspiration represents the
amount of applied water which is removed by plants
or water loss from surfaces via evaporation and hence
is unavailable for drainage. Thus, the extent of
evapotranspiration will strongly affect the water flux
below the root zone and therefore, the extent of
radionuclide leaching by mass flow. For soils not
receiving water input by irrigation, rainfall minus
evapotranspiration and runoff determines the net
amount of water infiltrating beyond the root zone.
Surface-Water/Ground-Water Interactions. Water
resource development has frequently been based on the
predominant use of either surface or ground water.
These two components of the total water resource,
however, are interdependent. Changes in one
component can have significant effects on the other.
In streams that are termed "gaining," stream flows are
sustained by ground water influx, whereas "losing"
streams replenish the ground water by seepage through
the stream bed. The hydraulic conductivity of the
stream bed sediments will be a contributing factor to
the rate at which water moves into or out of a stream.
Ground-Water Pumping/Injection. The injection or
withdrawal of water into or out of an aquifer can have
a pronounced effect on the hydraulic gradients. In the
case of withdrawal wells, capture zones are created
which will not allow contamination to migrate beyond
this zone.
Source Characteristics
The accurate portrayal of the contaminant source term
is one of the most difficult tasks in the modeling
process. All too often, there are no data that
characterize the nature and extent of the contamination
or the release history.
Some knowledge of the history of the waste disposal
activities can often provide valuable insight into the
probable nature of the contaminant source term. In
general, the longer the site has been active, the more
likely it is that the wastes have been discarded in many
different forms and dispersed over a larger area. The
presence of product and waste lines immediately
suggests that line-type sources are present.
Absorption beds and storage tanks indicate potential
point sources, whereas mill tailings, large lagoons,
and air emissions that carried and subsequently
deposited contaminants in the site vicinity would
generally represent area sources.
The distribution of measured contaminants in the soil
and ground water will also provide clues as to their
source. Contaminants that are wide-spread and of
similar concentrations suggest an areal source (or non-
point), while narrowly defined areas of contamination
indicate a more localized or point source.
As a general rule, it is best to keep initial assumptions
regarding the source term as simple as possible. The
large uncertainties of the initial scoping phase dictate
that the sensitivity analysis become a critical
component of the analysis to quantify associated
uncertainties. As more data become available during
the site characterization, source term characteristics
can be more accurately modeled. The characteristics
typically associated with the source term are discussed
below and include: source dimensions, release
mechanisms, radionuclide concentrations, and leaching
rates.
Areal and Vertical Extent. The vertical and areal
extent of contamination is a site-specific parameter
that will potentially have a high degree of associated
uncertainty for the scoping calculations. In practice,
scoping calculations often assume the source to be a
point source that includes 1 m of material. This
volume is subsequently scaled upwards to analyze
larger areas.
One of the primary objectives of the site
characterization program is to define the geometry of
2-14
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the contaminant source. Most numerical models allow
the source geometry to be accurately portrayed to the
degree desired.
Release Mechanism. Computer codes can simulate the
introduction of contaminants to the ground water as an
instantaneous pulse or as a continuous release over
time. A continuous release may either be constant or
vary with time. The two most common means of
simulating continuous or pulse releases are by either
specifying release concentrations or by specifying the
contaminant mass entering the system. In general,
both approaches have drawbacks and limitations and
require considerable thought and possibly a number of
independent calculations prior to selecting and
implementing the most appropriate method for the
modeling exercise. Furthermore, most ground-water
flow and transport codes do not explicitly account for
the physical degradation of waste containers, and
therefore, anticipated release rates must be estimated
through other means (e.g., waste package codes) and
input as boundary conditions into the flow and
transport model.
Concentration. The source term concentration is one
of the most critical parameters. It is recommended
that site-specific data be obtained because the initial
concentration will directly impact the predicted
concentration at a receptor. Source term
concentrations should also be varied as an integral part
of the sensitivity analysis. Frequently, the source
concentration is normalized: that is, the concentration
is set to one. This practice allows the predicted results
presented in terms of percentages of the actual
concentration.
Radioactive Decay. Radionuclides either decay to
stable products or to another radioactive species called
a daughter. In some species, several daughter
products may be produced before the parent species
decays to a stable element. In considering this process
over the transport path of radionuclides, one transport
equation must be written for each original species and
each daughter product to yield the concentration of
each radionuclide (original species and daughter
products) at points of interest along the flow path.
Radioactive source terms present special
considerations in that the activity of the parent isotopes
will diminish with time due to radioactive decay.
However, if the radionuclide release is solubility
controlled, or if the half-life is extremely long, the
concentration of the leachate may remain constant
despite the decay of the source term. The release
concentrations may remain constant until the source
term has decayed to concentrations where solubility
limits no longer dictate the amount of radionuclides
that may go into the solution.
Distribution Coefficient. As mentioned previously, the
distribution coefficient describes the soil-water
partitioning for a given compound. This relationship
is frequently used to predict the rate at which
radionuclides will leach from the source term, as
described in Appendix B.
2.4 Remedial Design and Implementation Phase
As the site characterization process ends and the
Remedial Design and Selection Phase is entered, data
that will assist in defining the remedial alternatives
have been acquired. The various remedial alternatives
can be conveniently grouped into the following three
categories:
Immobilization
Isolation
Removal
This section briefly describes each category, the types
of processes that need to be modeled to support each
category, and the special information needs for each of
these categories. The information is required not only
for implementation of the remedial design but also to
evaluate its effectiveness through numerical modeling.
Immobilization
Immobilization of the radioactive wastes refers to
physical, chemical, and/or biological processes used
to stabilize the radionuclides and preclude their
transport. A number of treatment options exist, each
having its own associated modeling needs, including:
Physical
vapor extraction
in-situ coating
grouting of fissures and pores
in-situ freezing
in-situ vitrification
2-15
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Chemical
induce secondary mineralization
induce complexation
alter oxidation-reduction potential
Biological
in-situ microbial activity
Physical/Chemical
alter surface tension relationships
alter surface charges
in-situ binding
adsorbent injection
radionuclide particle size augmentation
through clay flocculation
The following are the types of physical, chemical, and
biological processes that may need to be modeled to
support alternative remedies based on immobilization:
Physical Properties and Processes
unsaturated zone flow and transport*
heat energy transfer*
multiple layers*
vapor transport*
extreme heterogeneity*
temperature-dependent flow and transport*
Chemical Properties and Processes
density-dependent flow and transport*
oxidation-reduction reactions
system thermodynamics
chemical speciation*
ion-exchange phenomena
precipitation
natural colloidal formation
radiolysis
organic complexation
anion exclusion
Biotic Properties and Processes
biofixation
* indicates modeling codes are readily available
It would be ideal if these processes and properties
could be reliably described and modeled with
conventional and available models. However, many
of these properties and processes are not well
understood, and, in these instances, models do not
exist that yield reliable results.
The specialized data required to support ground-water
modeling of immobilization techniques include:
Determination of temperature-dependent flow
and transport parameters
Characterization of the geochemical
environment
Determination if the physical rock
properties that govern flow and transport
have been altered
Characterization of the microbial environment
Isolation
A common remedial alternative is to emplace
protective barriers either to prevent contaminated
ground water from migrating away from a
contaminated site or to divert incoming (i.e., clean)
ground water from the source of contaminants. These
barriers include walls, caps and lines. Several types
of materials are being used to construct such barriers,
including soil and bentonite, cement and bentonite,
concrete, and sheet piling. An alternative to the
physical emplacement of protective barriers is the use
of hydraulic containment which involves controlling
the hydraulic gradient through the use of injection
and/or withdrawal wells or trenches in order to
contain and treat the contaminant plume. Examples of
potential barriers include the following:
Physical
hydraulic containment
grout curtains, sheet piling, bentonite slurry
walls
low permeability caps (clay and/or synthetic)
liners
Chemical
ion-exchange barriers
Biological
microbial barriers
If properly designed and emplaced, such barriers can
last for several decades, barring any geological
disturbances, such as tremors, ground settling,
significant changes in hydraulic gradients, etc.
2-16
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Accordingly, such barriers can be useful in mitigating
the impacts of relatively short-lived radionuclides, or
to control the migration of long-lived radionuclides
until a more permanent remedy can be implemented.
Several mechanisms or processes can affect the long-
term integrity of such barriers. Once the installation
is complete, failures can occur due to cracking,
hydrofracturing, tunneling and piping, and chemical
disruption. Changes in the site's geological or
hydrological characteristics can also lead to
catastrophic failures, such as partial collapse, settling,
and breaking. If a barrier should fail following
installation, water may infiltrate the site, and
contaminated leachates may move beyond the site.
This type of failure could result in the dispersion of
contaminants in the environment.
The modeling approaches to simulating the effects of
flow barriers on the fate and transport of radionuclides
are closely tied to the ability of the code to
accommodate factors, such as: high permeability
contrasts, transient boundary conditions, and possibly
chemical and biological reactions. These
considerations will be discussed in greater detail in the
following sections.
The following are the types of physical, chemical, and
biological processes that may need to be modeled to
support alternative remedies based on isolation. Many
of these processes are very complex, and attempts at
modeling will meet with varying degrees of success.
Physical Properties and Processes
unsaturated zone flow and transport*
runoff*
multiple layers*
vegetative cover*
transient source term*
extreme heterogeneity*
areal recharge and zero flux capability*
Chemical Properties and Processes
localized ion exchange phenomena
Biotic Properties and Processes
localized biofixation
microbial population modeling
* indicates modeling codes readily available
Typical characterization data needs related to barrier
emplacement include:
Barrier dimensions
Barrier hydraulic conductivity
Geochemical environment
Structural integrity of barrier/barrier
degradation
Microbial environment
Detailed hydrogeology
Removal
Radioactively contaminated soil can result from the
disposal of both solid and liquid waste. Solid wastes
may have been buried in the past without sufficient
integrity of containment so that, eventually,
radioactivity intermingled with the contiguous soil.
Percolation of rain water through shallow burial sites
can contribute further to the migration of radionuclides
to lower depths as well as to some lateral movement.
Wider areas of contamination have occurred when
waste, stored temporarily at the surface, has lost
containment and has been dispersed by the wind. The
most common technologies for removing radionuclides
in solid, liquid, and vapor (e.g., tritium) form include
the following:
Physical
soil excavation (solid)
pump and treat (liquid)
in-situ vaporization (vapor)
Biological
injection and removal of biomass foam
The following are the types of physical, chemical, and
biological processes may need to be modeled to
support alternative remedies based on removal. Most
of these processes and properties are readily described
in mathematical terms and can be modeled reliably.
Obviously, modeling the biological activity associated
with the injection of a biomass will have the same
limitations that are common to other types of
biological modeling.
Physical Properties and Processes
transient source term*
unsaturated zone flow and transport*
matrix diffusion*
desaturation and resaturation of the aquifer*
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vapor transport* Typical characterization needs related to radionuclide
removal include:
Biological Properties and Processes
physical injection and withdrawal of the Air permeability of the unsaturated zone
biomass Unsaturated zone flow and transport parameters
microbial population modeling Areal extent of contaminated wastes
Depth to ground water
* indicates modeling codes readily available Saturated zone flow and transport properties
The degree to which these factors are addressed in the
modeling relies heavily upon the objectives as well as
the availability of the required data.
2-18
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CHAPTER 3 CONCEPTUAL MODEL DEVELOPMENT
3.1 Preliminary Conceptual Model
In the scoping phase, site-specific information is often
limited. Therefore, the modeling performed during
the early planning phase of most remedial
investigations is generally designed to support
relatively simple objectives which can be easily tied to
more ambitious goals developed during the later
phases of the investigation. The very nature of the
iterative process of data collection, analysis, and
decision making dictates that the preliminary
objectives will need to evolve to meet the needs of the
overall program. That is, it would be unreasonable to
assume that simplified modeling based upon limited
data would do little more than provide direction for
future activities.
An important issue that often arises during the scoping
phase is whether remediation and decommissioning
strategies can be selected during the scoping phase
based on limited data and simple screening models.
Such decisions can be costly at complex sites where
the nature and extent of the contamination and
transport processes are poorly understood. However,
at relatively simple sites, early remediation decisions
can help avoid the unnecessary delays and costs
associated with a possibly prolonged site
characterization and modeling exercise.
The formulation of a conceptual model is an integral
component of the modeling process. Sometimes,
components of the conceptual model may be simplified
to meet either limited objectives or limitations in the
data. That is, it is often useful to simulate only certain
components of the conceptual model. For instance,
even if there are data that indicate separable property
zones in the aquifer, it is common during the scoping
phase to evaluate ground-water flow and contaminant
transport as a function of average values using
homogeneous soil and rock properties. This
simplification of the conceptual model is a valid
approach because, in practice, early modeling focuses
upon assessing the significance of specific parameter
values and their effects on flow and transport, rather
than on modeling specific hydrogeologic transport
processes. Figure 3-1 illustrates typical conceptual
models in the early phase of the investigation.
While different aspects of the conceptual model may
be simulated in a variety of ways, the selected
approach must remain consistent with the objectives.
That is, the physical system cannot be overly
simplified to meet ambitious objectives, and less
demanding objectives should not be addressed with
sophisticated models. Hence, the development and
acceptance of a conceptual model is an evolutionary
process that depends upon the modeling goals and
availability of data. An important part of model
application in the early phase of the investigation is
understanding the project decisions that need to be
made, and identifying which of these decisions can be
supported by the use of specific codes when limited
data and the controlling hydrogeologic processes at the
site are incompletely understood.
Because general trends, rather than accuracy, are most
important during the scoping phase, the ground-water
modeler typically makes the following simplifying
assumptions early in the investigation:
Steady-State Assumptions
Restricted Dimensionality
Uncomplicated Boundary and Initial
Conditions
Simplified Flow and Transport
Processes
System Homogeneity
These conceptual assumptions, discussed in greater
detail below, generally translate into modeling
approaches that are consistent with the available data.
They are discussed in greater detail next.
Steady-State Solutions
In the scoping phase, the data generally available have
been collected over relatively short time intervals.
Therefore, modeling objectives would be limited to
those that could be met without a detailed
understanding of the temporal nature of processes
affecting flow and transport. For example, a typical
analysis not requiring detailed knowledge of the
temporal nature of recharge, source release rates, and
other flow and transport mechanisms would be the
estimation of the distance that radionuclides have
traveled since the beginning of waste management
activities. This analysis would use yearly average
3-1
-------
Conceptual Model
Cross-sectional Conceptual Model
Precipitation
ea
W
1
^
IplOf
ell f
Surface FluriGif ( EvafH5trai5spirat«o«i
Land Surface . ,
) Recharge )
No dilution In unsau
i One dunensioaal Ho
Unit hydraulic gradi
______ ^ - '
Water Table
Source
Steady stale
fe'easa oorre-;
lor tfccay-
onHorm
concentration
irated zone 1
W
jnl _J
"*| Two tiirnftssional ilow
Anuifer Base
Figure 3-1. Typical conceptual model(s) in the scoping phase
3-2
-------
values for the input parameters, such as ambient
recharge, stream flow stages, and source concentration
release rates. However, without accommodating the
transient nature of these processes, predictions of peak
contaminant concentrations arriving at downgradient
receptors would have a high degree of uncertainty.
The conceptual model could, therefore, be simplified
to a translation of the physics of the system into
relatively simple mathematical terms such as those
described by analytical expressions.
Restricted Dimensionality
Ground-water flow and contaminant transport are
seldom constrained to one or two dimensions.
However, during scoping, modeling objectives and
conceptual model development must take into account
that information is rarely sufficient to describe
mathematically the controlling flow and transport
processes in three dimensions. In reality, most of the
modeling analysis in the preliminary investigation will
focus upon centerline plume concentrations which are
essentially one- and two-dimensional analyses. One-
dimensional analyses of the unsaturated zone are
customarily performed in a cross-sectional orientation
because flow and transport are predominantly
vertically downward. Similarly, in the saturated zone,
vertical gradients are generally much smaller than
lateral gradients and, as a result, vertical transport
need not always be explicitly modeled. Therefore, the
assumption that flow is two-dimensional may be
appropriate for areal analyses.
Uncomplicated Boundary and Uniform Initial
Conditions
Boundary conditions are the conditions that the
modeler specifies as known values in order to solve
for the unknowns. Ground-water boundaries may be
described in terms of where water is flowing into the
ground-water system and where water is flowing out.
Many different types of boundaries exist, including:
surface-water bodies, ground-water divides, recharge,
wells, and geologic features such as faults and sharp
contrasts in lithology. Initial conditions are defined as
values of ground-water elevation, flow volumes, or
contaminant concentrations initially assumed to be
present in the area of interest.
Because of the lack of site-specific data in the scoping
phase, the system boundary and initial conditions
usually cannot be accurately defined; only very limited
calculations of approximate travel distances and
contaminant concentrations can be made.
Uniform Properties
Homogeneity describes a system where all of the
characteristics spatially are uniform within the aquifer,
whereas isotropy means that the hydraulic properties
are identical in all directions. A homogeneous system
may have anisotropic flow properties; for example, an
otherwise homogeneous sandstone aquifer may have a
greater hydraulic conductivity in the horizontal
direction than in the vertical. Therefore,
hydrogeologic units may have anisotropic qualities but
still be considered spatially homogeneous throughout,
provided the anisotropy does not vary within the unit.
Before site characterization, only the most general
assumptions may be made about the relative flow
properties of the aquifers. For example, it is often
assumed that the hydraulic conductivity in the
horizontal direction is ten times greater than that in the
vertical direction for sedimentary deposits. These
types of simplifying assumptions regarding the aquifer
properties would form the basis of the conceptual
model.
Simplified Flow and Transport Processes
Site-specific information describing the flow and
transport processes that dominate the migration of
radionuclides would not be available before detailed
site characterization occurs. Therefore, modeling
objectives would need to be limited to those that can
be addressed with only limited knowledge of the site
hydrogeology and geochemistry. In practice, this
means that uniform porous media flow would be
assumed in the conceptual model, and all of the
geochemical reactions that affect the radionuclide
transport would be lumped together as a single
parameter termed the distribution coefficient.
Discrete features, such as macropores, fractures, and
faults, would generally have to be excluded from the
mathematical expression of the conceptual model, and
conservative distribution coefficients would be selected
from conservative values found in the literature.
Movement through the unsaturated zone would be
simulated with simplified versions of more complex
equations describing the unsaturated flow and
transport.
3-3
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To demonstrate the potential effects that simplifying
assumptions may have on modeling results, the
discussion in Appendix B outlines the physical and
chemical processes that may affect the transport of
radionuclides (estimates of moisture content).
3.2
Evolution of the Conceptual Model
The conceptual model is based on the modeler's
experience and technical judgment and represents the
modeler' s understanding of the system framework and
behavior. The conceptual model will naturally
become more complex as more processes are
identified and interrelationships of important
components within the systems are considered. The
transformation of the conceptual model into a
mathematical model which, in reality, is only an
extrapolation of a basic understanding of the system,
will result in intrinsic simplifications of the system.
For example, the mathematical models assume that
there is a direct scaling between the model simulations
and the scale at which the data are collected. The lack
of knowledge about the system resulting from limited
information also contributes to inevitable
simplifications between the conceptual and
mathematical models.
Besides the simplifications inherent in the process,
there are deliberate simplifications in which the
modeler selects the physical characteristics and
processes relevant to the model's application.
Examples of these simplifying assumptions include:
Flow through the unsaturated zone is vertical and
in one dimension.
Chemical reactions are reversible and
instantaneous.
Soil or rock medium is isotropic and/or
homogeneous.
Flow field is uniform and under steady-state
conditions.
As more and more of these simplifying assumptions
are found to be significant, the complexity of the
model increases. Thus, the development of a dynamic
model allows for the neglected components of the
conceptual model to be integrated systematically.
Components of the conceptual model may be
simplified, either because of limited objectives or
because of constraints on data availability. Even when
the available data support the use of a more
sophisticated model, simplifications are sometimes
advisable. For instance, the site characterization
modeling described in Section 4.2 discusses the
application of complex numerical models. However,
after applying these complex models, it is frequently
possible to simplify the assumptions again and use less
sophisticated models to meet the objectives of the risk
assessment. This iterative process ensures that the
mathematical model is consistent with the modeling
objectives.
As discussed in Chapter 1, formulating a conceptual
model is an integral component of the modeling
process. The data obtained during site
characterization provides an opportunity to remove
some of the simplifications made in the scoping phase.
Generally, in the site characterization phase, sufficient
data are collected to formulate relatively complex
conceptual models. Therefore, the degree to which
the conceptual model is simplified frequently depends
more on the objectives, than on limitations in the data.
Figures 3-2 and 3-3 illustrate typical conceptual
models in the site characterization phase of the
investigation.
The following assumptions are typical of the
conceptual model in the site characterization phase:
Steady-State Flow/Transient Transport
Multi-Dimensionality
Steady-State Boundary and Non-uniform Initial
Conditions
Complex Flow and Transport Processes
System Heterogeneity
These conceptual model assumptions generally
translate into modeling approaches that are consistent
with the data available during the site characterization
phase. They are discussed in greater detail next.
Steady-State Flow/Transient Transport
The data obtained during the site characterization
program are generally collected over a relatively short
time and frequently do not reflect the temporal nature
of the hydrogeologic system. Unfortunately,
objectives in the site characterization phase often
involve the prediction of temporal trends in the data.
3-4
-------
-.7
_
rr-Ai..!!> t. n.'..-4:i 111 nri
! i
Figure 3-2. Representative conceptual model of the unsaturated zone.
WEST
(ACE
AHBIEk'3
S I
_ J,
Figure 3-3. Representative conceptual model of the saturated zone.
3-5
-------
For instance, the risk assessment generally includes an
analysis of the peak arrival times of radionuclides at
downgradient receptors. This incompatibility between
the objectives and the data available gives rise to some
of the greatest uncertainties associated with the entire
remedial investigation. However, one of the principal
utilities of mathematical models is their ability to
extrapolate unknown values through time.
The modeling approach during site characterization
generally assumes a steady-state flow field and
accommodates the transient nature of the system
through the contaminant transport analysis. Steady or
transient leaching rates are used in conjunction with
the existing plume concentrations for initial conditions.
Therefore, the system is actually modeled as a steady
flow system with possibly a transient or pulse-like
source term. However, the transient nature of the
plume is generally used as a model calibration
parameter and is not carried forward into the
predictive analysis for future radionuclide
concentrations. That is, rarely are there sufficient
data to describe the temporal nature of the source
release. Exceptions to this occur when records are
available pertaining to the volumes of radioactive
liquids that were dumped over time into infiltration,
recharge or evaporation trenches or when correlations
between rainfall events and source leaching rates may
be extrapolated.
The validity of the steady-state assumption depends on
the features of the flow pattern, which are in turn
dictated by the nature of boundary conditions and sinks
or sources existing in the flow domain. Whether or
not such an assumption is justified also depends on the
time scale of interest and, perhaps most important, the
conservative objectives of the modeling study.
For example, as mentioned above, ground-water flow
modeling performed in conjunction with contaminant
transport modeling is usually based on an assumption
of steady-state flow. This is done to reduce the
complexity and cost of the time-dependent transport
simulation. Indeed, all analytical solutions and most
of the numerical solutions of the contaminant transport
equation likely to be used in the modeling study will
be based on the assumption of a steady-state velocity
field. Such an assumption is valid, provided that
during the time period of the transport simulation the
flow pattern or velocity distributions do not change
significantly. A common pitfall is a situation where
the modeler deals with the ground-water flow system
containing internal sinks or sources (e.g., pumping or
injection wells) but ignores drastic changes in the
velocity distribution due to changes in the well
operation or flow rates. Another situation involves
gradual or sudden changes in conditions at the flow
boundaries which lead to reversal of flow directions
during the period of the transport simulation. Such
changes must be taken into account to obtain reliable
predictions of contaminant migration.
As a cautionary note, one modeling report identified
in the EPA CSMoS study (LEE95) assumed steady-
state conditions based on only several months of
ground-water monitoring data. This assumption
resulted in a predicted ground-water gradient to the
west, whereas contaminant data indicated that the
plume was migrating northward. Obviously, steady-
state assumptions must be based on a sufficient
monitoring period.
Multi-Dimensionality
The site characterization program should be designed
to gather sufficient data to develop a three-dimensional
conceptual model. It is only after the three-
dimensional system is relatively well understood that
it can be determined whether one-, two-, or three-
dimensional modeling is necessary. If one or two
dimensions are eliminated from the analysis, careful
consideration needs to be given to what impact
restricting the dimensions will have on the model' s
capability to simulate existing field conditions.
The magnitude of flow and transport in any direction
relative to the other directions provides the criteria for
which dimension(s) should be included or excluded.
In most instances, flow and transport in the
unsaturated zone are assumed to be predominantly
downward with smaller horizontal components. If the
flow components are found to have two dominant flow
directions, a two-dimensional cross section may allow
for an adequate representation of the flow field.
Modeling and field validation studies of the vadose
zone (the unsaturated zone) have yielded mixed results
both in model calibration and in the comparison of
transport predictions against measured field values. In
modeling the vadose zone, as well as the saturated
zone, the question is always how much uncertainty in
the results is acceptable, considering the objectives.
Two-dimensional simulations of the saturated zone are
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usually performed when the horizontal flow
components are far greater than the vertical flow
components, allowing the vertical components to be
ignored. However, in much of the modeling
performed for site characterization, the vertical
components of flow are important because many
natural features, such as surface water bodies, often
have strong vertical flow components associated with
them. Furthermore, particular care must be taken in
eliminating the third dimension because attempts to
simulate three-dimensional processes in two
dimensions can lead to difficulties in model
calibration, as well as in producing defensible
modeling results.
Water-level data collected from closely spaced wells
that penetrate the same aquifer and are screened at
different depths provide excellent information on
vertical hydraulic gradients. This information may be
used during the site characterization program to
determine the effective hydraulic basement of any
contamination present, as well as recharge and
discharge areas. If there are strong vertical gradients,
the capability to simulate the vertical movement of
ground water within the hydrogeologic system
becomes very important in defining the nature and
extent of the contaminant plume.
It should also be kept in mind that two-dimensional
planar modeling will average the contaminant
concentrations over the entire thickness of the aquifer,
and the vertical definition of the contaminant plumes
will be lost. This vertical averaging of contaminants
will result in lower downgradient concentrations and
may not provide a realistic or conservative baseline
risk assessment. Again, this example illustrates that
the decision as to how many dimensions to include in
the modeling must be related to the objectives and the
need to be aware of the limitations in the results if one
or more dimensions are eliminated.
The recent development of more sophisticated pre- and
post-processors greatly facilitates data entry and
processing. These advances, in conjunction with the
rapid increase in computer speeds over the past several
years, have greatly reduced the time involved in
performing three-dimensional modeling. In general,
it is better to include the third dimension, even if
many of the parameters in the third dimension have to
be estimated than to constrain the analysis to two
dimensions.
Two-dimensional analyses during the site
characterization program are most valuable for
modeling the unsaturated zone and for performing
sensitivity analyses of selected cross-sections through
a three-dimensional model. Two-dimensional
approaches are also useful for performing regional
modeling from which the boundary conditions for a
more site-scale modeling study may be extrapolated.
Steady-State and Non-uniform Initial Conditions
In general, boundary conditions are known or
estimated values that are assigned to surfaces and
planes that either frame the perimeter of the modeled
area or define the nature of release from the
contaminant source. The different types of flow
boundary conditions are: (a) head (ground-water
elevation) is known for surfaces or planes bounding
the modeled region; (b) ground-water flow volumes
are known for surfaces or planes bounding the
modeled region; and (c) some combination of (a) and
(b) is known for surfaces or planes bounding the
region. Boundary conditions could also be assigned to
interior features of the modeled region where ground-
water elevations or flow volumes are known, such as
lakes, rivers, canals, lagoons, or marshes.
The most common types of contaminant source
boundaries either specify the source concentration or
prescribe the mass flux of contamination entering the
system. The concentration is generally prescribed
when the solubility limits of the contaminant largely
controlled the release. The mass flux type boundary
is typically used when a leaching rate is known or
estimated. Specialized source boundaries have also
been formulated which allow for radioactively decay
in the source. The ability of the code to treat source
decay may not be important if the parents and
daughters have a relatively long half-life compared to
the expected travel time to the nearest receptor.
One of the primary objectives of the site
characterization program is to identify the presence
and location of ground-water flow and contaminant
source boundaries so that they may be incorporated
into the conceptual model. These boundaries are
generally quantified in terms of the volume of ground
water and contamination moving through the system.
The physical boundaries are then translated into
mathematical terms as input into the computer model.
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Initial conditions are defined as values of ground-water
elevation, flow volumes, or contaminant
concentrations, which are initially assigned to interior
areas of the modeled regions. At least for the flow
modeling performed during the site characterization,
initial conditions are generally set to uniform values.
This is because the temporal nature of the flow system
is usually poorly defined. In addition, if the flow
analysis is performed to steady-state, which is usually
the case, the initial conditions assigned to the model
domain are irrelevant as identical solutions will be
reached for these values regardless of the values
initially assigned. This occurs because these steady-
state values depend solely on the values assigned to the
boundaries of the model.
Non-uniform initial values (i.e., contaminant
concentrations) are routinely used in the contaminant
transport analysis to depict the geometry and varying
contaminant concentrations within the plume, as well
as to define the contaminant concentrations leaching
from the contaminant source. The ability of a code to
allow non-uniform initial conditions would be essential
to a full description and simulation of the contaminant
plume(s).
Complex Flow and Transport Processes
Site-specific information describing the flow and
transport processes that dominate the migration of
radionuclides is not available during the scoping phase
of the investigation. As the site characterization
activities progress, greater attention is focused on the
physical, chemical, and biological processes that affect
ground-water flow and contaminant transport. Up
until this time, attention has been paid primarily to
estimating parameter ranges and variance s within these
ranges via sensitivity analyses. This approach has
limitations and needs to be broadened during the site
characterization phase if ground-water flow and
contaminant transport are to be well described. This
parameter-based approach is expanded by using
computer codes that mathematically accommodate the
dominant flow and transport processes. These
processes could include flow and transport through
fractures, density-driven flow, matrix diffusion,
fingering, surface-water/ground-water interactions,
and geochemical reactions. If active, each of these
processes can invalidate the output of models based on
the assumption that uniform flow and transport are
occurring through a homogeneous porous media.
Even at this stage, all of the geochemical reactions
that affect radionuclide transport are likely to be
lumped together into a single parameter, termed the
distribution coefficient. However, a better delineation
of any geochemical facies would allow for the
distribution coefficient to vary from layer to layer as
well as within the units themselves. If this simplified
means of simulating geochemical processes is found to
be inadequate, it may be necessary to use
thermodynamically based geochemical models in order
to address specific geochemical reactions.
Movement through the unsaturated zone can be
simulated in a number of different ways, depending
upon the objectives. If the unsaturated zone is
relatively thin and travel times are short, simplified
versions of more complex equations describing the
unsaturated flow and transport may suffice. However,
if the travel time through the unsaturated zone is
significant and accurate flow and transport predictions
are required, then mathematical methods, which
account for complex processes associated with flow
and transport through the unsaturated zone, may be
necessary.
The modeling objectives need to be defined prior to
the characterization; only then can the modeler be sure
that data are sufficient to perform modeling at the
necessary level of complexity. All too often,
limitations in the data, rather than the modeling
objectives, drive the sophistication of the modeling.
System Heterogeneity
One of the primary objectives of the site
characterization program is to identify heterogeneity
within the system and to delineate zones of varying
hydraulic properties. System heterogeneity is one of
the leading causes of a poor understanding of the
physical system controlling flow and transport.
If an accurate simulation of heterogeneous rocks is
required to meet the modeling objectives, a modeling
approach which allows for zones with different porous
rock properties is required; however, relatively few
codes can simulate discrete features, such as faults,
fractures, solution features, or macropores.
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3.3 Remedial Design and Implementation
As the site investigation proceeds into the remedial
phase, data are acquired that will be useful in
identifying feasible remedial alternatives. In
combination with models, these data are used to
simulate flow and transport. By predicting the
behavior of ground-water flow and the transport of
radionuclides the data and models in the selection and
design of the remedy and can be used to demonstrate
that the chosen remedy will achieve the remedial
goals.
Once remedial action alternatives have been identified,
their design may be refined as part of the development
of a conceptual design. Optimizing a design involves
evaluating alternative screen depths, pumping rates,
and well locations to identify the most effective
configuration.
Numerous studies have demonstrated the benefits of
using models to evaluate the flow of ground water and
the transport of radioactive substances. Models have
been used in the detailed analysis of alternative actions
to identify actions that would be ineffective or would
fail to meet the site's remediation goals. The
quantitative measures of performance derived from
simulation provide a useful basis for comparison with
other factors like the costs of remedial action.
However, if the travel time through the unsaturated
zone is significant and accurate predictions of flow and
transport are required, then mathematical methods
may be necessary to account for the complexity of
flow and transport through the unsaturated zone.
The modeling objectives associated with remedial
alternative design generally are more ambitious than
those associated with the site characterization phase.
Therefore, it often is necessary either to select an
advanced computer code or to modify the existing
model to simulate the more complex conditions. The
following are specific examples of processes that may
not be important to assessing baseline risk or to site
characterization but are often essential to the remedial
design:
three-dimensional flow and transport
matrix diffusion (pump-and-treat)
desaturation and resaturation of the aquifer
(pump-and-treat)
heat-energy transfer
vitrification/freezing)
(in-situ
sharp contrasts in hydraulic conductivity (barrier
walls)
multiple aquifers (barrier walls)
movement from confined to unconfined
conditions (pump-and-treat)
simulation of complex flow conditions (pumping
wells, trenches, injection wells)
From a modeling standpoint, the remedial design and
implementation is the most challenging phase of the
investigation. Frequently, it is the first time that data
are sufficient to verify the model' s predictions. The
many potential remedial actions (e.g., pump and treat)
provide excellent information on the temporal
response of the flow and transport to hydraulic
stresses. These data allow continuous refinement to
the calibration, making the model a powerful
management tool.
The modeling approaches taken at various sites
generally would have the following characteristics of
the conceptual model in common:
Transient Flow and Transport
Multi-Dimensionality
Prescribed Boundary and Non-uniform Initial
Conditions
Specialized Flow and Transport Processes
System Heterogeneity
Transient Solutions
By the time of the remedial design phase, the available
data usually span a relatively long period, which often
allows the temporal nature of the hydrogeologic
system to be well defined. The objectives of remedial
design can involve many criteria that could not be met
during the site characterization phase. Many of these
additional criteria may require that the code simulate
transient flow and transport which is necessary to
evaluate the effectiveness of remedial alternatives.
One such alternative is the placing of earthen covers
and a broad range of natural and synthetic barriers,
which are engineered to cap the surface and
subsurface soil. The cover prevents rainwater from
percolating through contaminated soil and carrying
3-9
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radionuclides
to the ground water. In the site characterization
program, the objectives could probably have been met
by assuming a constant areal recharge over the
modeled area. However, this steady-state approach
would not account for varying recharge rates, which
would be needed to simulate the deterioration of the
cap and the subsequent effect on the leaching rates of
radionuclides.
Excavation of radioactively contaminated soil will
leave some residual radioactivity in the soil contiguous
to the removal operations. It also could redistribute
contaminants in the unsaturated zone. Without the
ability to perform transient simulations, with the
source now largely removed, it would not be possible
to determine how long it would take for the remedial
actions to have a noticeable effect on downgradient
receptors.
Multi-Dim ensionality
The need to perform three-dimensional modeling
during the remedial phase will largely depend on the
remedial alternatives being considered and how their
effectiveness will be evaluated.
The remedial alternatives that are most commonly
supported by three-dimensional and quasi-three-
dimensional modeling are those that impart a strong
artificial stress to the hydraulic flow field, such as
pumping wells and extraction trenches. Often, before
these stresses are imposed, vertical ground-water
gradients are up to several orders of magnitude less
than the horizontal gradients and, therefore, can be
ignored; this simplification allows the system to be
modeled using a one- or two-dimensional flow
analysis. However, when imposed stresses
significantly alter on the hydraulic gradients, three-
dimensional flow fields generally develop. Without
the ability to simulate a three-dimensional flow field,
it would be very difficult to determine capture zones
and influent contaminant concentrations largely
because vertical leakage from units above and below
the screened interval of the extraction well would be
ignored, as would vertical concentration gradients.
Another remedial alternative that generally creates
three-dimensional flow fields is installation of physical
barriers to ground-water flow. Whether the barriers
consist of grout injection techniques, sheet pile cutoff
walls, or bentonite slurry walls, all have a common
problem: the hydraulic head builds up behind the
structures and induces vertical gradients, allowing
ground water to flow under the barriers. In these
cases, the analysis of vertical flow component is
essential in determining probable leakage rates and the
volume of water that would potentially flow beneath
the structure.
Transient Boundary and Non-Uniform Initial
Conditions
Most of the analysis up until the remedial phase can be
modeled with steady-state boundary conditions; i.e.,
physical features, such as the water elevations of
surface water bodies and areal recharge, can be
simulated with values that are constant in time. The
objectives of the remedial phase, however, may
demand that the transiency of these boundaries be
considered in the analysis. Time-weighted averages
may no longer apply. For instance, water bodies,
such as radioactively contaminated waste lagoons,
probably would have been treated as constant
boundaries, and their water-surface elevations would
have been held constant. However, if one of the
remedial activities involved withdrawing contaminated
water from one or more of the lagoons, the effect that
the change in water-surface elevations would have on
the ground-water gradients could be evaluated only by
simulating the drop in surface elevations with time.
This would be simulated by prescribing progressive
changes in the lagoon water level over time.
The ability to prescribe boundaries within the model
domain also would be important in the evaluation of
in-situ soil flushing techniques, which are used to
enhance the mobility of contaminants migrating
towards recovery points. In this case, recharge would
be varied with time to reproduce the effects that
various rates of flushing would have on the ground-
water flow and transport of contaminants.
Protective barriers to ground-water flow are
constructed of very low permeability material and
emplaced either to prevent contaminated ground-water
from migrating away from a site or to divert incoming
clean ground water away from the source of
contaminants. Potentially, barriers can last for several
decades, barring any geological disturbances, such as
tremors, ground settling, or significant changes in
hydraulic gradients. However, if a barrier should fail,
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water may infiltrate the site, and contaminated
leachate may move beyond the site. Therefore, the
effects of failure of a barrier to ground-water flow and
transport should be evaluated. The failure of the
barrier can be simulated in several ways. The most
straightforward method is to use transient boundaries
to simulate additional flow through the barrier as well
as a reduction in the difference between water-level
elevations in front of and behind the barrier.
Therefore, the selected code should be able to
incorporate transient boundaries.
Specialized Flow and Transport Processes
The design and evaluation of remedial alternatives
frequently involve consideration of flow and transport
processes that probably were not explicitly modeled
during site characterization. These processes include
complex geochemical reactions, matrix diffusion, heat
flow, and possibly, biological reactions.
As discussed, few numerical models satisfactorily
couple ground-water flow and contaminant transport
to complex geochemical reactions. The complex
geochemical models are based upon the laws of
thermodynamics, so they predict the potential for a
particular reaction to occur within a closed system.
Consequently, it is important that the controlling
geochemical reactions are examined, possibly in
laboratory benchscale or field studies. This is
particularly significant when physical/chemical
stabilization processes are considered as a remedial
alternative. In these processes, physical or chemical
agents are added to and mixed with a waste (typically
sludge in pits, ponds, and lagoons), to improve the
handling or leaching characteristics of the waste
destined for land disposal.
A detailed understanding of geochemistry also can be
very useful in estimating leach rates for uranium mill
tailings which otherwise might be associated with
unacceptably high uncertainties.
Matrix diffusion is the process by which concentration
gradients cause contaminants either to move into or be
drawn out of low-permeability areas where diffusion
rather than advection and dispersion governs
contaminant transport. Pump-and-treat systems tend
to draw water from the more permeable units, which
may leave large volumes of contaminants stored in the
clays and other fine-grained materials, which will
eventually diffuse out. Many computer codes do not
adequately simulate this very slow process. If matrix
diffusion is not accounted for, the movement of the
contaminant will be based solely on ground-water
velocities rather than the diffusion term. Ground-
water velocity generally will move the contaminant
much more rapidly than diffusion, and clean-up times
may be dramatically underestimated.
In-situ vitrification (ISV) of soils is a destructive
thermal treatment that converts contaminated soil and
waste into a chemically inert, stable glass and
crystalline product that resembles obsidian. Predicting
the effectiveness of ISV requires the modeling of
several specialized processes. One such process
would be vapor transport of radionuclides, such as
tritium, which would be an important health
consideration if the media was heated.
Microbial fixation appears to affect the transport of
radionuclides under some conditions. Radionuclides
may be either immobilized or mobilized by organisms
or plants. Immobilization may occur if radionuclides
are incorporated in the cells of microorganisms or
plants that are relatively stationary. On the other
hand, radionuclides may be mobilized by forming
biocolloids with bacteria, spores, and viruses.
Modeling microbial processes requires a code that, at
a bare minimum, allows a degradation rate to be
assigned to the contaminant(s).
System Heterogeneity
The ability of a code to accommodate severe contrasts
in the properties of soils and rocks is particularly
important in designing and evaluating physical barriers
for protecting ground water. If the application
involves extending the barrier down to a low-
permeability strata to form a seal and deter underflow
leakage, it is important that the code can incorporate
multiple stratigraphic layers, as well as sharp contrasts
in hydraulic conductivity. Only in this way can the
modeler show how contaminant flow and transport
affects leakage through the barrier wall and basement
strata.
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CHAPTER 4 -^MODEL APPLICATION
Figure 4-1 shows the three primary sources of ground-
water contamination by radioactivity that lead to four
possible contamination scenarios: (1) placing wastes
beneath the water table in direct contact with the
ground water, (2) placing wastes above the water
table, (3) disposing of wastes in surface impoundments
and seepage basins, and (4) recharge from surface
water bodies (steams, rivers, lakes) to the ground
water. Once the radioactivity reaches the ground
water, it may be reintroduced to the ecosystem in the
following ways: (1) discharge to rivers, lakes, and
other surface water bodies, (2) pumped to the surface,
(3) brought to the surface through plant transpiration
(phreatophytes), and (4) transport of vaporized
radionuclides undergoing phase transformations (e.g.,
3H, 14C, 85Kr, and222Rn).
The complexity of the analysis of the fate and
transport of radionuclides will depend upon the
objectives of the evaluation as well as on the
availability of the necessary data. It is important to
understand the uncertainty associated with the data
used for the analysis because this uncertainty will play
a role in evaluating the results of the analysis. As a
general rule, it is best to start with the simplest means
of evaluating the data and progress towards more
complex techniques. Appendix B presents simplistic
but useful calculations that may be made for release
rates, and the fate and transport of ground water and
radionuclides. Appendix B also gives some examples
of problems where common analytical methods, which
are less complex than numerical methods, have been
used to estimate the fate and transport of
radionuclides.
Prior to scoping calculations, the appropriate release
mechanisms for the movement of radionuclides to the
ground water must be determined. If it is relatively
certain that one or more components of the conceptual
model (e.g., unsaturated zone) is unimportant, then it
may be neglected. The physical and chemical
processes affecting the fate and transport of
radionuclides at the site (e.g., fracture flow, vapor
transport) also needs to be determined. An earlier
report issued as part of this interagency agreement
outlines how to determine what site-related
characteristics may be important (EPA94a).
After these determinations have been made, it must
then be decided how accurate the results need to be
and what level of analysis is appropriate to obtain the
desired results. If the physical and chemical processes
at the site are too complex to be satisfactorily
predicted by simplistic data analysis, then experts in
the field should be consulted regarding how to
proceed. It is not practical to perform complex
analyses without the use of computer programs and
considerable expert help.
The calculational methods presented in this chapter
and Appendix B are focused on scoping level analyses.
They have been divided into two parts: the release
analysis and the fate analysis. The equations given in
the release analysis section are used to estimate
contaminant release concentrations and volumetric
release rates. The fate analysis section deals with the
processes influencing radionuclide transport and how
to estimate radionuclide concentrations in the ground
water.
lie-low Winer Tiibk1
1
CVnlJiuiiiiiiiiun
t-
S'Urf;icc: Water Rndy
fLakc. River or Streams)
1
V
Ground Wni
Figure 4-1. Modes in which ground water may become contaminated
4-1
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4.1
Scoping Calculations
One of the primary goals of mathematical modeling is
to synthesize the conceptual model, as discussed in
Chapter 3, into mathematical expressions, which in
turn are solved with either a hand calculator or a
microcomputer. However, accurate modeling of all
aspects of the conceptual model is not always
necessary; only certain components require modeling.
In practice, early modeling focuses upon assessing the
significance of specific parameter values and their
effects on flow and transport, rather than modeling
specific hydrogeologic transport processes. For
instance, it is common during the scoping phase to
evaluate transport as a function of a range of hydraulic
conductivities; however, it is unlikely that more
complex processes, such as flow and transport through
fractures, would be considered.
This section, in conjunction with Appendix B,
describes methods used to calculate the volume and
concentration of radioactivity that may be expected to
reach the ground water from several types of
contaminant sources (Figure 4-1). Also described are
methods for estimating transit times and concentrations
of radionuclides. To impart a sense of the
uncertainties inherent in the calculations, the primary
mechanisms that affect the fate and transport of
radionuclides are discussed. In particular, it is
important to keep in mind that all of the complex
geochemical reactions that influence radionuclide
transport generally are lumped into a single parameter
termed the distribution coefficient (Kd). Without a
basic understanding of how these processes can affect
radionuclide transport, even the most simplistic
analysis may have fundamental flaws. Therefore, the
geochemical processes affecting radionuclide transport
are discussed briefly in Appendix A. In Appendix B,
a series of screening calculations are given with
several examples of problems illustrating the
mathematical methods that may be used to estimate
radionuclide transit times and concentrations.
4.1.1 Release Analysis - Ground Water
When analyzing a radionuclide release to ground
water, the potential release mechanisms should be
evaluated first. This evaluation involves a qualitative
screening approach aimed at determining the sources
of release. The factors and mechanisms that may
significantly affect the potential for release are then
analyzed; this may require more detailed field
investigations or numerical modeling.
Several mechanisms may release radionuclides to the
ground water either directly or indirectly, as follows:
Direct discharge (e.g., on-site release from
treatment processes)
Generation of leachate (e.g., from buried wastes,
surface impoundments, and absorption beds)
Overland flow (e.g., from impoundment
overflow or failure, drum leakage)
Generation of leachate and direct discharges are the
mechanisms most likely to affect ground water
directly. Overland flow might affect ground water
indirectly, and is discussed in Appendix B.
Several factors affect leachate release:
Physical/chemical/radiological properties of the
radionuclides;
Type of waste form and container (e.g., steel
drums, wooden or cardboard boxes, plastic bags,
absorption beds);
Length of time that the wastes have been stored
or buried;
Hydrogeologic framework of the system (e.g.,
depth to water table, soil/rock properties);
Quantity of wastes;
Climatological considerations (e.g., pre-
cipitation).
Information about the physical, chemical, and
radiological properties of the radionuclides may help
to determine the associated disposal practices, which
may, in turn, assist in estimating the potential for
release. Furthermore, the physical and chemical
properties of a particular radionuclide will dictate its
fate and transport processes.
A clear understanding of the physical system (e.g.,
climatology, hydrogeology) is necessary as a basis
from which to predict migration rates and exposure
pathways.
Information about when the wastes were disposed or
emplaced and the quantities involved is important
4-2
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when estimating concentrations of radionuclides that
could potentially reach sensitive receptors.
Ground-water contamination results from
radionuclides leaching from surface or subsurface
soils; from treatment, storage, and disposal areas; and
from the direct migration of liquid wastes.
Radionuclides can be leached from disposal areas by
precipitation and runoff that percolates through the soil
or by direct submersion of the waste in ground water.
Unlined lagoons and surface impoundments may
introduce contaminants directly into ground water.
The containers for radioactive materials stored above
ground may leak and percolate to ground water. The
potential for release from all of these sources must be
evaluated; any that are significant generally undergo
a more quantitative assessment.
Estimation of release involves two quantifications: (1)
of radionuclide concentrations in waste and/or
leachate, and (2) of the volume of the leachate or
direct-discharge release rates. The procedures vary,
depending upon the characteristics of the release site.
Section B.I of Appendix B describes procedures to
calculate steady-state releases of radionuclides to
ground water. The calculations focus on release
concentrations and volumetric release rates. For
ponds or lagoons, the concentration of the radionuclide
in the lagoon or impoundment is considered to be the
concentration of the leachate. This assumption ignores
the geochemical reactions that may be occurring at the
base of the pond where sediments may tie up high
concentrations of radionuclides when oxidizing
conditions prevail (summer) and release pulse
concentrations of contaminants when reducing
conditions are favored (winter). Methods for
estimating volumetric release rates are presented for
both solid and liquid wastes.
4.1.2 Fate Analysis - Ground Water
The nature of the ground-water environment restricts
the number of processes that control the fate of
radionuclides as they are transported from their source
to the receptor area. These processes fall into two
categories: transport processes and radioactive decay.
Transport-related processes (i.e., sorption, ion
exchange, and precipitation of solids) can facilitate or
retard the movement of ground-water contaminants,
but radioactive decay always results in a loss of
activity (disintegrations or decays per second) of the
original radionuclide. However, as the parent
radionuclide disintegrates, radioactive or chemically
toxic daughter products can increase.
Calculational screening methods do not directly
simulate fate processes that influence the transport of
radionuclides. Generally, the effects of these
processes on transport are combined into a single
term, designated the distribution coefficient. In
Appendix A, these fate processes are discussed
qualitatively to provide a general understanding of
distribution coefficients, which are later used in
Section B.2 (Appendix B) to determine quantitative
retardation factors. Fate processes associated with
radionuclide transport must be explicitly simulated,
geochemical and/or hydrochemical computer models
will be needed.
4.1.3 Analytical Methods for Aquifer Flow and
Transport
Analytical ground-water transport models can be used
for certain analyses where the available data do not
warrant a more complicated numerical analysis. Such
models are useful for scoping the transport problem
and may frequently be adequate for regulatory needs
if the model and corresponding input data are chosen
conservatively.
Analytical transport solutions are generally able to
simulate only systems that assume steady-state flow
conditions. However, because the available data
rarely support transient simulations during the scoping
phases, common analytical methods may often be used
more effectively than numerical methods. It is much
easier to conduct bounding and sensitivity analyses
with analytical rather than numerical models.
Examples of such calculations are presented in
Appendix B.
4.1.4 Uncertainty Analysis
In the scoping phase, the uncertainty in the analysis
should be emphasized. Uncertainty is inherent in
models of the behavior of a hydrogeologic system
because our knowledge is incomplete. Many
parameters used as inputs to a model are obtained by
data collection. Investigators knowledgeable about the
data they collect make a finite number of observations,
choosing the parameters, and, how, where, and when
to measure them. However, the collection process
itself can introduce uncertainty through errors in
measurements, the system's inherent randomness, and
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limited sampling of the variable physical, chemical,
and biological properties of the system. In many
aspects of data collection, the professional judgment of
an analyst with expertise in the area of investigation
often enters into the scientific process. For example,
selecting methods to collect data, interpreting data,
developing conceptual models, and selecting model' s
parameters all require professional analysis and
judgment. Therefore, the uncertainty in input
parameters used in predictive models may result from
several sources, including incomplete data, intrinsic
spatial variability of the property in question,
uncertainties in measurement, and uncertainties
resulting from differences in scale between acquisition
of the data and application of the model.
In practice, much of the early modeling focuses on
assessing the significance of the uncertainty associated
with specific parameters and their effects on flow and
transport, rather than on modeling specific processes
of hydrogeologic transport. For instance, it is
common during the scoping phase to evaluate transport
as a function of a range of hydraulic conductivities;
however, it is unlikely that more complex processes,
such as flow and transport through fractures, would be
considered.
In general, the uncertainty associated with each of the
parameters is expressed by a probability distribution,
which yields a likely range of values for each
parameter. It is important to select a model where
individual parameter values can be selected
systematically from the range and easily substituted
into the governing mathematical equations that
describe the dominant flow and transport processes at
the site. In this manner, the effects of a single
parameter or a multitude of parameters on the rate of
contaminant movement and concentrations may be
evaluated. This technique of substituting one value for
another from within a range of values is called a
sensitivity analysis.
In many cases, the possible range of values of
important parameters is unknown or very large.
Consequently, the analyst has little alternative but to
evaluate the sensitivity of the results to a very broad
range of possible values. Many of these early results
will be unrealistic but cannot be ruled out until reliable
site data are obtained. These types of analyses are
useful because they help to direct the field work.
However, they also can be used incorrectly. For
example, individuals unfamiliar with the scoping
process could reach grossly inappropriate conclusions
about the potential public-health impacts of the site
based on these scoping analyses. Accordingly, care
must be taken to assure that the results of scoping
analyses are used for their intended purpose.
Sensitivity analyses identify the main contributors to
the observed variation in the results. These techniques
typically are applied iteratively. The first iteration can
include rather general assumptions leading to
preliminary results that help focus these techniques in
subsequent iterations. Thus, the resources required
for the uncertainty reduction techniques can be
directed at the areas of the site characterization where
the benefits of understanding uncertainty and reducing
it (where possible) are greater.
However, sensitivity analyses alone will rarely identify
a flawed conceptual model. For example, the failure
to identify and include a fault(s) in the conceptual
model subsequently would not account for preferential
pathways that could, potentially, underestimate
receptor concentrations.
An alternative to the detailed sensitivity analysis is a
conservative bounding approach. In this less
demanding analysis, values are selected from the
range of parameters to provide the highest probability
that the results are conservative, i.e., that the
migration rates and concentrations of the contaminant
would not be underestimated. For example, high
values of hydraulic conductivity combined with low
effective porosities and distribution coefficients would
maximize the predicted migration rates of the
contaminant although its concentrations at receptors
may be underestimated.
Tables 4-1 and 4-2 give preliminary guidance on the
general effects that various parameters have on the
modeling results. Table 4-1 indicates whether the high
end or low end of the parameter's distribution should
be used if a conservative estimate (i.e., maximum
value) of the extent of contamination is desired. The
same concept has been used in Table 4-2 to show the
effect of various parameters on the maximum
concentrations arriving at a downgradient receptor.
As shown in these tables, when estimating the
maximum extent of contamination, the dispersivity
should be maximized. The opposite is true when
estimating maximum concentrations; in this case
dispersivity should be minimized.
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Table 4-1. Bounding Analyses: Extent of Contamination
Parameter
Maximum
Minimum
Source Term
Constant Concentration
Mass Flux
Insensitive
Insensitive
Unsaturated Zone
Infiltration Rate
Moisture Content
Total Porosity
Bulk Density
Distribution Coefficient
High
Low
Low
Low
Low
Low
High
High
High
High
Saturated Zone
Aquifer Thickness
Gradient
Distribution Coefficient
Dispersivity
Effective Porosity
Hydraulic Conductivity
Well Location and Intake Depth
Insensitive
High
Low
High
Low
High
Shallow/Plume Centerline
Low
High
Low
High
Low
Deep/Off Plume Centerline
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Table 4-2. Bounding Analyses: Concentration of Contaminants
Parameter
Source Term
Highest
Constant Concentration
Lowest
Mass Flux
Unsaturated Zone
Infiltration Rate
Moisture Content
Total Porosity
Bulk Density
Distribution Coefficient
High
Low
Low
Low
Low
Low
High
High
High
High
Saturated Zone
Aquifer Thickness
Gradient
Distribution Coefficient
Dispersivity
Effective Porosity
Hydraulic Conductivity
Well Location and Intake Depth
Thin
Low
Low
Low
Low
Low
Shallow/Plume Centerline
Thick
Low
High
High
High
High
Deep/Off Plume Centerline
4.2 Site Characterization Modeling
In the scoping phase of the investigation, data
limitations impose a simple modeling approach which
uses conservative parameter estimates. One of the
primary objectives of the site characterization program
is to obtain sufficient data to enable the conservative
modeling approach to be replaced by a defensible and
more realistic approach which incorporates site-
specific data.
Conservative analysis by itself cannot meet many of
the objectives defined for the site characterization
phase of the investigation. If parameter values are not
known, it may be necessary to make conservative
estimates; however, the effects that a conservative
approach may have on other aspects of the remedial
program must be considered. For example, if, during
the baseline risk assessment, conservatively high
hydraulic conductivities are used in order to ensure
that the downgradient contaminant arrival times are
not underestimated, several problems may occur.
First, it would be difficult to calibrate the model to
known parameters (e.g., potentiometric surface), and
adjustments to other parameters would be required in
order to match measured field values. The end result
would be a model that poorly predicts system
responses to hydraulic stresses (e.g., extraction wells).
A second problem would involve contaminant
concentrations. A conservative increase in hydraulic
conductivity would predict more ground-water flow
through the system than is actually occurring, which
might result in an underestimate of the contaminant
concentrations at downgradient receptors. More
problems may arise during the remedial design. If the
modeling results are used to estimate clean-up times,
the model may predict that water and contaminants are
flowing faster than they actually are and at lower
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concentrations. This would result in an underestimate
of both the amount of time required for remediation as
well as the contaminant breakthrough concentrations.
The major impact that a more specific conceptual
model of the site will have on model application is that
now parameter ranges have been narrowed by
additional data acquisition, and sensitivity analyses can
become more focused. Refinement of the parameter
values diminishes the need to perform so many
sensitivity analyses. In view of the increased demand
for more accurate simulation of the controlling flow
and transport processes, the advantages of scoping
type analyses are outweighed by their inability to
simulate more complex conditions. Therefore, model
application in the site characterization phase will
generally be performed with numerical models.
After the conceptual model is formulated and the
modeling objectives are clearly defined in terms of the
available data, the investigator should have a good
idea of the level of sophistication that the anticipated
modeling will require. Now, one or more computer
code(s) can be chosen that have the attributes
necessary to describe mathematically the conceptual
model at the desired level of detail. Selecting the code
requires a detailed analysis of the conceptual model to
determine the degree to which specific waste and site
characteristics need to be explicitly modeled. The
selection process and evaluation criteria are described
in a joint agency publication (EPA 94).
During the site characterization phase, application of
the model is generally quite sophisticated and typically
experienced modelers are in charge of the modeling
process.
Therefore, this section will not describe the modeling
process step-by-step, but, rather, will provide the
Remediation Manager and support personnel with
sufficient information to allow the remedial team to
make informed decisions about ground-water
modeling.
Accordingly, this section has three goals:
1. to impart a general understanding of how ground-
water modeling is carried out in the site
characterization phase and what objectives are
achievable.
2. to outline the topics that should be covered in the
application report and how results should be
presented to facilitate peer review.
3. to provide relatively simple methods that can be
used as reality checks on modeling performed by
others.
This section is organized into subsections that follow
a path typical of model application strategies. Specific
guidance for each subsection is tabulated in Table 5-1.
4.2.1 Code Selection
After formulating the conceptual model, it is necessary
to select one or more computer code(s). Three basic
choices are available: analytical, semi-analytical, or
numerical codes. Analytical and semi-analytical
methods, which are limited to simplified
representations of the physical setting and flow and
transport processes, are ideally suited for performing
sensitivity and conservative bounding analyses because
they are computationally efficient (i.e., fast) and
require relatively little data as input. As discussed in
Section 4.1.3, analytical models are typically designed
for easy performance of sensitivity analyses. In
contrast, numerical methods do not lend themselves to
the same kind of "simplified" applications. Primarily
because numerical models are difficult to set up and
require substantial data input to calibrate the model,
and multiple parameter substitutions are generally very
cumbersome. However, as the modeling objectives
become more complex and can no longer be addressed
by simple bounding and sensitivity analyses, additional
field data and more sophisticated analysis methods
(e.g., numerical models) become necessary.
The greatest difficulty in selecting the most
appropriate computer code is not in determining which
codes have specific capabilities, but rather, which
capabilities are required to support remedial decision-
making during each phase of the remediation at a
specific site. The necessary degree of sophistication
of the modeling can be evaluated in terms of both site-
related issues and objectives, as well as the qualities
inherent in the computational methods for solving
ground-water flow and transport equations.
A contaminant fate and transport model results from
the application of a previously written or new
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computer code to a specific problem via the collection
of input data and the parameterization of site
characteristics. The resultant model is, therefore, a
merger of a mathematical formulation, solution
methodology, data, and ancillary information which
enhances or controls the use of the model. In addition
to selection criteria for the modeling objectives which
were presented in the previous section, the code
evaluation process must also consider attributes that
are integral components of the computer code(s)
including;
Source Code Availability
History of Use
Code Documentation
Code Testing
Hardware Requirements
The development of selection criteria presented in this
section takes an approach consistent with industry
standards by relying on published reports pertaining to
the quality assurance and quality control in the
development and application of computer codes.
Source Code Availability
To facilitate a thorough review of the generic code,
detailed documentation of the code and its
developmental history is required. Also, the source
code must be available for inspection. In addition, to
ensure independent evaluation of the reproducibility of
the verification and validation results, the computer
source code as well as the compiled version of the
code (i.e., computer code in machine language) should
be available to the reviewer, together with files
containing the original test data used in the code's
verification and validation.
History of Use
Much of the information needed for a thorough code
evaluation can be obtained from the author or
distributor of the code. In fact, inability to obtain the
necessary publications may indicate that the code is
either not well documented or that the code is not
widely used. In either case, the inaccessibility of the
documentation and related publications should be
strong grounds for deciding that the code is
unacceptable.
The acceptance and evaluation process should rely on
user opinions and published information in addition to
hands-on experience and testing. User opinions are
especially valuable in determining whether the code
functions as documented or has significant errors or
shortcomings. In some instances, users independent
of the developer have performed extensive testing and
bench-marking or are familiar with published papers
documenting the use of the code. Users will also have
first-hand knowledge about how easy it is to use the
code and what level of experience is required.
Quality Assurance
Code selection should be closely tied to the quality
assurance criteria followed during the development of
the computer code. These criteria will determine the
adequacy of the code testing and documentation.
Quality assurance in modeling is the procedural and
operational framework put in place by the organization
managing the modeling study, to assure technically
and scientifically adequate execution of all project
tasks included in the study, and to assure that all
modeling-based analysis is verifiable and defensible
(TAY85).
The two major elements of quality assurance are
quality control and quality assessment. Quality control
refers to the procedures that ensure the quality of the
final product. These procedures include the use of
appropriate methodology in developing and applying
computer simulation codes, adequate verification and
validation procedures, and proper usage of the selected
methods and codes (HEI92). To monitor the quality
control procedures and to evaluate the quality of the
studies, quality assessment is applied (HEI89).
Software quality assurance (SQA) consists of the
application of procedures, techniques, and tools
through the software life cycle, to ensure that the
products conform to pre-specified requirements
(BRY87). This requires that in the initial stage of the
software development project, appropriate SQA
procedures (e.g., auditing, design inspection, code
inspection, error-prone analysis, functional testing,
logical testing, path testing, reviewing, walk-through),
and tools (e.g., text-editors, software debuggers,
source code comparitors, language processors) need to
be identified and the software design criteria be
determined (HEI92).
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Quality assurance for code development and
maintenance implies a systematic approach, starting
with the careful formulation of code design objectives,
criteria, and standards, followed by an implementation
strategy. The implementation strategy includes the
design of the code structure and a description of the
way in which software engineering principles will be
applied to the code. In this planning stage, measures
are to be taken to ensure complete documentation of
code design and implementation, record keeping of the
coding process, description of the purpose and
structure of each code segment (functions,
subroutines), and record keeping of the code
verification process.
Records for the coding and verification process may
include: a description of the fundamental algorithms
describing the physical process(es) which are to be
modeled; the means by which the mathematical
algorithms have been translated into computer code
(e.g., Fortran); results of discrete checks on the
subroutines for accuracy; and comparisons between
the codes' numerical solutions and analytical or other
independently verified numerical solutions.
Code verification or testing ensures that the underlying
mathematical algorithms have been correctly translated
into computer code. The verification process varies
for different codes and ranges from simply checking
the results of a plotting routine to comparing the
results of the computer code to known analytical
solutions or to results from other verified codes.
Traceability describes the ability of the computer
analyst to identify the software which was used to
perform a particular calculation, including its name,
date, and version number, while retrievability refers
to the availability of the same version of the software
for further use.
Code Documentation
Detailed guidelines for the preparation of
comprehensive software documentation are given by
the Federal Computer Performance Evaluation and
Simulation Center (FEDSIM81). This publication
discusses the structure recommended for four types of
manuals providing model information for managers,
users, analysts, and programmers. According to
FEDSIM81, the manager's summary manual should
contain a model description, model development
history, an experimentation report, and a discussion of
current and future applications. Currently, ASTM
(American Society for Testing and Materials) is
developing a standard ground-water code description
for this specific purpose (HEI92).
As discussed in van der Heijde (HEI92), the code
documentation should include a description of the
theoretical framework represented by the generic
model on which the code is based, code structure and
language standards applied, and code use instructions
regarding model setup and code execution parameters.
The documentation should also include a complete
treatment of the equations on which the generic model
is based, the underlying mathematical and conceptual
assumptions, the boundary conditions that are
incorporated in the model, the method and algorithms
used to solve the equations, and the limiting conditions
resulting from the chosen approach. The
documentation should also include user's instructions
for implementing and operating the code, and
preparing data files. It should present examples of
model formulation (e. g., grid design, assignment of
boundary conditions), complete with input and output
file descriptions, and include an extensive code
verification and validation or field testing report.
Finally, programmer-oriented documentation should
provide instructions for code modification and
maintenance.
An integral part of the code development process is
the preparation of the code documentation. This
documentation of QA in model development consists
of reports and files pertaining to the development of
the model and should include (HEI92):
A report on the development of the code
including the (standardized and approved)
programmer's bound notebook containing
detailed descriptions of the code verification
process;
Verification report including verification
scenarios, parameter values, boundary and initial
conditions, source-term conditions, dominant
flow and transport processes;
Orientation and spacing of the grid and
justification;
Time-stepping scheme and justification;
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Changes and documentation of changes made in
code after baselining;
Executable and source code version of baselined
code;
Input and output (numerical and graphical) for
each verification run;
Notebook containing reference material (e.g.,
published papers, laboratory results,
programmer's rationale) used to formulate the
verification problem.
Furthermore, the software should be documented in
sufficient detail to (GAS79):
record technical information that enables system
and program changes to be made quickly and
effectively;
enable programmers and system analysts, other
than software originators, to use and to work on
the programs;
assist the user in understanding what the program
is about and what it can do;
increase program sharing potential;
facilitate auditing and verification of program
operations;
provide managers with information to review at
significant developmental milestones so that they
may independently determine that project
requirements have been met and that resources
should continue to be expended;
reduce disruptive effects of personnel turnover;
facilitate understanding among managers,
developers, programmers, operators, and users
by providing information about maintenance,
training, and changes in and operation of the
software;
inform other potential users of the functions and
capabilities of the software, so that they can
determine whether it serves their needs.
The user's manual should, at a minimum, consist of:
an extended code description;
code input data description and format;
type of output data provided;
code execution preparation instructions;
sample model runs;
trouble shooting guide; and
contact person/affiliated office.
The programmer' s manual should, at a minimum,
include;
code specifications;
code description;
flow charts;
descriptions of routines;
data base description;
source listing;
error messages; and
contact person/affiliated office.
The analyst's manual should, at a minimum, present:
a functional description of the code;
code input and output data;
code verification and validation information; and
contact person/affiliated office.
The code itself should be well structured and internally
well documented; where possible, self-explanatory
parameter, variable, subroutine, and function names
should be used.
Code Testing
Before a code can be used as a planning and decision-
making tool, its credentials must be established
through systematic testing of the code's correctness
and evaluation of the code's performance
characteristics (HEI89). Of the two major approaches
available, the evaluation or review process is
qualitative in nature, while code testing results can be
expressed using quantitative performance measures.
Code testing (or code verification) is aimed at
detecting programming errors, testing embedded
algorithms, and evaluating the operational
characteristics of the code through its execution on
carefully selected example test problems and test data
sets. ASTM84 defines verification as the examination
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of the numerical technique in the computer code to
ascertain that it truly represents the conceptual model,
and that there are no inherent problems that prevent
correct solutions.
At this point, it is necessary to point out the distinction
between generic simulation codes based on an
analytical solution of the governing equation(s)
(Appendix C) and codes that include a numerical
solution. Verification of a coded analytical solution is
restricted to comparison with independently calculated
results using the same mathematical expression, i.e.,
manual calculations, using the results from computer
programs coded independently by third-party
programmers. Verification of a code formulated with
numerical methods might take two forms: (1)
comparison with analytical solutions, and (2) code
intercomparison between numerically based codes,
representing the same generic simulation model, using
synthetic data sets.
It is important to distinguish between code testing and
model testing. Code testing is limited to establishing
the correctness of the computer code with respect to
the criteria and requirements for which it is designed
(e.g., to represent the mathematical model). Model
testing (or model validation) is more inclusive than
code testing, as it represents the final step in
determining the validity of the quantitative
relationships derived for the real-world system the
model is designed to simulate.
Attempts to validate models must address the issue of
spatial and temporal variability when comparing model
predictions with limited field observations. If
sufficient field data are obtained to derive the
probability distribution of contaminant concentrations,
the results of a stochastic model can be compared
directly. For a deterministic model, however, the
traditional approach has been to vary the input data
within its expected range of variability (or uncertainty)
and determine whether the model results satisfactorily
match historical field measured values. This code-
testing exercise is sometimes referred to as history
matching.
Konikow and Bredehoeft (KON92) argue compellingly
that computer models cannot be truly validated but can
only be invalidated. As reported by Hawking
(HAW88), any physical theory is only provisional, in
the sense that it is only a hypothesis that can never be
proven. No matter how many times the results of the
experiments agree with some theory, there is never
complete certainty that the next test will not contradict
the theory. On the other hand, a theory can be
disproven by finding even a single observation that
disagrees with the predictions of the theory.
From a philosophical perspective, it is difficult to
develop selection criteria for a model validation
process which may be intrinsically flawed. However,
the average strategy presented in this chapter provides
some assurance that the code selected has the highest
probability of most accurately representing the
conceptual model.
Hardware Requirements
In general, hardware requirements should rarely be a
discriminatory factor in the selection of a computer
code. However, a number of the available codes
require very sophisticated hardware, not so much
because of the intrinsic requirements of the code but
because the simulated processes may be very complex
and require time-consuming solution methods.
Therefore, hardware requirements should be clearly
identified for the code itself and be consistent with the
hardware available to the user.
An earlier report prepared by this interagency working
group details the conditions under which specific
features and capabilities of the model are needed to
support remedial decision-making (EPA94).
A final consideration, true for all phases of the
project, is the need to select codes that have been
accepted by technical experts and used within a
regulatory context.
4.2.2 Model Construction
One primary goal of mathematical modeling is to
synthesize the conceptual model into numerical terms
from which flow and transport processes may be
investigated under specified conditions. This process
entails several discrete steps: (1) partitioning the
conceptual model into units of time and space; (2)
assignment of boundary conditions; and (3)
specification of the values of parameters. The
following sections discuss the relevance of each of
these topics to the modeling process. Then guidelines
for modeling and, where appropriate, modeling review
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criteria are suggested.
4.2.2.1 Layering and Gridding
In a numerical model the region of interest is
partitioned into a series of blocks (i.e., elements)
which are arranged in layers (Figures 4-2 and 4-3).
This practice, termed discretization, effectively
replaces the continuous problem domain with an array
of blocks and nodes. The basic concept used is to
divide up the section as realistically as practical.
When possible, geologic logs and other information
typically are used to identify geologic unit contacts.
Information on formational dip, depositional, and
erosional features may have a pronounced effect on
unit contact elevations and, therefore, will also have
a significant impact on ground-water flow and
contaminant transport. To accommodate variations in
unit thickness, some finite element models allow the
use of curvilinear elements to allow the model' s
planes to trace the unit contacts more precisely.
The determination of how many layers to include
depends on both the conceptual model and the
objectives of modeling. Typically, multiple layers are
used to accommodate the vertical variation of
hydrologic parameters that represent the hydrogeo logic
units within the modeled region.
One of the critical steps in applying a ground-water
model is selecting the size of the nodal spacing. The
more finely the grid is spaced, the more accurate the
numerical solution. However, the desire for accuracy
must be balanced against the impracticality of solving
for large numbers of nodes and the long computer run
times that may be involved.
The most quantitative guidance for selecting the nodal
spacing applies only to modeling contaminant transport
not to ground-water flow modeling. These criteria are
related to the fact that the value of dispersion
coefficients (Section 2.4) varies with the absolute
value of the Darcy velocity (Section B.3).
This relationship, as expressed below, defines a
dimensionless Pec let number.
p =
D
where:
Pe = Peclet Number
V = Darcy Velocity
D = Dispersivity
The numerical solution of the transport equation
becomes unstable if the Peclet number becomes too
large. Price et al (PRI66) have shown that the
stability of the transport solution is ensured if the
Peclet number is less than 2.
Figure 4-2. Three-dimensional view of model grid.
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Rawer
Uytr 3
Lr:
Figure 4-3. Cross-sectional view of model grid.
4.2.2.2 Definition of Boundary and Initial Conditions
To obtain a unique solution for the governing equation
of ground-water flow and contaminant transport,
additional information is required about the physical
state of the ground-water system. This information is
described by boundary and initial conditions.
Boundary conditions are the conditions the modeler
specifies as known values to solve for the unknowns in
the problem; these values may be associated with
either ground-water flow or contaminant transport.
One of the primary objectives of the site
characterization program is to identify the presence
and location of ground-water flow and contaminant
source boundaries so that they may be incorporated
into the conceptual model. These boundaries
generally are quantified in terms of the volume of
ground water and contamination moving through the
system. The physical boundaries are then translated
into mathematical terms, as input into the computer
model. The initial conditions are simply the values of
hydraulic head or contaminant concentrations at a
reference initial time. For steady-state problems, only
boundary conditions are required, whereas for
transient problems, both conditions are required.
Ground-Water Flow
Boundary Conditions
Ground-water boundaries may be described in terms
of where water is flowing into and out of the ground-
water system. There are many different types of
boundaries, including: surface-water bodies, ground-
water divides, rainfall, wells, and geologic features,
such as faults and sharp contrasts in lithology.
In general, boundary conditions are known or
estimated values that are assigned to surfaces and
planes that either frame the perimeter of the modeled
area or define the release from the contaminant
source. The different types of flow boundary
conditions are: (a) head (ground-water elevation) is
known for surfaces or planes bounding the modeled
region; (b) ground-water flow volumes are known for
surfaces or planes bounding the modeled region; and
(c) some combination of (a) and (b) is known for
surfaces or planes bounding the region. Boundary
conditions could also be assigned to interior features
of the modeled region where ground water elevations
or flow volumes are known, such as lakes, rivers, or
marshes. In practice, these types of boundaries result
in three conditions including: (1) specified value of
hydraulic head; (2) specified flux; and, (3) head-
dependent flux. Table 4-3 briefly describes the three
boundaries and their examples.
Just as the physical ground-water system is idealized
as a continuum in deriving the mass balance
differential equations, it also is expedient to idealize
the conditions on the boundaries of the system so that
they too can be approximated by a mathematical
expression. In nature, the boundary conditions of
ground-water systems are of several kinds. One of the
most common would be at a well. Since the porous
medium terminates at the well face, the aquifer not
only has a boundary around its perimeter, but the
outline of each well also is considered a
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Table 4-3. Boundary Conditions of Ground-Water Flow Equations
Type
Description and Examples
I. Specified head
Value of hydraulic head is specified along the boundary. Typical
examples include: (1) constant head condition on the boundary in
direct hydraulic contact with a river or lake or spring outlets; and (2)
boundary condition at a pumped well operating at constant or specified
drawdown.
II. Specified flux
Flux or flow rate of water is specified along the boundary. The flux is
usually expressed as qn = -K 3h/3n, where n refers to the direction
perpendicular to the boundary. Typical examples include: (1)
condition of zero flow across impermeable rock boundaries or across a
water divide or a streamline; and (2) boundary condition on the water
table receiving prescribed rate of accretion.
III. Head-dependent flux
Flux or flow rate of water is dependent on the head difference across a
semi-impervious layer adjacent to the aquifer. This boundary
condition is encountered when the flow domain is intercepted by a
river bed clogged by a thin layer of silt or clay. The leakage flux is
given by qn = K/b (h-H) where K is the hydraulic conductivity, b is
the thickness of the thin layer, and (h-H) is the pressure difference
across the layer. A similar situation is encountered when the aquifer
is overlain by a water table aquitard layer.
boundary to the aquifer. The boundary conditions at
wells may also be treated as a constant or variable
specified flux, or constant head, depending on which
best describes the physical conditions.
Impermeable or nearly impermeable boundaries are
formed by underlying or overlying beds of rock, by
contiguous rock masses (as along a fault or along the
wall of a buried rock valley), or by dikes or similar
structures. Permeable boundaries are formed at the
bottom of rivers, canals, lakes, and other bodies of
surface water. These permeable boundaries may be
treated as surfaces of equal head (specified) if the
volume of surface water is large, so that its level is
uniform and independent of changes in ground water
flow. However, the uniform head on a boundary of
this type may change with time due to seasonal
variation in the surface-water level. Other bodies of
surface water, such as streams, may form boundaries
with non-uniform distributions of head which may be
either constant or variable with time. For example, a
small stream might be affected by a nearby withdrawal
of ground water if that withdrawal occurred at a rate
similar to the flow in the stream. Then, the boundary
condition would depend on the ground-water flow; that
is, it would be a head-dependent flux.
In an analysis of ground-water flow, it is common to
assume a simple geometry for sinks and sources
existing at the boundary, or inside the flow system. In
areal flow simulations, points often are used to
represent individual wells, whereas lines are used to
represent rivers, lakes, and other surface-water
bodies. These representations are justified, provided
that detailed information about potentiometric head
and velocity distributions in the immediate vicinity of
the individual sources or sinks is not a concern. If
such information is required, then the actual geometric
features of the source or sink must be incorporated
into the flow system. For instance, if the modeler is
using flow analysis to assess the performance (e.g.,
specific capacity or maximum yield) of a pumping
well or evaluate its drawdown versus time data, then
the well must be represented as a cylindrical boundary
of specified diameter and specified screened length.
Other assumptions related to sinks and sources are
variations of volumetric flow rates of flux (flow rate
per unit length) distributions. For a point source or
sink, it is common to assume a constant flow rate
unless the field data indicate drastic variations
necessitating a more accurate treatment. For a line
source or sink, uniform flux distribution along the line
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also is assumed to simplify the analysis. Again, the
justification of this assumption depends on actual field
conditions.
Initial Conditions
Initial conditions are defined as values of ground-water
elevation which are initially assigned to interior areas
of the modeled regions. The initial conditions for
steady-state flow models are generally set to uniform
values because the temporal nature of the flow system
is not simulated. The initial conditions assigned to the
model domain are irrelevant as identical solutions will
be reached regardless of the values initially assigned.
This occurs because these steady-state values depend
solely on the values assigned to the boundaries of the
model.
The initial and boundary conditions for the variably
saturated water flow equation can be stated in the
same manner as those for the saturated ground-water
flow equation. The solution of the former problem
enables the analyst to obtain the head distribution, as
well as the water saturation (or water content)
distribution, and details of the velocities and flow rates
for an analysis of the migration of contaminants in
variably saturated porous media.
Initial conditions for a transient flow model may be
interpolated from water-level data or may be set from
a previous steady-state flow simulation. Ideally, the
initial heads should come from a steady-state
simulation. Interpolated initial conditions are often
not consistent with the model boundary conditions and
parameterization. In this case, heads computed during
early time steps are inaccurate.
Contaminant Source
Boundary Conditions
The most common contaminant source boundaries
either specify the source concentration or prescribe the
mass flux of contamination entering the system. The
former generally is prescribed when the release rate is
largely controlled by the solubility limits of the
contaminant. The mass-flux type boundary typically
is used when a leaching rate is known or estimated.
Specialized source boundaries also have been
formulated which allow for radioactively decay in the
source. The ability of the code to treat source decay
may not be important if the parents and daughters have
a relatively long half-life compared to the expected
time to travel to the nearest receptor.
Contaminant transport should not be analyzed until
after the ground-water flow model has been calibrated
(Section 4.2.3). As previously mentioned in assigning
initial and boundary conditions to problems of ground-
water flow, the solution of a contaminant transport
problem is not unique unless the initial and boundary
conditions associated with the governing transport
equation are given. Generally, the initial
concentration is specified for each node in the flow
domain at some initial time, t= 0. This results in a
concentration distribution that forms the basis of the
initial conditions. In addition, boundary conditions
must also be specified at all times. Three types of
boundary conditions are commonly encountered in
practice: (1) specified value of concentration; (2) zero
normal concentration gradient; and (3) specified mass
flux of solute. Table 4-4 briefly describes the three
boundary conditions and gives examples.
In simulating areal transport, it also is common to
assume a simple geometry for a contaminant' s source
(or sink). As in the areal simulation of ground-water
flow, points are used to represent individual injection
and pumping wells and waste disposal areas, such as
landfills or recharge ponds. Lines are used to
represent rivers, creeks, and leaking sewer or
pipelines. Again, it is emphasized that point and line
representations of sources are justified as long as
attention is focused on contamination over an areal
scale that is much larger than the area of the sources
(greater than 10 or 100 times). If local information
very near the sources is required, then the source
geometry must be described more accurately as part of
the flow region under consideration. Also, when
analytical methods such as those given in Appendix B
are used to simulate the transport problems, one of the
following assumptions often is made about the source
input: (1) the assumption of constant concentration
during a continuous injection period or during a finite
injection period; (2) the assumption of constant
injection rate during a continuous injection period; or
(3) the assumption of instantaneous injection of a slug
of contaminant. The validity of these assumptions
certainly depends on field conditions.
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Table 4-4. Boundary Conditions of Solute Transport Equations
Type
Description and Examples
I. Specified concentration
Value of concentration is specified along the boundary. Typical examples
include: (1) specified concentration condition on the boundary in direct
hydraulic contact with a surface water body that is recharging into the aquifer
system; (2) zero concentration condition on the boundary located at a great
distance from the contaminant source; and (3) specified concentration condition
at injection wells.
II. Zero concentration
gradient, normal to
the boundary
This type of boundary condition can be expressed as dc/dn = 0. Typical
examples include: (1) zero normal concentration gradient on impervious
boundaries; and (2) zero normal concentration gradient on outflow boundaries
(e.g., river and spring outlets, drains, and pumped wells) where the
contaminant leaves the aquifer system.
III. Specified total mass
flux of contaminant
normal to the
boundary
Total (advective and dispersive) flux is specified on the boundary. This
boundary condition is usually expressed as -(Dij5c/5Xj)ni = qn(c(-c), where n; is
the unit vector in the direction outward and normal to the boundary, j is the
index for the principal axis, and c is the concentration. Typical examples
include: (1) specified mass flux of contaminant at injection wells (in this case,
qn corresponds to the volumetric fluid injection rate per unit area of the aquifer
(L/T), and c" corresponds to concentration of injected fluid); and (2) specified
mass flux of contaminant at the boundary receiving influx of contaminant from
sources such as landfills and disposal ditches.
Initial Conditions
Initial conditions are defined as values of contaminant
concentrations initially assumed to be present in the
area. Non-uniform initial values (i.e., contaminant
concentrations) are routinely used in the contaminant
transport analysis to depict the geometry and varying
concentrations of contaminant within the plume, as
well as to define the concentrations leaching from the
contaminant source. The ability of a code to allow
non-uniform initial conditions is essential to fully
describing and simulating the contaminant plume(s).
4.2.2.3 Specification of Time Steps
Transient simulations of flow and/or transport require
the use of time steps. There is a direct relationship
between numerical accuracy and stability, grid
density, and time-step size. The time-step size should
be selected to ensure that the Courant criterion is less
than or equal to one as shown below:
C =
vat
Ax
where:
Cr =
A*1 =
V
A*
Courant criterion
Time step interval
= Darcy velocity
= Grid spacing
That is, the time step should be selected so that it
would take longer than the specified time to move the
distance of the grid block.
4.2.2.4 Specifying Parameter Values in the Model
Table 2-1 shows the data typically required for
modeling ground-water flow and transport, which can
be obtained from previous and ongoing field studies.
Data input into the numerical model is a painstaking
process; therefore, to identify where a significant
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effort should be made to ensure accuracy, the
uncertainties inherent in the data must be considered.
Such uncertainties will give an early indication of
which data are most likely candidates for modification.
For example, in all likelihood, the assumed recharge
will be changed considerably from initial
approximations when calibrating the model.
Therefore, it would make sense to assign uniform
values, rather than to estimate zones of recharge that
most probably will be changed later. Modeling
uncertainties include:
Measurements of flow and transport usually are
taken at only a few sampling points. To obtain a
complete picture, it is necessary to interpolate
between these points, or make inferences from
point data to apply to conditions over a larger
area.
There may be an inherent difference between the
scale on which the processes are mathematically
described and the scale at which the data are
obtained. For example, laboratory measurements
may exhibit scale-dependence when extrapolated
to a field site.
Data are never complete; in particular, those data
pertaining to values of dispersivities and
contaminant input rates usually are not available.
Therefore, the modeling discussion should include
a data limitation section. These gaps will have to
be filled using estimates made during the
calibration and sensitivity analyses.
The various methods for collecting samples and
measuring parameters all have some error or
uncertainty. Typical sources of error include an
improper type of test, poor specification of test
procedures, poor instrumentation, incorrect
measurement, and incorrect interpretation of the
results. Hence, the reliability of data obtained
from various sources must be weighted.
4.2.3 Calibration of the Model
Traditionally, the term "model calibration" is used to
refer to the trial-and-error adjustment of parameters of
the ground-water system by comparing the model' s
output (calculated values of hydraulic head or
concentration) and the measured output (observed
values of hydraulic head or concentration). In
essence, such a calibration procedure involves the
following routines: (1) operating the model, using
initial estimates of the values of parameters; (2)
history-matching or comparing computed and observed
values of hydraulic head or concentration; and (3)
adjusting the values of the parameters and repeating
the simulation.
Calibration of the model is aimed at demonstrating
that it can produce realistic, and to a certain extent,
accurate and reliable predictions. The model is
calibrated by determining a set of parameters,
boundary conditions, and hydraulic stresses that
generate simulated potentiometric surfaces and fluxes
that match field-measured values to within an
acceptable range of errors.
The end result of the process of model calibration is
an optimal set of values for parameters that minimize
the discrepancy between the model's output and the
observed data. Several major causes of discrepancy
should be recognized; they are listed below, with
general comments on actions that can be taken to
rectify the problem.
1. Poor estimate of values for flow or transport
parameters, or incorrect assessment of initial and
boundary conditions of the ground-water system.
In this case, the problem may be corrected by
adjusting values of the physical parameters and
rerunning the model.
2. Use of incorrect or inappropriate data on
potentiometric head or concentration in the
history-matching or the comparison of output. A
common pitfall is comparing the potentiometric
head computed for a point located at coordinates
(x,y, z) in a three-dimensional flow field with
data measured from an observation well located
at (x,y) but screened over a significant portion of
the aquifer thickness. Such a comparison is not
valid unless the vertical flow component is
negligible because the observed head represents
the vertically averaged head over the screened
length of the observation well, and not the head
at the point (x,y,z).
Such a discrepancy between the model's result and
observed head can be avoided by carefully checking
and interpreting the data.
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3. Use of inadequate spatial and temporal
discretizations. As pointed out earlier, it is
important for the user of a numerical model to
select grid and time steps that are sufficiently
refined to give acceptable accuracy.
4. Use of an inappropriate conceptual model. In
some instances, lack of information about some
features of the ground-water system may have led
to the use of an oversimplified or a wrong type of
conceptual model, or to a poor definition of the
flow and transport problems. Usually when this
happens, it is necessary to return to the early
stage of the simulation process and formulate a
new conceptual model, or alter the existing
model. Alternatively, additional data may be
acquired that better define the flow and transport
problems.
For a cost-effective calibration of the model, a
systematic approach should be taken to the problem of
identifying parameters. In general, a checklist should
be kept of those parameters that are being varied and
those that are being held constant. Various
simplifying assumptions made in the conceptual model
and formulation of the problem should be noted,
together with the levels of uncertainties associated
with all critical input parameters.
The iterative process of matching calculated values
with observed (historical) data by adjusting the
model's input can be a manual trial-and-error
procedure or can be automated. The calibration
process, also known as history-matching, is closely
related to estimating parameters. This process might
result in the refinement of initial estimates of aquifer
properties (parameters), the establishment of the
location of the boundaries (areal and vertical extent of
aquifer), and the determination of flow and transport
conditions at the boundaries. Trial-and-error
calibration is a highly subjective, intuitive procedure.
As the quantity and quality of data are often limited,
no unique set of parameters results, leaving the
modeler with a subjective choice. For example, the
simulated potentiometric surface could be raised either
by increasing areal recharge or by increasing the
amount of leakage from a stream.
The success of such a procedure depends very much
on the experience of the analyst. Another type of
technique is based on the use of a formal optimization
procedure (e.g., Gauss-Newton parameter
optimization algorithm), in conjunction with the
ground-water flow model. Such techniques can be
coded into a computer program that performs
automatic history-matching, and calculates the optimal
set of parameter values that minimize a certain
objective function (e.g., the total sum of the square of
the difference between computed and observed head
values at all observation points).
Both the manual and automatic adjustment techniques
may be categorized as the "indirect approach" to the
problem of identifying parameters (See FRE79,
pp. 357-359). This statement implies that calibration
can be carried out via a "direct approach." Such an
approach requires the inverse formulation of the
problem of the ground-water flow. In other words,
the flow problem is set up in such a way that the
piezometric head and well discharges are known, but
the transmissivity, storage, and recharge parameters
are unknown. When posed in this manner, the
identification problem is referred to specifically as the
"inverse problem." Several techniques recently were
developed to solve the inverse problem.
Automation calibration is based on the use of
prescribed algorithms, which are completed when
preset matching criteria are met. Because of the
formal approach taken in adjusting input to the model,
automatic procedures are less subjective than trial-and-
error procedures. However, automatic procedures
effectively eliminate the judgment of the modeler, and
because the calibrated solution is not unique it may not
be the best fit to the overall conceptual model.
However, if numerous calibrations are required, it
would make sense to use automated calibration tools
to obtain probability distributions.
In contrast to the calibration of the ground-water flow
model, calibration of a contaminant transport model
usually is more subjective using the manual procedure
of history-matching and trial-and-error adjustments of
the parameter. There are several reasons for this; the
most important is that data on concentration are
usually completely lacking, or insufficient to permit an
accurate calibration. Another important reason is that
the transport equation contains more parameters and is
more complex than the ground-water flow equation.
Thus, it is more difficult to identify the parameters by
an automatic estimation technique or by the direct
(inverse formulation) approach.
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Calibration can be performed to steady-state or
transient data sets. Although most flow model
calibration exercises involve steady-state data, in some
hydrogeologic settings, assumption of steady-state
conditions may be inappropriate due to large
fluctuations in the water table or boundary conditions.
In this case, the model may be calibrated against long-
er short-term trends in water levels, stream and lake
elevations, and possibly, system responses resulting
from imposed stresses such as pumping wells. A
transient flow calibration is necessary to calibrate
values of storage parameters, which are needed if
transient flow is to be modeled.
The results of the model calibration need to be
evaluated both qualitatively and quantitatively. At
present, no established protocol exists for determining
whether a model has been satisfactorily calibrated.
However, there are several common ways of reporting
the calibration results. The simplest way is to list the
measured and simulated heads together with their
differences and some average of the differences.
Comparison between contour maps of measured heads
versus contoured maps of simulated heads provides a
visual means of assessing the "goodness of fit" and
gives some idea of the spatial distribution of error in
the calibration.
The simulated heads and concentrations will have
some degree of error arising from how the model was
discretized in both space and time. In some instances
(e.g., if large mass balance errors are occurring), it
may be necessary to perform a grid-convergence test
to determine if the grid spacings in the model are fine
enough; this can be time-consuming and painstaking
unless a good pre-processor is available. The ready
accessibility of high-speed computers may warrant
over-discretizing the model grid at the beginning of
the project, rather than having to redesign the grid
during calibration. If a transient calibration is to be
performed, the appropriateness of the selected time
steps should be checked by comparing the model' s
results against an identical simulation with the time
steps set at very small intervals. If the results with the
large time steps do not diverge significantly from
those with the smaller steps, the larger ones would
provide a comparable result.
The mass balance error is calculated by comparing all
of the water entering the modeled system with all of
the water exiting the domain. Transient simulations
also will consider water going into and out of storage.
The mass balance error should typically be less than
one percent.
4.2.4 Uncertainty and Sensitivity Analyses
After the model has been satisfactorily calibrated,
sensitivity analyses should be performed to determine
the sensitivity of the model' s output to variations (or
uncertainties) in physical parameters. For instance,
what would be the effect of a 10 percent error in
recharge values on the potentiometric head and
velocity distributions?
The common practice for carrying out the sensitivity
analysis is to repeat the simulations using a series of
selected values for the physical parameters and to
compare the results with those obtained using the
calibrated values. Usually, the selected values of each
varied parameter are within a range that depends on
the degree of associated uncertainty. Output from the
sensitivity runs can be expressed in actual units or in
a dimensionless form. Using dimensionless variables
often allows systematic conclusions to be drawn from
the sensitivity study. In particular, if the model is an
analytical model, important dimensionless parameters
can often be easily identified by carefully examining
the analytical solution.
Sensitivity analyses identify the main contributors to
the observed variation in the results. These techniques
typically are applied iteratively. The first iteration can
include rather general assumptions leading to
preliminary results that help focus the subsequent
iterations.
Uncertainty in input parameters used in predictive
models may result from several sources, including
incomplete data, intrinsic spatial variability of a
property, uncertainties in measurements, and
uncertainties resulting from differences in scale
between data acquisition and model application.
However, uncertainty in input parameters is not the
only potential source of uncertainty in modeling
ground-water flow and contaminant modeling;
additional uncertainty may enter the analysis through
the choice of conceptual models used to represent the
system.
The following definitions will be useful in this
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discussion of uncertainty and sensitivity analyses.
Conceptual Model: A set of qualitative assumptions
used to describe a system or subsystem for a given
purpose. At a minimum, these assumptions concern
the geometry and dimensions of the system, initial
and boundary conditions, time-dependence, and the
type of physical and chemical processes. The
assumptions should be consistent with one another
and with existing information within the context of
the purpose.
Alternative Conceptual Models: Alternative sets of
assumptions that describe the same system for the
same purpose, where each set of assumptions is
consistent with the existing information.
Conceptual Model Uncertainty: The lack of
knowledge about the system resulting from the
limited information available to support or refute
alternative conceptual models.
Uncertainty also may exist in the computational
models used for quantitative analyses, based on the
chosen conceptual models. In this discussion,
computational models refer to mathematical models
used to represent the physical processes, numerical
models used to solve the mathematical models, and
computer codes used to implement the solution.
The selection of scenarios to be analyzed also may
introduce uncertainty into the estimated performance.
Still more uncertainty may exist in the completeness of
the scenarios considered, in the way in which
computational results are aggregated to represent the
consequences of scenarios, and in the probabilities
associated with their occurrence.
Sensitivity analyses identify the main contributors to
the observed variation in the results. These techniques
typically are applied iteratively. The first iteration can
include rather general assumptions, leading to
preliminary results that help focus these techniques in
subsequent iterations. In this manner, the resources
and techniques required to reduce the uncertainty can
be directed at the areas of the modeling study where
the benefits of understanding uncertainty and reducing
it (where possible) are greater. However, sensitivity
analyses alone will rarely identify a flawed conceptual
model. For example, the failure to identify and
include a fault(s) in the conceptual model would lead
to an analysis that would not account for preferential
pathways that potentially could result in higher than
predicted risks.
Modeling the behavior of a hydrogeologic system
necessarily will be uncertain because knowledge about
its real behavior is limited. Many of the parameters
used as inputs to a model of the system are obtained
only by collecting data. Investigators knowledgeable
about the data they collect make a finite number of
observations, choosing which parameters to measure
and how, where, and when to measure them.
However, the collection process itself can introduce
uncertainty through errors in measurement, the
system's inherent randomness, and limited sampling of
the variable physical, chemical, and biological
properties of the system. The professional judgment
of an expert in the area of investigation often enters
into the scientific process. For example, selecting
methods to collect data, interpreting data, developing
conceptual models, and selecting parameters all
require professional analysis and judgment. The
analyst' s final data set is based on available data, use
of the parameter in the computational model, behavior
of analogous systems, and the analyst's own expert
judgment.
Uncertainties arising from the numerical solutions of
a mathematical model are resolved when verifying
(checking for numerical accuracy) the computer
programs. Uncertainty resulting from the scenarios
selected for modeling is best addressed by a
systematic, thorough examination of a scenario's
possible components, based on probability,
consequence, physical reasonableness, and regulatory
guidance, and by assigning probability through
techniques used for evaluation or estimation.
Monte Carlo techniques may be used for uncertainty
and sensitivity analyses. Uncertainty analyses evaluate
uncertainty in performance estimates that result both
from the existence of alternative conceptual models
and from imprecise knowledge input variables.
Sensitivity analyses determine the contribution of
individual input variables to the uncertainty in model
predictions.
Monte Carlo analyses involve five steps: (1) selection
of variables to be examined and the ranges and
distributions of their values; (2) generation of the
samples to be analyzed; (3) propagation of the samples
through the analysis; (4) uncertainty analysis; and (5)
sensitivity analysis.
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4.3 Predictive Simulations
The final stage of the process is to perform predictive
simulations using the optimal set of parameter values
obtained from the model calibration. In general, these
simulations address specific issues of the existing
problem of ground-water contamination and provide
guidance for policy decisions. Each simulation usually
corresponds to one specified set of data pertaining to
natural boundary conditions, pumping operation for
the ground-water reservoir, or proposed remedial
measures for the contamination. Typical objectives of
the predictive simulation study are outlined below,
alone with pertinent comments.
1. To learn more about the existing contamination
and to predict future behavior of the ground-
water system under natural conditions. Among
the questions that may be addressed are the
following: Where is the contaminant plume
located presently? In which direction is the
plume moving? Will the contamination at the site
pose serious danger to the public if no action is
taken? Do the contaminant concentrations pose
unacceptable risks? If the risks are unacceptable,
how extensive must the clean-up be to reduce the
concentration of the contaminant to an acceptable
level?
2. To evaluate and compare alternative remedial
schemes for the existing contamination. Typical
remedial schemes that may be considered are (1)
hydraulic barriers (e.g., pumping and recharge
wells to flush the contaminant out of the aquifer,
and (2) subsurface barriers to inhibit the
contaminant from leaving the site. The
predictive simulations are also aimed at the level
of risk reduction offered by each alternative
measure. Results from the comparative study are
useful in the selection and implementation of a
suitable remedial scheme for the site. These
decisions usually are made in conjunction with
economic considerations. In some instances, the
predictive simulations can be directly
incorporated into an economic analysis, allowing
the most cost-effective remedy to be selected.
3. To predict the responses of the ground-water
system to various management alternatives
including, for example, different pumping or
recharge operations that may be applied to the
wells existing at the site or groups of wells in the
areas surrounding the site. The following are
typical questions: How will different well
operations affect the contaminant plume? Can
the existing contamination problem be contained
to the site and recovery efforts maximized by
modeling review criteria for selecting a proper
pumping schedule?
Modeling review criteria for the second two topics
have been considered in previous chapters. The first
topic is related to the baseline risk assessment,
discussed in the next section.
4.4 Baseline Risk Assessment
The Baseline Risk Assessment typically accomplishes
the following three objectives:
Assesses the magnitude and sources of current
and potential future risks to humans and the
environment.
Assists in the scoping ongoing site
characterization.
Identifies contaminants of potential concern and
assumptions of exposure for developing risk-
based preliminary remediation goals (PRGs).
As the remedial investigation/feasibility study
proceeds, action levels will evolve from the PRGs
which ultimately will become part of the final
objectives for remedial action. These action levels
will entail consideration of applicable or relevant and
appropriate requirements (ARARs) for site-wide
baseline risk assessments and, potentially, operable
unit-specific risk assessments.
Although risk assessments generally include several
receptors (e.g., future resident farmers, plant
workers), the following discussion is targeted at
pathways and receptors related to ground water.
Risk-based PRGs for ground water are frequently
developed in accordance with the Human Health
Evaluation Manual, Part B (EPA91). This approach
is a first-tier type of analysis and typically is very
conservative. The methodologies outlined in
Appendix B for predicting radionuclide transport rates
and concentrations are designed for conservative
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analyses, yet they also are designed to be more
realistic than the HHEM Part B approach.
In many cases, estimating flow and transport through
the unsaturated zone is an integral component of the
risk assessment, particularly if the compliance point is
relatively near the contaminant source. In these
instances, the release rates, concentrations and hold-up
times within the unsaturated zone will influence
receptor concentrations far more than flow and
transport in the saturated zone.
Under these circumstances, a practical approach would
be to use one of the many risk-based computer models
that emphasize flow and transport in the unsaturated
zone. However, if flow and transport processes in the
saturated zone are deemed very important to the
analysis, or if the receptor is located off the centerline
of the plume or far from the source, it may be
worthwhile to use a risk-based type code or
calculations from Appendix B to give a transient
source-type boundary. This can then be used in a
more complex numerical model to simulate the flow
and transport in the saturated zone more accurately.
One of the primary differences of the site modeling
with risk-based codes is that the model is not
typically calibrated. In general, this is not a problem
because the required data from the unsaturated zone
rarely are collected during site characterization.
Therefore, evaluation of the parameters during the
sensitivity analysis is crucial.
The conceptual model sub-components of the risk-
based codes related to ground-water flow and transport
processes consist of the following:
Infiltration
Source Release Rate
Source and Leach Strength
Fate and Transport in the Unsaturated Zone
Fate and Transport through the Saturated Zone
4.5 Exposure EstimationGround Water
Appendix B describes methods for estimating ground-
water concentration at the point of receptor exposure.
With these estimated concentrations, the assessor can
estimate exposure based on the equations and
parameter values presented in the Exposure Factors
Handbook (EPA89).
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CHAPTER 5 - SUMMARY AND CONCLUSIONS
In this chapter, the modeling evaluation criteria and a
checklist containing the major steps within these
procedures are reviewed. With this checklist (Table
5-1), a person analyzing a specific project can identify
potential problems in the model application.
MODELING OBJECTIVES
REQUIREMENTS
AND DATA
The purpose and scope of the modeling exercise
should be clearly indicated.
The purpose of the modeling should be kept in
mind when reviewing the modeling report. The
reviewer should determine whether the analyses
performed are consistent with the purpose of the
project. Common problems are overkill (the
modeling analyses are much too complex for the
purpose of the project) and over-simplification
(the modeling analyses are too simplistic to
achieve the project objectives).
The modeling objectives should be identified and
related to the decision-making needs.
In the Summary and Conclusions of the model
application report, each of these objectives should
be discussed separately in the context of how the
modeling was used to meet the objective and the
degree to which the objective was met.
The data required to construct a conceptual model
should be described, and the relevance of the data
to ground-water flow and contaminant transport
should be discussed.
The data should be related to the processes being
simulated, (e.g., groundwater flow, contaminant
transport, variably-saturated flow and transport,
etc.). It is important to ascertain that no
important processes are overlooked and that all
data types required to simulate a particular
process have been identified. In addition to data
related to the physical system, a detailed waste
disposal history should be provided. The latter is
a key element in determining the source term(s)
for the contaminant transport modeling.
The source of the data should be presented.
Data should be categorized based upon source
type: (1) site-specific data collected in the field,
(2) data obtained from the scientific literature, and
(3) data values estimated through model
calibration. Data obtained from literature should
be thoroughly cited and should be representative
of the same geologic and hydrologic conditions
found at the site. Data values obtained through
calibration should also be consistent with
anticipated ranges of values (see the model
calibration section).
The uncertainties associated with the data should
be discussed.
Are some field collection methods better than
others? How reliable are literature values? A
probable range and distribution in which the
parameters will fall should be assigned, prior to
the modeling analysis.
The general sensitivity of the data to the
determination of ground-water flow and
contaminant transport calculations should be
discussed.
This discussion should enable the field
characterization program to be more focused. For
example, bulk density is used for the transport
calculations although the modeling results are
typically insensitive to their values. Therefore,
time and resources would be better spent obtaining
site-specific distribution coefficients which may be
critical to the analysis.
Limitations and weaknesses in the data base
should be presented as well as plans to enhance
the data base.
Data gaps should be reviewed with the modeling
objectives in mind. For example, scoping
calculations may be performed with relatively
little site-specific data. Detailed simulation of
remedial measures, however, would require
numerous field measurements of key hydraulic
5-1
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and geochemical parameters (e.g., hydraulic
conductivity, storage coefficient, distribution
coefficients, etc.).
Recommendations should be presented, detailing
additional data needed to increase confidence in
the modeling results.
CONCEPTUAL MODEL DEVELOPMENT
The physical and hydrogeologic frameworks of
the system should be described in detail.
The physical and hydrogeologic framework
includes lithologic contacts, facies changes,
discrete features such as fractures, and spatial
variations of geologic units and their hydraulic
properties. The thickness, top elevation, and
bottom elevation should be described in detail for
areally continuous hydro stratigraphic units
(aquifers and aquitards). The rationale for the
variability of the properties should be explained
(e.g., depositional history).
The geometry of the system should be presented
in three dimensions with a rationale for possible
simplification. For example, the analysis of the
unsaturated zone may be reduced toone or two
dimensions. The saturated ground-water system
may also be simplified to two-dimensions in plan
view if vertical gradients are negligible.
The boundaries of the system should be described
in a water budget analysis.
The primary components of the water budget
include recharge, evapotranspiration, runoff,
pumping rates, and flow to other sources and
sinks including rivers and lakes. The
methodology for determining individual
components of the water budget should also be
included. Boundaries of the system should be
identified based upon regional hydraulic features.
Aquifer boundaries are seldom constrained to the
immediate vicinity of the site and may extend far
beyond the area of interest. It is important to
characterize these regional features, however.
The contaminant source term should be described
in detail.
Source terms should be described in terms of
geometry (in three dimensions), radionuclide
concentrations leached from the source, timing of
the release, and the release mechanism. The site
waste disposal history described in the first section
of the modeling report should be helpful in
determining this information.
The conceptual model should be consistent with
the field data.
One of the fundamental problems in modeling is
a poor conceptual model. Synthesizing the field
data into a coherent picture of the relevant
physical and chemical processes is critical to the
subsequent modeling analyses. Errors in the
conceptual model will propagate throughout the
modeling. It is important to review the conceptual
model and the raw data to determine whether
there are significant errors at this early stage in
the project.
The rationale for any simplifications made to the
conceptual model should be presented.
Examples of simplifications include (1) modeling
ground-water flow at steady-state conditions, (2)
simulating the unsaturated zone in one or two
dimensions, and (3) approximating the source
term at a constant concentration. Each
simplification should be reviewed for consistency
with the conceptual model of the site, the
availability of data, and the potential impact on
the accuracy of the modeling results.
Uncertainties in the conceptual model should be
presented and related to earlier discussions of data
limitations and uncertainties.
Uncertainties can be related to the variability in
field data or interpretations or simplifying
assumptions required to evaluate the field data.
Uncertainties can be evaluated through a
sensitivity analysis in subsequent model phases,
but should be discussed in the conceptual model
portion of the report.
Are sufficient data available to meet the modeling
objectives?
Data in this context refer to site-specific data.
5-2
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Data can always be obtained from the scientific
literature; however, as the objectives of the modeling
become more detailed, site-specific data requirements
increase. It is important to relate the quantity and
quality of site-specific parameter values to the types of
analyses performed.
Have database deficiencies been clearly identified
and modeling implications discussed?
Figures and Tables
The following are illustrations and tables that should
be included in the conceptual modeling report. Some
figures may not be required; however, justification
should be given if figures and tables are omitted.
Map showing location of study area.
Maps and cross sections showing the thickness of
the unsaturated zone.
Maps, hydrographs, and/or tables of water-budget
information, including evapotranspiration, runoff,
ground-water recharge, ground-water pumping,
and gains/losses between ground water and
surface water.
Maps, cross sections, or tables indicating
transportparameters, including effective porosity,
dispersion coefficients, and distribution
coefficients.
Areal and cross sectional isoconcentration maps of
primary contaminants in soil and ground water.
Time-series graphs of contaminant concentrations
measured over time at monitoring wells or surface
water locations.
Relevant source-term inventory information.
MODEL APPLICATION
Geologic map and cross sections indicating the
areal and vertical extent of the system.
Topographic map indicating surface water bodies.
Contour maps showing the tops and/or bottoms of
the aquifers and confining units.
Isopach maps of hydrostratigraphic units.
Maps showing extent and thicknesses of stream
and lake sediments.
Maps indicating any discrete features (e.g.,
faults).
Maps and cross sections showing the unsaturated
zone properties.
Potentiometric surface maps of aquifer(s) showing
hydraulic boundaries.
Maps, cross sections, or tables showing storage
properties of the aquifers and confining units.
Maps, cross sections, or tables showing hydraulic
conductivity of the aquifers, confining units, and
stream and lake sediments.
Scoping Analysis
The results of any scoping analyses that are
performed to support the modeling should be
presented. These results should be able to support
the approach taken for more complex modeling.
Even though scoping analyses represent sim-
plified modeling approaches whereby model
parameters are chosen to be conservative. Review
of the scoping calculations should concentrate on
whether the chosen parameter values and models
(or equations) are conservative from a regulatory
perspective.
Code Selection
Selection criteria should be clearly presented for
the selected code(s).
Criteria used in selecting computer codes
generally include (1) degree of code testing and
documentation, (2) ease of use, (3) whether the
code is proprietary or public, (4) physical and
chemical processes to be solved, and (5)
application history. Even if only one computer
code is used in the project, a series of codes
should be presented as possible candidates and
rationale should be presented to justify the
5-3
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selection of the chosen code.
inaccuracies or numerical problems.
The general features of the code should be
discussed.
Code features include whether the code is a
proprietary version of a publicly available code,
solution methodologies for the flow and transport
equations, hardware requirements, degree of code
testing, and availability of source code and
documentation.
The code assumptions and limitations should be
described.
Of particular interest are those assumptions
pertaining to the conceptual model. These would
include code dimensionality, ability to simulate
heterogeneities, and flow and transport through
the unsaturated zone. The code should be capable
of simulating all pertinent processes identified in
the site conceptual model.
The basis for regulatory acceptance should be
discussed.
Regulatory acceptance criteria may include a
history of use, particularly for applications in a
similar regulatory context and degree of code
testing. The code should also be well documented
and the source code should be available for
inspection. Code testing should ideally follow the
three-level procedures advocated by the
International Ground Water Modeling Center
(IGWMC) or those used in the various
international code testing studies (INTRACOIN,
INTRAVAL, etc.).
Documentation on the source code should be
included, with an executable version of the code
and data sets relevant to the problem. This allows
the reviewer to independently verify the results
presented in the modeling report and to review
details from the individual model output files.
Layering and Gridding
The rationale for the selection of the grid spacing,
number of model layers, and the resulting number
of nodes and elements should be given.
The grid should be evaluated in terms of potential
Common features to evaluate include: (1) the
model boundaries should not be too close to the
area of active remediation (wells etc.), (2) model
nodes should coincide with pumping centers
otherwise the effects of these stresses will be
offset, (3) the grid should be alligned with the
principal axes of hydraulic conductivity, (4) in
finite-difference models, the grid spacing between
adjacent cells should not vary by more than a
factor of 1.5, (5) in transport modeling, the Peclet
number should not be greater than 2, (6) the
maximum aspect ratio of the grid should not be
greater than 100:1.
Other aspects of grid design are more subjective.
For example, the degree of discretization (number
of rows and columns) should be appropriate for
the problem being solved. Areas of sharp contrast
in hydraulic properties should be more finely
discretized. Model layering should be consistent
with the magnitude of vertical gradients and the
degree of vertical heterogeneity. If matrix
diffusion is important, confining units should be
discretized into multiple layers.
Figures and Tables
The grid should be presented as an overlay of a
map of the area to be modeled.
A vertical cross section of the modeled area which
displays the vertical layering of the model with
respect to its hydrogeology should be included.
Horizontal and vertical grid coordinates and
elevations should be identified clearly on all
figures.
Boundary and Initial Conditions
Selection of all boundaries and initial conditions
should be justified.
The justification would involve a discussion of
how a natural feature was simulated (e.g., a river
or ground-water divide) including any assumptions
related to the choice of bounary type and location.
Of particular concern are boundaries that do not
coincide with natural features but are somewhat
5-4
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arbitrary. Careful scrutiny should be given to such
artificial boundaries. Ideally, a sensitivity analysis
would be performed on these arbitrary boundaries.
Descriptions of model boundaries should include
whether the boundaries are transient or steady-
state. For transient boundaries, the report should
discuss how the boundary condition changes with
time and how these changes were determined. In
the natural system, boundaries may shift with
time, and the effect that these positional changes
may have on the results of modeling should be
considered.
Boundaries should also be chosen to ensure that
future simulations will not be adversely effected
by pumping wells or other features that stress the
system. Justification of the chosen boundaries
should address this potential problem.
Uncertainty surrounding boundaries and initial
conditions should be discussed.
There is usually significant uncertainty in the
selection of boundary conditions. These
uncertainties include the type of boundary chosen
to simulate a natural feature, the position of the
boundary, the value of head or concentration at
the boundary, the assignment of conductance
properties to the boundary, and the transient
response of the boundary. All of these factors
should be addressed by the model report. A
sensitivity analysis on boundary conditions should
also be included.
The following specific examples may be useful in
reviewing the model boundary conditions:
Under steady-state conditions, the areal recharge
should not exceed the saturated hydraulic
conductivity of the surficial soil through which it
must travel; otherwise ponding would occur.
Potentiometric lines on streams that are gaining
water should point upstream, whereas the lines
should point downstream along losing streams.
Contaminant source release rates should be
discussed.
Ephemeral streams generally should not be
modeled as constant head boundaries.
Streams are frequently modeled as ground-water
divides, that is, all ground-water flowing towards
the stream is assumed to be captured by the
stream. The modeler should justify this
assumption, as not all streams fully penetrate the
aquifer.
Surface-water/ground-water interactions should be
discussed.
Recharge and evapotranspiration are difficult to
determine, and therefore, recharge as a flux
boundary is often used as a calibration parameter.
The method for determining recharge should be
presented.
Interpretation and extrapolation methods (e.g.,
Kriging) should be described.
Boundaries between two types of porous media
should always coincide with element boundaries.
Figures and Tables
The report should clearly identify assigned
boundaries and initial conditions in figures and
tables. A typical figure would be a plot of the
model grid for each layer clearly illustrating
boundary cells. Each type of boundary (e.g.,
constant head, constant flux, and head-dependent
flux) should be labeled using a different symbol or
color. For transient boundaries, multiple figures
representing different times may be used or tables
of values may be more appropriate.
The boundary condition sensitivity analysis should
be illustrated using figures and tables.
Time Steps
The Courant criterion outlined in Section 4.2.2.3
should be satisfied for transport simulations.
Even for flow models, the time steps should be
small at the start of the simulation and gradually
increase. Time step size should be decreased
when major changes in stresses are simulated.
Calibration
The calibration process should be described in
5-5
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detail, including any assumptions and limitations.
Proper justification should be given if the model
was not calibrated. In some cases, such as a
screening analysis, calibration may not be
required. However, an uncalibrated model is not
as reliable a decision-making tool as a well-
calibrated model.
Documentation of the calibration process should
include flow diagrams illustrating the approach
that was taken to calibrate the model. The
objectives or criteria used to calibrate heads,
flows, and radionuclide concentrations should be
presented. The method of calibration (inverse
model, trial-and-error, or a combination of both)
should be documented. If special calibration
software is used (e.g. an inverse model), it should
be documented and described under the code
selection section.
If both steady-state and transient calibrations are
performed, their similarities and differences
within the results should be discussed. The
rationale and selection of time steps for the
transient calibration should be discussed.
The sources and magnitudes of errors should be
described.
All calibrated models have errors. These errors
are often called residuals and represent the
difference between a model-computed value and
a value measured in the field (usually head or
concentration). The errors should be described in
detail in the report. The review should
concentrate on potential effects on the predictive
simulations which will be performed later (e.g.,
risk assessment).
Modifications to the parameter values, boundary
conditions, and imposed hydraulic stresses should
be discussed in detail.
The calibration process is an exercise in parameter
estimation where key model parameter and
boundary values are adjusted within reasonable
bounds to achieve the calibration objectives. The
review should focus on the response of the
modeled system to the altered values and the
rationale for the parameter changes made during
the calibration. The calibrated parameter values
should be compared with the initial range of these
parameters. Particular emphasis should be placed
on parameters that fall outside their originally
estimated range. The final values should be
compared to those identified in the conceptual
model.
The rationale for the convergence criterion for the
heads and concentrations should be presented, in
addition to a discussion of the overall mass
balance results.
Problems that arose due to failure of the code to
converge or numerical instabilities should be
described. The mass-balance results should be
discussed in relation to any convergence
problems. Overall, the water balance should be in
error by less than one percent.
The user-specified error or convergence criterion
will result in a level of accuracy that is one to two
orders of magnitude greater than the criterion.
This difference should be evaluated with respect
to the desired level of accuracy.
The calibrated model should be a good match with
the conceptual model, such as flow directions and
parameter values.
Specific examples to look for in reviewing the
calibration include:
The calibrated parameters, especially hydraulic
conductivity, should not appear patchworked.
Unless there is evidence indicating that hydraulic
conductivity values change substantially from one
grid block to the next, it should be assumed that
large percentages of the modeled area are
relatively homogeneous.
Areal recharge should be uniform unless there is
sufficient justification to vary the recharge rates
locally.
Well logs and aquifer stress test data should be
reviewed to ensure that the hydraulic
conductivities assigned to that area are
compatible.
The volume of water entering or exiting local
streams, lakes, or rivers should be consistent with
the field data.
5-6
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It should be kept in mind that head and
concentration values computed at a node are
representative of an area rather than a point.
Therefore, if drawdown values during aquifer
tests are used as calibration points, even a well-
calibrated model would not match the field data
exactly because data collected from wells
represent points in space.
Vertical gradients within an aquifer in which the
well is not fully penetrating should be considered
when the model is calibrated.
Figures and Tables
Areal and cross-sectional diagrams of the error
(residual) between computed and measured
hydraulic head and radionuclide concentrations
should be presented. The errors should not show
significant spatial bias. For example, if all of the
targets in the western half of the model are
computed too high and those in the east are too
low, there is a bias in the calibration. In this
example, the gradients would be inaccurate.
A list and a figure indicating the final calibrated
values for parameters and boundary conditions
should be included.
The match to the calibration targets should be
shown in figures as well as in tables. Sections
within the model should be outlined and discussed
according to their "goodness of fit" to the
calibration targets.
Particle tracks or calibrated plumes should be
shown in planar and cross-sectional views.
Sensitivity Analysis
The approach undertaken for the sensitivity
analysis should be described in detail.
There are a variety of ways to perform a
sensitivity analysis. According to ASTM, the
sensitivity analysis should evaluate both the
calibration and the predictions (if any).
Sensitivity of predictions to model parameters and
boundary conditions is important in evaluating the
degree of uncertainty in the model. Both
parameter values (e.g. hydraulic conductivity) and
boundary conditions should be evaluated in the
sensitivity analysis.
The sensitivity of model calibration quality and
model predictions to variations in parameter
values, including grid spacing, time steps, and
boundary conditions, should be discussed,
emphasizing parameters in which there is a large
degree of uncertainty and the results are very
sensitive.
The rationale for selecting parameters for the
sensitivity analysis and for determining whether
there were sufficient simulations investigating
single or multiple parameters should be presented.
The relevance of the overall uncertainty and
sensitivity with respect to the objectives of the
predictive simulations should be discussed.
Figures and Tables
The results of the sensitivity analysis should be
displayed in a graph or table.
A typical sensitivity graph plots a calibration
statistic (sum of squared residuals for example)
versus a range in parameter values for each
parameter. Multiple parameters may be plotted
on one graph. Model sensitivity coefficients may
be computed as the change in a model-computed
value (head or concentration) divided by the
parameter change. These sensitivity coefficients
may then be contoured to illustrate changes in
model sensitivity through space. Where many
parameters are involved, a table may be presented
to identify the most sensitive parameters.
Modeling to Support Baseline Risk Assessment
The objectives of the risk assessment should be
stated.
The modeling approach, in addition to any
inherent limitations, should be clearly indicated.
The conceptual model should be presented, in
conjunction with the validity of and rationale for
any simplifying assumptions.
The method used to calculate infiltration rates and
other relevant parameters should be included.
5-7
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A discussion of the source term should be
presented, including its dimensions, strength, and
composition.
The means by which release rates and leachate
concentrations are calculated should be described.
The treatment of daughter-ingrowth in the source
term in unsaturated and saturated zones should be
described.
The fate and transport processes active in the
unsaturated and saturated zones should be
presented.
The processes by which the leachate becomes
diluted along the transport path from the source
term to the receptor should be discussed.
If the output from the risk-based code is coupled
to a more sophisticated code, this process should
be described in detail.
The process by which remedial action goals were
determined from the results of the risk assessment
should be discussed.
The methods of calculation outlined in Appendix
B can be used for independent verification of the
results of the risk assessment.
Figures and Tables
An areal and cross-sectional representation of the
conceptual model should be shown, including the
locations of the assumed receptor.
Radionuclide breakthrough concentration plots
should be included for each receptor and
radionuclide of interest.
Selected areal isoconcentration plots should be
given.
Preliminary Remedial Design
The report should follow the guidelines given
earlier in this chapter and include discussions and
similar presentations on developing the conceptual
model, selecting the parameters, designing the
grid, calibrating the model, and carrying out the
sensitivity analyses.
The assumptions and calculational procedures
used to determine the specific assumptions
associated with the remedial design should be
presented, such as the locations of recovery wells
and failure rates for barriers.
In addition to the review of the grid design and
time-stepping schemes previously presented, are
there other relevant processes which may be
important and should have been considered (e.g.
matrix diffusion)?
If a pump-and-treat scenario is modeled, does the
model accurately simulate the rise and fall of the
water table?
If the model also was used for risk assessment,
have conservative assumptions been removed and
the model recalibrated? For instance, a
conservatively low hydraulic conductivity would
yield high well concentrations, which may be
acceptable for the risk assessment, but would
overestimate the capture zones and influent
concentration of the remedial design.
Figures and Tables
In addition to the figures and tables previously
discussed that are relevant to the remedial design
presentation, additional figures such as areal and
cross sectional views of barrier walls, capture
zones, and/or recovery wells should be included.
5-8
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Table 5-1. Major Steps in Modeling Evaluation Procedures
MODELING AND EVALUATION CRITERIA
APPRAISAL
Yes
No
Comments
CHAPTER 2
OBJECTIVES AND DATA REQUIREMENTS
Are the purpose and scope outlined?
Are the objectives consistent with decision-making needs?
Are the objectives satisfactory?
Are a site description and waste disposal history provided?
Are the data requirements for the proposed modeling outlined?
Are the sources of data adequately presented?
Are data uncertainties discussed?
Is the probable sensitivity of the future modeling results presented for the
data?
Are the potential data limitations and weaknesses provided?
Are the plans to resolve data limitations discussed?
CHAPTER 3
CONCEPTUAL MODEL DEVELOPMENT
Is the physical framework discussed in detail?
Both regional and local?
Is the hydrogeologic framework described in detail?
Both regional and local?
Is the nature of the contaminant source term described?
Are the hydraulic boundaries described in detail?
Are data base deficiencies clearly identified and modeling implications
discussed?
Is the conceptual model consistent with the field data?
Are the uncertainties inherent in the conceptual model discussed?
Are the simplifying assumptions outlined?
Are the assumptions justified?
Are the natural boundaries or the aquifer system described?
Are the following figures and/or tables included:
Map showing location of study area.
In some instances tabular representation of the data may be appropriate.
5-9
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Table 5-1 (Continued)
MODELING AND EVALUATION CRITERIA
Geologic map and cross sections indicating the areal and vertical
extent of the system.
Topographic map with the surface water bodies.
Contour maps showing the tops and/or bottoms of the aquifers and
confining units.
Isopach maps of hydrostratigraphic units.
Maps showing extent and thicknesses of stream and lake sediments.
Maps indicating discrete features (e.g., faults), if present.
Maps and cross sections showing the unsaturated zone properties
(e.g., thickness, Ksat).
Potentiometric surface maps of aquifer(s) and hydraulic boundaries.
Maps and cross sections showing storage properties of the aquifers
and confining units. '
Maps and cross sections showing hydraulic conductivity of the
aquifers, confining units and stream and lake sediments.
Maps and hydrographs of water-budget information.
Maps and cross sections indicating transport parameters (e.g., Kd).'
Areal and cross sectional isoconcentration maps of primary
contaminants in soil and ground water.
Time-series graphs of contaminant concentrations.
Relevant source-term inventory information.
APPRAISAL
Yes
No
Comments
CHAPTER 4
MODEL APPLICATION
Section
(4.1) SCOPING ANALYSIS
Are scoping analyses performed?
Do scoping results lead to proposed modeling approach?
(4.2) SITE CHARACTERIZATION MODELING
(4.2.1) Code Selection
Is the rationale for the selection clearly presented for proposed code(s)?
Are the general features of the code(s) presented?
Are the assumptions and limitations of the code(s) presented and
compared to the conceptual model?
5-10
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Table 5-1 (Continued)
MODELING AND EVALUATION CRITERIA
Is the basis for regulatory acceptance presented?
Is the source documentation for the code included?
Is an executable version of the code included?
Is the source code readily available for inspection?
Does the code have a history of use?
Is the code well documented?
Is the code adequately tested?
Are the hardware requirements compatible with those available?
(4.2.2) Model Construction
(4.2.2.1) Layering and Gridding:
Is the domain of the grid large enough so that the boundaries will not
interfere with the results?
Do the nodes fall near pumping centers on existing and potential future
wells and along the boundaries?
Is the grid oriented along the principal axes of hydraulic conductivity?
Is the grid discretized at the scale appropriate for the problem?
Are areas of sharp contrasts (e.g., hydraulic conductivity, concentration,
gradient) more finely discretized?
Is the Peclet number less than 2?
Do adjacent elements vary in size by a distance less than a factor of 1.5?
Are strong vertical gradients within a single aquifer accommodated by
multiple planes or layers of nodes?
If matrix diffusion is important, are the confining units adequately
discretized in the relevant regions of the model?
Is the grid more finely spaced along the longitudinal direction of
simulated contaminant plumes?
Is the aspect ratio less than 100:1?
Are the following figures included:
Grid presented as an overlay of a map of the area to be modeled.
A vertical cross section(s) which displays the vertical layering of the
model grid.
(4.2.2.2) Boundary and Initial Conditions
APPRAISAL
Yes
No
Comments
5-11
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Table 5-1 (Continued)
MODELING AND EVALUATION CRITERIA
Is justification provided for the selection of all boundary and initial
conditions?
Are model boundaries consistent with natural hydrologic features?
Are the boundary and initial conditions consistent with the conceptual
model?
Are the uncertainties associated with the boundaries and initial conditions
addressed?
Are the boundaries far enough away from any pumping/injection centered
to prevent "boundary effects"?
Are transient boundaries discussed?
Is the rationale given for simplifying the boundaries from the conceptual
model discussed?
Are the values for the assigned boundaries presented?
(4.2.2.3) Specification of Time Steps
Is the Courant criterion satisfied?
(4.2.2.4) Model Parameterization
Are data input requirements fully described?
Is the discussion of the data well founded with respect to Objectives and
Data Review Section?
Are the interpretation and extrapolation methods (e.g. , Kriging)
adequately presented?
Do the figures and tables completely describe the data input with respect
to discrete components of the model?
Are the model parameters within the range of reported or measured
values?
(4.2.3) MODEL CALIBRATION
Has calibration been attempted?
Is the rationale for model calibration approach presented?
Are the calibration procedures described in detail?
Are the calibration criteria presented?
Does the calibration satisfactorily meet specified criteria?
Is the rationale presented for selecting convergence criteria?
Are code convergences and numerical instabilities discussed?
Do the calibrated parameters fall within their expected ranges?
APPRAISAL
Yes
No
Comments
5-12
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Table 5-1 (Continued)
MODELING AND EVALUATION CRITERIA
Are discrepancies explained?
Has the calibration been tested against actual field data?
Are the differences between steady-state and transient calibrations
presented?
Could other sets or parameters have calibrated the code just as well? Is
this discussed?
Are areal and cross-sectional representations of the final calibrated
results included for both hydraulic heads and radionuclide plume(s)?
Does calibration of the model take into account the inconsistency between
point measurements at wells and areal averages of model output?
Is the match between the calibration targets and final parameters shown
diagrammatically?
Were calibrating errors presented quantitatively through the use of
descriptive statistics?
If particle-tracking was performed, are these results shown?
Is the calibrated model consistent with the conceptual model?
Are any changes to the conceptual model discussed and justified?
Is non-uniform areal recharge applied? Is this approach justified?
Does the calibration properly account for vertical gradients?
Is the calibrated hydraulic conductivity field consistent with the geologic
logs and aquifer stress tests?
Are the convergence criteria appropriate?
Was a mass balance performed?
Is the water-balance error less than 1%?
Are the mass balance results for the calibrated model discussed?
Is the model's water balance consistent with known flows of rivers and
levels of lakes?
APPRAISAL
Yes
No
Comments
5-13
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Table 5-1 (Continued)
MODELING AND EVALUATION CRITERIA
(4.2.4) SENSITIVITY ANALYSES
Was a sensitivity analysis performed?
Is the approach to the sensitivity analysis detailed?
Were all input parameters selected for investigation?
If not, was rationale presented for excluding parameters?
Was a sensitivity analysis performed on the boundary conditions?
Are the ranges of parameters appropriate?
Were sufficient simulations performed? Was justification provided?
Was the relevance of the sensitivity analysis results to the overall project
objectives discussed?
Are the results presented so that they are easy to interpret?
Were sensitivity analyses performed for both the calibration and the
predictive simulations?
APPRAISAL
Yes
No
Comments
5-14
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6. REFERENCES
ASTM84 Am. Soc. for Testing and Materials (ASTM), 1984. Standard Practaices for
Evaluating Environmental Fate Models of Chemicals. Annual Book of ASTM
Standards, E 978-84, Am. Soc. for Testing and Materials, Philadelphia,
Pennsylvania.
BRY87 Bryant, J.L., and N.P. Wilburn, 1987. Handbook of Software Quality Assurance
techniques Applicable to the Nuclear Industry. NUREG/CR-4640, Off. of
Nuclear Reactor Regulation, U.S. NUclear Regulatory Commission, Washington,
D.C.
D-5447 Standard Guide for Application of a Ground-Water Flow Model to a Site-Specific
Problem. ASTM (in preparation).
D-5490 Standard Guide for Comparing Ground-Water Flow Model Simulations to Site-
Specific Information. ASTM (in preparation).
D-5609 Standard Guide for Defining Boundary Conditions in Ground-Water Modeling.
ASTM (in preparation).
D-5610 Standard Guide for Defining Initial Conditions in Ground-Water Modeling.
ASTM (in preparation).
D-5611 Standard Guide for Conducting a Sensitivity Analysis for a Ground-Water Flow
Model Application. ASTM (in preparation).
D-5718 Standard Guide for Documenting a Ground-Water Flow Model Application.
ASTM (in preparation).
D-5719 Standard Guide for Simulating Subsurface Air Flow using Ground-Water Flow
Modeling Techniques. ASTM (in preparation).
DAS77 Dass, P., G.R. Tamke, and C.M. Stoffel. 1977. "Leachate Production at Sanitary
Landfills," J. Environ. Eng. Division.. Proc. ASCE 103 (EEG).
EPA87 Environmental Protection Agency, 1987. The Use of Models in Managing
Ground-Water Protection Programs, EPA 600/8-87/003, January 1987.
EPA88a Environmental Protection Agency, 1988. Selection Criteria for Mathematical
Models Used in Exposure Assessments: Ground-Water Models, EPA/600/8-
88/075, May 1988.
EPA88b Environmental Protection Agency, 1988. Groundwater Modeling: An Overview
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EPA89
and Status Report, EPA/600/2-89/028, December 1988.
Environmental Protection Agency, 1989. Exposure Factors Handbook,
EPA/600/8-89/043.
EPA91 Environmental Protection Agency, 1991. Risk Assessment Guidance for
Superfund. Volume 1, Human Health Evaluation Manual. Part B, Development
of Risk Based Preliminary Remediation Goals, PB92-963333.
EPA94a Environmental Protection Agency, 1994. Report of the Agency Task Force on
Environmental Regulatory Modeling, EPA 500-R-94-001, March 1994.
EPA94b Environmental Protection Agency, 1994. Assessment Framework for Ground-
Water Model Applications, EPA 500-B-94-003, July 1994.
EPA94c Environmental Protection Agency, 1994. Ground-Water Modeling Compendium
- Second Edition. Model Fact Sheets, Descriptions, Applications and Cost
Guidelines, EPA 500-B-94-004, July 1994.
EPA94d Environmental Protection Agency, 1994. Guidance for Conducting External Peer
Review of Environmental Regulatory Models, EPA 100-B-94-001, July 1994.
FEDSIM81 Federal Computer Perfromance Evaluation and Simulation Center (FEDSIM),
1981. Computer Model Documentation Guide. NBS Special Publ. 500-73, Inst.
for Computer Science and Technology, Nat. Bur. of Standards, U.S. Dept. of
Commerce, Washington, D.C.
FEI75 Fein, D.G., K.J. Hanley and T.V. DeGeare. 1975. Use of the Water Balance
Method for Predicting Leachate Generation from Solid Waste Disposal Sites.
SW-168, U.S. Environmental Protection Agency, Washington, D.C.
FRE79 Freeze, R.A. and J. Cherry, 1979. Groundwater. Prentice-Hall, Englewood Cliffs,
N.J.
GAS79 Gass, S.I., 1979. Computer Model Documenation: A Review and an Approach.
NBS Special Publ. 500-39, Inst. for Computer Science and Technology, Nat. Bur.
of Standards, U.S. Dept. of Commerce, Washington, D.C.
HEI92 van der Heijde, P.K.M., and O.A. Elnawawy, 1992. Compilation of Ground-
Water Models. GWMI 91-106. International Ground Water Modeling Center,
Colorado School of Mines, Golden, Colorado.
HEI89 van der Heijde, P.K.M., 1989. Quality assurance and Quality Control in Ground-
Water Modeling GWMI 89-04. Internal. Ground Water Modeing Center,
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Holcomb Research Inst, Indianapolis, Indiana.
KON92 Konikow, L.F., and J.D. Bredehoeft, 1992. Ground-Water Models Cannont be
Validated. Advances in Water Resources SWRENI 15(1): 75-83.
LEE95 Lee, Sang B., V. Ravi, J.R. Williams, D.S. Burden, 1995 "Evaluation of
Subsurface Modeling Application at CERCLA/RCRA Sites," Subsurface Fluid-
Flow (Ground-Water and Vadose Zone) Modeling. ASTM STP 1288. Joseph D.
Ritchey and James O. Rumbaugh, Eds., American Society for Testing and
Materials, Philadelphia, 1996.
PRI66 Price, H.S., J.C. Cavendish, and R.A. Varga, 1966. Numerical methods of higher
order accuracy for diffusion convection equations. Society of Petroleum
Engineers Journal pp. 293-303.
SCH83 Schroeder, P.R., A.C. Gibson, and M.D. Smolen, 1983. The Hvdrologic
Evaluation of Landfill Performance (HELP) Model. U.S. Environmental
Protection Agency, Cincinnati, Ohio.
SCH84 Schroeder, P.R., A.C. Gibson, and M.D. Smolen, 1984. The Hvdrologic
Evaluation of Landfill Performance (HELP) Model. U.S. Environmental
Protection Agency, Cincinnati, Ohio.
TAY85 Taylor, J.K., 1985. What is Quality Assurance? In: J.K. Taylor and T.W. Stanley
(eds.), Quality Assurance for Environmental Measurements, pp. 5-11. ASTM
Special Technical Publication 867, Am. Soc. for Testing and Materials,
Philadelphia, Pennsylvania.
THO55 Thornthwaite, C.W., and J.R. Mather, 1955. "The Water Balance." Publications
in Climatology. Vol. VIE, No. 1. Drexel Institute of Technology, Laboratory of
Climatology, Centerton, New Jersey.
THO57 Thornthwaite, C.W., and J.R. Mather, 1957. "Instructions and Tables for
Computing Potential Evapotranspiration and the Water Balance." Publications in
Climatology. Vol. X, No. 3. Drexel Institute of Technology, Laboratory of
Climatology, Centerton, New Jersey.
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APPENDIX A
FATE AND TRANSPORT OF RADIONUCLIDES
-------
Contents - Appendix A
A. 1 Physical Transport and Retardation Processes A-2
A.2 Chemical Transport and Retardation Processes A-5
A.3 References A-14
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APPENDIX A. FATE AND TRANSPORT OF RADIONUCLIDES
The following discussion is intended to explicate the potential effects that simplifying
assumptions (e.g., K^) may have on the modeling results.
The nature of the ground water environment restricts the number of processes that control the
fate of radionuclides as they are transported from their source to the accessible environment.
These processes fall into two categories: radioactive decay and those processes related to
transport. Transport-related processes (e.g., sorption, ion exchange, and precipitation) can
facilitate or retard the movement of ground-water contaminants, but radioactive decay always
results in a loss of activity of the original radionuclide. Radioactive decay, however, can result in
an increase in radioactive or chemically toxic daughter products as the original radionuclide
disintegrates.
Various mechanical and geochemical processes will affect the transport of contaminants by flow
through either a porous matrix or a fracture system in a porous matrix. The mechanical processes
considered are advection, dispersive effects (hydrodynamic dispersion, channeling), and
diffusion. Among the chemical processes considered in this paper are adsorption on mineral
surfaces, including the kinetics of adsorption, and processes leading to precipitation. Although
vapor transport is not always directly associated with ground-water flow, the two processes are
closely related; therefore, gaseous transport is also briefly discussed. Furthermore, some
radionuclides can occur and migrate in the gaseous phase. Gas phase migration can therefore be
an important mechanism for some radionuclides to migrate from the repository to the accessible
environment. For example, radioactive carbon-14 can be in the form of carbon dioxide gas, and
tritium (radioactive hydrogen) can be in the form of hydrogen gas or tritiated water vapor.
A. 1 Physical Transport and Retardation Processes
In both saturated and partially saturated conditions, ground water can carry material along in
solution or as suspended solids. The rate at which the transported material moves is affected by a
variety of factors, the most important being the velocity of the flowing water and the partitioning
of the material between liquid (i.e., water) and solid (i.e., rock) phases. The dominant physical
processes are advection, dispersive effects, and diffusion.
(a) Advection and hydrodynamic dispersion. The process by which solutes are
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transported by the bulk motion of water is known as advection. There is a tendency, however,
for the solute to spread out from the path that it would be expected to follow according to the
advective hydraulics of the flow system. This spreading phenomenon is called hydrodynamic
dispersion. It causes dilution of the solute and occurs because of spatial variations in ground-
water flow velocities and mechanical mixing during fluid advection; molecular diffusion, due to
the thermal-kinetic energy of the solute particles, also contributes to the dispersion process.
Dispersion can result from diffusion, channeling, and turbulent flow, but dispersion by itself does
not affect the average rate at which the transported material moves. It can, however, cause some
of the contaminant (in a diluted state) to move faster than the average ground-water flow
velocity. This may be significant for radionuclides with relatively short decay half-lives; if some
of the radionuclide arrives relatively quickly at the accessible environment due to dispersion, it
will have higher radioactivity because of the shorter decay time.
Dispersion is generally responsible for the shape of the tracer-test breakthrough curves. The low
concentrations of the radionuclides that mark the first arrival at the withdrawal well have been
dispersed ahead of the bulk or average ground-water velocity. The peak of the breakthrough
curve represents the average velocity of the contaminant (advection dominated), and the tail of
the breakthrough curve is formed by radionuclides that have been dispersed along longer flow
paths and at slower velocities than the average rate of ground water.
Dispersion is the primary mechanism responsible for dilution (mixing) processes in ground water
but is generally less significant than in air and in surface water. In both air and surface water,
dispersive dilution is often a major phenomenon because the flow can be turbulent. Turbulent
flow means that all the flow paths are not essentially parallel to the gross direction of motion.
Flow components that are perpendicular to the bulk fluid motion cause the plume to spread
laterally not longitudinally, thus reducing the concentration in the plume, as the volume of
contaminated air or surface water increases. In ground water, however, the magnitude of dilution
is usually much smaller, partly because turbulent flow rarely exists. The slow speed of ground
water, coupled with the effects of small channels in the intergranular pore space, tends to keep
the flow smooth and laminar. In an idealized conceptual model, the interconnecting pore spaces
can be thought of as forming flow channels or tubes; any tendency for the flow to eddy is resisted
by the sides of the flow channel. However, since the interconnecting pore spaces do not make a
continuous flow channel in real materials, there is some lateral mixing due to branching of flow
channels and spatial variation in flow velocity.
A-2
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Dispersion (neglecting molecular diffusion) is not significantly affected by laminar eddy currents.
If molecular diffusion is momentarily disregarded, dispersion in porous or fractured media is
caused by five principal phenomena: anisotropic permeability, varying pore sizes, varying path
length, variation in the velocity gradient across pore space, and flow splitting around soil
particles with mixing within the pore space. These five phenomena all contribute to longitudinal
dispersion; anisotropic permeability and flow splitting around the soil particles can cause lateral
dispersion. In nearly all ground-water systems, longitudinal dispersion effects are much larger
than lateral dispersion effects. Researchers have reported longitudinal dispersivity values
ranging from about 1 to 25 times higher than transverse dispersivity values (GELS5). In
fractured systems, the longitudinal dispersivity would likely be much greater than the transverse.
(b) Molecular diffusion. Diffusion in solutions is the process whereby ionic or molecular
constituents move under the influence of their kinetic activity in the direction of their
concentration gradient. Molecular diffusion is a relatively slow process but also contributes to
the overall dispersion process, primarily through micro-scale mixing within individual pore or
fracture channels which leads to large-scale bulk dilution and spreading in very slow moving
ground water.
The diffusion of radionuclides from water moving within fractures, or coarse-grained material,
into the finer-grained rock matrix (i.e., matrix diffusion) can be an important means of slowing
the transport of the dissolved radionuclides, particularly for non-sorbing or low-sorbing soluble
species. The apparent diffusion coefficient for a given radionuclide depends on properties that
are intrinsic to the chemical species (e.g., ion mobility), as well as properties of the rocks (such
as porosity, tortuosity, and sorption ratios).
A radionuclide introduced into a fractured porous medium will migrate through the fracture
openings by means of advection as well as hydrodynamic dispersion. The radionuclide may also
diffuse slowly into the porous matrix. If molecular diffusion is occurring, it will dominate flow
and transport within the porous matrix because the fluid velocity in the porous matrix is usually
very low. When introduced into a fractured aquifer, the radionuclide moves rapidly within the
fracture network. As time passes, the zone of contamination will diffuse farther into the porous
matrix. Since the porous matrix has a very large capacity to store the contaminant, it plays a
significant role in retarding the advance of the concentration front in the fractures. If the source
of contamination is discontinued and the water-bearing unit is flushed by non-contaminated
water, the contaminant mass in the fractures will be removed relatively quickly, whereas the
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contaminant in the porous matrix will be removed very slowly via diffusion back into the fracture
openings.
(c) Gaseous transport. A limited number of radionuclides can form volatile species capable
of being transported in a moving vapor or gas. Among these are tritium, carbon-14, and iodine-
129. On a macroscopic scale, factors that affect transport in flowing ground water also affect
transport in flowing gas (i.e., the velocity of the gas determines the potential for advective
transport). In the absence of flow, diffusion is the only mechanism for transport in the gaseous
state. The processes of partitioning of the volatile species between the gaseous, liquid, and solid
state and isotopic exchange must also be considered when assessing the impact of gaseous
transport.
A.2 Chemical Transport and Retardation Processes
In addition to the physical processes, the transport of radionuclides is affected by a wide range of
chemical processes. Many of these reactions are poorly understood and are the subject of on-
going research. From a practical view, the important aspect is the removal of solute from
solution irrespective of the process. For this reason, most computer codes simply lump all of the
cumulative effects of the geochemical processes into a single term (i.e., distribution coefficient-
Kd) which describes the degree to which the radionuclide is retarded relative to the ground water.
Thus, the distribution coefficient relates the radionuclide concentration in solution to
concentrations adsorbed on the rocks. The following paragraphs summarize the primary
geochemical processes that can play a role in the transport of radionuclides.
(a) Sorption. An important mechanism in retarding the migration of radionuclides in ground
water is sorption, which is defined to include all solute-rock interactions that cause the
radionuclides to migrate at a slower rate than the ground water itself. The amount of sorption is
dependent on both the chemistry of the water and of the rocks; because some of the chemical
reactions are relatively slow, it can be a function of time as well.
Sorption coefficients are usually obtained using a standard batch test where rocks or soils are put
in contact with ground water in which small amounts of dissolved radionuclides have been
mixed. A problem with this technique is that more detailed geochemical data are necessary to
support the validity of applying the sorption measurement to the real-world physical and
chemical conditions and expected travel time of the radionuclides (which may be of the order of
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hundreds to thousands of years). Such mechanisms as dissolution/ precipitation, complexing,
adsorption/desorption, phase transformations, and solubility should be understood for
radionuclides of interest in the geochemical environment.
The tendency of a radionuclide dissolved in ground water to be sorbed by the aquifer's solid
phase can be expressed in terms of the soil/solution partition coefficient, Kd, also referred to as a
distribution coefficient. Kd is the simplest mathematical approach to adsorption and may be
derived from the Freundlich isotherm equation
x/m = KdC1/n (1)
where x/m is the amount adsorbed (Ci chemical per gram of soil), and C is the concentration of
chemical Ci/ml in the aqueous phase. The value of 1/n depends on the sorbate and sorbent being
studied and is usually close to 1 (LYM82).
Sorption of radionuclides in the saturated zone will be due primarily to the high surface area (per
unit mass) of minerals such as clays. Lipophilic substances tend to form films at water/solid
interfaces just as they do at the air/water interface. Thus, if the saturated zone contains clays or
other high surface area minerals, the ground water is presented with a large water/solid interface
on which some types of contaminants can form a surface film. Adsorption isotherms in which
sorption can be correlated with the surface area of the adsorbent are called Langmuir isotherms.
Adsorption phenomena of this type are not linear and can reach a saturation limit after which
further adsorption will not occur, even from water with greater concentrations of radionuclides.
Several variations of the adsorption isotherm equation are available for the fitting of empirical
data from experimental sorption studies (KIN86). The means for calculating sorption retardation
of dissolved inorganic radionuclides is similar to the method used for computing retardation of
organic contaminants by soil carbon content and octanol/water partition coefficients. The
distribution coefficient for a specific radionuclide depends in part on chemical composition of
aqueous solutions. Thus, for a given geologic material, a radionuclide can have a wide range of
distribution coefficients, depending on the total chemical characteristics of the water.
Limitations of the distribution-coefficient approach to geologic investigations include:
The assumption of a linear sorption isotherm. The terms "sorption isotherm," "Freundlich
isotherm," or "Langmuir isotherm" are generally used to define the relationships between
A-5
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sorption and the concentration of the element being sorbed at a constant temperature.
Total reversibility in the sorption/desorption reaction is assumed. However, the
distribution between the solid phase and the aqueous phase may include precipitation or
irreversible reactions or both.
Sorption/desorption reactions are generally assumed to be instantaneous. However, in
some cases the reaction rate may be too slow to justify that assumption.
The aqueous-phase speciation is not well known for many of the radionuclides.
These limitations do not necessarily apply to all of the radionuclides. For example, the solution
chemistry of the alkaline earths (Cs, Ra, and Sr) is well known, and the aqueous-phase species
can be predicted with relatively high certainty provided that the nature of the soil and rocks is
known. When measured sorption values are very high or very low, the range of individual
measurements may be quite large. A very high or very low Kd indicates that one phase, either the
solution or the solid, has very little of the radionuclide present; therefore, very few detectible
radioactive disintegrations occur, giving rise to relative high potential counting errors compared
to those obtained where sorption ratios are close to unity.
(b) Ion exchange phenomena. Ion exchange is one of several possible sorption processes. It
is a particularly important process for many common cationic radionuclides and therefore
deserves separate focus. The primary retardation mechanisms for both organic and inorganic
ionic contaminants in ground-water systems are ion exchange and precipitation. Ion exchange is
primarily effective on cations (positively charged ions), although in certain hydrogeochemical
environments anions are also retarded by ion exchange. Ion exchange capacity within a geologic
material is almost exclusively limited to colloidal clay and silica particles (diameters in the range
10~3 to 10~6 mm), because these particles have a large ionic charge relative to their surface areas.
This charge is the result of (1) cationic substitutions within the crystal lattice and (2) ionic
dissociation at the surface. To neutralize this charge, an adsorbed layer of cations and anions
forms a zone adjacent to the hydroxylated layer (PAR67). The net charge of this zone can be
negative or positive, depending on the pH of the immediate environment. At low pH, a
positively charged surface prevails; at neutral to high pH, a negatively charged surface develops
(FRE79). The tendency for sorption of either cations or anions therefore depends on the pH.
Most natural ground-water systems have a pH in the neutral to positive range. Therefore, most
systems tend to have a stronger tendency for cation exchange than for anion exchange.
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Neutralized colloidal particles maybe transported in the ground water with organic and inorganic
contaminants on their surfaces. Additionally, humic substances can exist as colloidal particles
and also serve as ion exchangers. Some radionuclide species such as plutonium and other
transuranic elements have been reported to exist as suspended colloids (CLE81). Some
contaminants that might otherwise be sorbed to stationary material in the aquifer could be
transported in the sorbed layers of these mobile colloids. Sorption in this case has facilitated
transport. The cation exchange capacity (CEC) of soils and other geological materials is usually
expressed as the number of milliequivalents of cations that can be exchanged in a sample with
dry mass of 100 grams.
(c) Speciation. The solubility of the waste elements can influence their transport by limiting
the maximum concentration of the elements dissolved in the aqueous phase. Speciation, defined
as the formation of various complexes and oxidation states in the aqueous phase, in turn affects
the solubility and mobility. Speciation and solubility of individual waste elements depend on the
chemical properties of the waste element, on the state of the local water (composition, pH,
oxidation state, and temperature), and, if nonequilibrium processes are important, on factors such
as precipitation and dissolution kinetics, oxidation-reduction kinetics, the identity of the solids
present, water-flow conditions, and colloid formation.
Elements dissolved in water can exist as various chemical species such as different oxidation
states or complexes with other ions in water (STU81). Solubility generally increases as the
variety and concentration of complexes of an element increase; thus, the solubility is influenced
by the tendency of a given element to form complexes and the concentrations of species with
which it can complex. Sorptive behavior depends on the size and charge of the sorbing species;
both of these quantities vary among the complexes of a given element. Thus, speciation can
influence sorption. Plutonium, for example, can exist at several different oxidation states in
either a cationic or anionic form. It can also be complexed with various other ionic species such
as carbonate. Each of the various species may have different solubility and mobility
characteristics.
Aqueous species of most elements can be experimentally detected in solution by a number of
techniques; spectroscopy is most commonly used. However, concentrations of aqueous species
are normally calculated from a knowledge of the overall composition of the solution (total
concentrations of the elements in solution) and the formation constants of possible aqueous
species using equilibrium thermodynamic methods. Equilibrium thermodynamic methods work
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well (given the proper data) for the various complexes of a particular oxidation state, but may
yield inaccurate results for the distribution of the element among possible oxidation states
(LIN84).
(d) Precipitation (phase separation from ground water). After a radionuclide has entered
the ground water, changes in the temperature, pH, and other chemical constituents may bring
about precipitation (i.e., phase separation) of the intruding radionuclide. The solubility of most
radionuclides varies directly with the temperature, and the groundwater's level of acidity can also
affect the solubility (SPO81). If an element can also exist in various oxidation states, variables
that control oxidation-reduction behavior also influence solubility. Unlike complex formation,
which is essentially always an equilibrium process, precipitation processes are often not in
equilibrium or in metastable equilibrium (STU81). If dissolution and precipitation kinetics are
relatively rapid, over the time frame of interest, equilibrium behavior can be assumed. If kinetics
are very slow, a metastable equilibrium may exist where the aqueous phase is in metastable
equilibrium with some solid other than the most stable (least soluble) one. In some intermediate
cases, the dissolution or precipitation rates maybe comparable to the time scale of interest; for
these cases, kinetic data are required to describe aqueous concentrations accurately.
Coprecipitation refers to a group of processes whereby more than one compound precipitates at
one time (SPO81). Three examples are mixed-solid formation, adsorption during precipitation,
and inclusion during precipitation.
(e) Natural colloid formation. A number of actinides, plutonium in particular, can form
natural colloids under conditions of near neutral solutions of low ionic strength (AVO84).
These colloids are optically clear in solution, show a characteristic adsorption spectrum, and do
not settle out of solution. Colloidal plutonium shows x-ray diffraction patterns similar to
crystalline plutonium dioxide; higher order lines are missing, indicating small crystalline size (20
to 30 angstroms). There is also some indication that americium may form colloids under similar
conditions (OLO84).
A possible mode of radionuclide transport involves the movement of radioactive particles
suspended in the ground water. Colloidal particles (up to 0.5 micrometers in diameter) remain
suspended for long periods and hence may migrate with the ground water. As the solid waste
form is leached, particles containing radionuclides may form by the sorption of dissolved
radionuclides on nonradioactive particles.
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To estimate the amount of radionuclides that can be transported by colloidal suspension, it is first
necessary to determine whether colloidal-sized particles exist in the ground water. Then, the
sorption ratios for waste elements on these particles must be measured or estimated from the
composition of the particles. In addition, the conditions under which colloids can form from the
waste elements or from the waste and their stability after formation must be determined. Finally,
the conditions necessary for the filtration or sorption of the particles by the rock matrix itself
must be defined.
Matter in the colloidal state has a relatively large surface area; thus the most important properties
of colloids are those that depend on surface interactions, such as adsorption. Drever (DRE82)
discusses the nature and geochemistry of colloids, with emphasis on the charge surrounding
colloids and its effect on suspension stability.
Olofsson et. al. (OLO81) classify radiocolloids (colloids containing radionuclides) as true
colloids or pseudocolloids depending on their formation process. True colloids are formed by
condensation of molecules or ions as a result of hydrolytic or precipitation processes. Colloids
consist mostly of hydroxides or polymers formed by hydrolysis, and they have a very rapid
formation rate. Pseudocolloids, on the other hand, are formed as a result of adsorption on
impurities in the solution and tend to be much larger than true colloids. Pseudocolloids can be of
two types, reversible and irreversible. The formation rate of pseudocolloids is basically
determined by the sorption rate on colloidal impurities (OLO82).
Radiocolloids are believed to be a significant factor for the transport of radionuclides in some
environments and might facilitate their transport away from the source area (AVO82).
Radiocolloids may arise from a variety of sources. The corrosion of metal containers can lead to
the formation of absorbent colloids. Degradation of engineered backfills may also lead to
colloidal formation. If the waste form is leached by ground water, naturally occurring colloids
derived from smectites, vermiculites, illites, kaolinite, and chlorite present in groundwater may
also adsorb radionuclides. Champ et. al. (CHA82) have demonstrated experimentally the
existence of rapid transport of plutonium colloids using core samples and ground water.
Transport of particulates in geologic media will depend on aqueous flow rate, on pore and
fracture size in the rock, on ions carried in the water, and on the nature of the particulate matter.
Several mechanisms may remove colloidal particulates from ground water such as mechanical
filtration by the rock matrix, sorption on the surface of the rock pores (van der Waals forces), and
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neutralization of the repulsive charges on the colloids, thus allowing them to coagulate. In
addition, colloids will be subject to gravitational settling for particles larger than about 0.1
micrometer (TRA87).
(f) Radiolysis. Radiation can affect the solubility of waste elements by altering the
composition of the water or by influencing the crystallinity of the solids that form (AUS69). The
primary effect of gamma radiation will be a reduction in water pH and a trend toward more
oxidizing conditions as long as air is present; secondary effects will be the production of nitrate
(or nitrite) anions if nitrogen is present. Gamma-emitting radionuclides tend to be relatively
short-lived and will be most important early in the life of a repository and, if ground water travel
rates are slow, will have the greatest effect on the water near the waste.
Alpha radiation can affect water compositions in ways that are similar to the effects of gamma
radiation. The primary effects of alpha radiation are a decrease in pH and a trend toward more
oxidizing conditions in the water. Solids composed of alpha-emitters tend to show self-
irradiation damage to their crystal structure; the solubilities of solids like the actinide oxides and
hydroxides are affected in that amorphous solids, which are generally more soluble than
crystalline solids, are more likely to be the natural precipitation products (NIT85). Some alpha-
emitters are relatively long-lived and maybe of concern for hundreds or thousands of years.
Neutron radiation can have effects on other elements through neutron capture reactions.
However, these effects are generally only significant near the source of strong neutron-emitters
and would not be expected to affect ground-water chemistry or the migration of radionuclides
downgradient, away from the waste source. Beta radiation is relatively weak and similarly would
not be expected to have significant effects on ground-water chemistry or radionuclide migration
away from the immediate vicinity of the source.
(g) Biofixation. A mechanism that appears to affect the transport of radionuclides under
some conditions is microbial fixation (WES84). Radionuclides may be immobilized and/or
mobilized by organisms in the repository environment. Immobilization may occur if
radionuclides are incorporated into the cell structure of microorganisms or plants that are
relatively stationary. On the other hand, radionuclides may be mobilized by forming biocolloids
with bacteria, spores, and viruses.
(h) Natural organic matter interactions. Organic matter, in some instances, plays a
significant role in the transport of radionuclides (LEV79). Wastes in the repository will contain a
A-10
-------
significant quantity of organic matter. The most important transport-related interactions between
organic matter and radionuclides are:
Mobilization - Decomposition of organic material raises the partial pressure of carbon
dioxide (CO2) in ground water and soil and adds organic CO2 and organic acids which
leach and mobilize certain radionuclides (e.g., uranium).
Transportation - Uranium can then be transported as bicarbonate anion or as soluble
organic complex in ground water.
Concentration - The humic acid can precipitate in ground water when pH becomes more
acidic or where increased salt content is encountered. The humic acids can exchange or
chelate uranium. Concentration factors greater than 10,000 times that of the ground water
have been observed on the organic material.
(i) Anion exclusion. The negative charge present on many mineral surfaces can repel the
approach of anions. The exclusion may limit the diffusion of anions into the matrix, thereby
allowing the anions to move at the higher velocity of the water moving in the center of the
fractures or intergranular pore space, away from the surface film. The same phenomenon can
restrict the entry of anions into the smaller pores. This process is significant to the transport of
radionuclides because negatively charged radiocolloids could potentially move faster than the
average rate of the ground water. Under some geochemical conditions (i.e., very low pH),
mineral surfaces may assume a more positive charge and thus repel cations rather than anions
and cause the cations to move faster than the average ground- water rate.
(j) Organic complexation. Natural and anthropogenic organic colloids occurring in the
subsurface can act as a sorbent for radionuclides in adsorption-desorption and cation exchange
processes. This association of radionuclides with organic matter is a relationship that has been
well documented in both the field and laboratory. Due to the large surface area per unit mass and
anionic surface functional groups associated with some organic colloidal material, radionuclides
have a significant potential to be adsorbed. If the radionuclide is adsorbed onto mobil colloidal
matter, the radionuclide may be transported as a colloid.
The chemical and/or physical reaction which influences the radionuclide complexation with
organic colloids will vary considerably with a number of experimental variables. Complexation
increases at higher pH's and high humic substance concentrations and decreases at high ionic
strengths.
A-ll
-------
The chemical and/or physical reaction which influences the radionuclide complexation with
organic colloids is a reversible process. Parameters that influence reversibility include:
pH, ionic strength, and radionuclide and organic compound concentrations. Complexation
reversibility may be an important factor when ground water from various flow regions mix
together in common hydrogeological units. When complexation reactions are reversed, the fate
and transport mechanisms associated with the complexation may change accordingly.
A-12
-------
A. 3 References
AUS69 Ausloos, P., 1969. Fundamental Processes in Radiation Chemistry. Interscience
Publishers, New York, pp. 651-685.
AVO84 Avogadro, A., and G. DeMarsily, 1984. "The Role of Colloids in Nuclear Waste
Disposal," in Scientific Symposia Proceedings. Boston. Massachusetts. November
1983. G.L. McVay (ed.), Vol. 25, North-Holland, Elsevier Science Publishing
Co., Inc., New York, pp. 405-505.
AVO82 Avogadro, A., and F. Lanza, 1982. "Relationship Between Glass Leaching
Mechanism and Geochemical Transport of Radionuclides," in Scientific Basis
Proceedings. Berlin. Germany. June 7-10. 1982. W. Lutze (ed.), Vol. 5, North-
Holland, Elsevier Science Publishing Co., Inc., New York, pp. 103-112.
CHA82 Champ, D.R, W.F. Merritt, and J.L. Young, 1982. "Potential for the Rapid
Transport of Plutonium in Groundwater as Demonstrated by Core Column
Studies," in Scientific Basis for Nuclear Waste Management V. Materials
Research Society Symposia Proceedings. Berlin. Germany. June 7-10. 1982. W.
Lutze (ed.), Vol. 5, North-Holland, Elsevier Science Publishing Co., Inc., New
York, pp. 745-754.
DRE82 Drever, J.L, 1982. "Colloid Properties." The Geochemistry of Natural Waters.
Prentice Hall, New Jersey, pp. 78-79.
FRE79 Freeze, R.A. and J. Cherry, 1979. Groundwater. Prentice-Hall, Englewood Cliffs,
N.J.
GEL85 Gelhar, L.W., A. Montoglou, C. Welty, and K.R. Rehfeldt, 1985. A review of
field-scale physical solute transport processes in saturated and unsaturated porous
media. Palo Alto, CA: Electric Power Research Institute. EPRIEA-190. As
cited in EPA88b.
KIN86 Kinniburgh, D.G., 1986. General purpose adsorption isotherms, Environ. Sci. Technol.
20:895-904.
LEV79 Leventhal, J.S., 1979. Organic Matter and Sandstone-type Uranium Deposits: A
Primer, Open File Report 79-1310.
LIN84 Lindberg, R.D., and D.D. Runnells, 1984. "Ground Water Redox Reactions: An Analysis
of Equilibrium State Applied to Eh Measurements and Geochemical Modeling," Science.
Vol. 225, pp. 925-927.
A-13
-------
NIT85 Nitsche, H., andN.M. Edelstein, 1985. Determination of Solubilities and Complexation
of Waste Radionuclides Pertinent to Geologic Disposal at the Nevada Tuff Site, LBL-
18900, Lawrence Berkeley Laboratory, Berkeley, Calif.
OLO81 Olofsson, U., B. Allard, K. Andersson, and B. Torstenfelt, 1981. Formation and
Properties of Radiocolloids in Aqueous SolutionA Literature Survey. National
Council for Radioactive Waste Report Prav 4.25, Department of Nuclear
Chemistry, Chalmers University of Technology, Goteborg, Sweden.
OLO82 Olofsson, U., B. Allard, B. Torstenfelt, and K. Andersson, 1982. "Properties and
Mobilities of Actinide Colloids in Geologic Systems," in Scientific Basis for
Nuclear Waste Management V. Materials Research Society Symposia
Proceedings. Berlin. Germany. June 7-10. 1982. W. Lutze (ed.), Vol. 5, North-
Holland, Elsevier Science Publishing Co., Inc., New York, pp. 755-764.
OLO84 Olofsson, U., M. Bengtsson, and B. Allard, 1984. "Generation and Transport
Properties of Colloidal Tri- and Tetravalent Actinide Species in Geologic
Environments," in Scientific Basis for Nuclear Waste Management VII. Materials
Research Society Symposia Proceedings. Boston. Massachusetts. November 1983.
G.L. McVay (ed.), Vol. 26, North-Holland, Elsevier Science Publishing Co., Inc.,
New York, pp. 859-866.
PAR67 Parks, G.A., 1967. Aqueous surface chemistry of oxides and complex oxide
minerals. Equilibrium concepts in natural water systems. Washington, D.C:
American Chemical Society, pp. 121-160. As quoted in Freeze and Cherry
(1979).
SPO81 Sposito, G., 1981. The Thermodynamics of Soil Solutions. Oxford Clarendon
Press, New York, pp. 66-69, 95.
STU81 Stumm, W., and J.J. Morgan, 1981. Aquatic Chemistry. A Wiley-Interscience
Publication, John Wiley & Sons, New York, pp. 8-9, 68-73, 94-95, 230-237, 248-
263, 274-275, 323-325, 332-333, 348-351, 366-367, 418-421.
TRA87 Travis. B.J. andH.E. Nuttall, 1987. Two-Dimensional Numerical Simulation of
Geochemical Transport in Yucca Mountain. LA-10532-MS, Los Alamos National
Laboratory, Los Alamos, N. Mex.
WES84 West, J.M., and I.G. McKinley, 1984. "The Geomicrobiology of Nuclear Waste
Disposal," in Scientific Basis for Nuclear Waste Management VII. Materials
Research Society Symposia Proceedings. Boston. Massachusetts. November 1983.
G.L. McVay (ed.), Vol. 26, North-Holland, Elsevier Science Publishing Co., New
York, pp. 487-494.
A-14
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APPENDIX B
SCOPING ANALYSIS PROCEDURES
-------
Contents - Appendix B
Page
B. 1 Release Analysis - Ground Water B-l
B.2 Fate Analysis - Ground Water B-5
B.2.1 Estimations of Ground-water Concentration B-5
B.3 Quantitative Fate Estimation B-6
B.4 Analytical Methods for Aquifer Flow and Transport B-l9
B.4.1 Point Concentration Model B-21
B.4.2 Flux Models B-26
B.4.3 Source Released from a Vertical Plane (x = 0) B-28
B.4.4 Horizontal Area Source B-28
B.4.5 Generalization of Instantaneous Models B-29
B.5 Simplified Analytical Methods for Minimum Dilutions B-30
B.5.1 Dilution at Downgradient Wells in Confined Aquifers for an
Instantaneous Point Source at the Surface B-30
B.5.2 Ground Water-Surface Water Interface-Instantaneous Source B-33
B.5.3 Quantity of Released Radioactivity Crossing a Vertical Plane B-33
B.5.4 Direct Ground-Water Usage B-34
B.6 References B-43
Figure B. 1 Idealized ground-water system for point concentration model, point
source (COD83) B-23
Figure B.2 Idealized ground-water system for point concentration model, horizontal
line source (COD83) B-23
Figure B.3 Vertically averaged ground-water dispersion model (COD83) B-25
Figure B.4 Concentration in downgradient wells for Example B.3 B-27
Figure B.5 Ground-water surface-water interface, flux model B-27
Figure B.6 Flux of pollutant into river for Example B.4 B-29
Figure B.7 Mixing factor for confined aquifers B-33
B-i
-------
APPENDIX B - SCOPING ANALYSIS PROCEDURES
Prior to scoping calculations, the appropriate release mechanisms for the movement of
radionuclides to the ground water need to be determined. If it is relatively certain that one or
more transport mechanisms (e.g., unsaturated zone) is unimportant, than it may be neglected.
The physical and chemical processes that may affect the fate and transport of radionuclides at the
site (e.g., fracture flow, vapor transport) also need to be determined. A previous report issued as
part of this interagency agreement outlines how to determine what site related characteristics may
be important (EPA94).
Having made these determinations, it must then be decided how accurate the results need to be
and what level analysis is appropriate to obtain the desired results. If the physical and chemical
processes at the site are complex and will not be satisfactrily predicted with simplistic data
analysis, then it may be necessary to consult with experts in the field regarding how to proceed.
It is not practical to perform complex analyses without the use of computer programs and
considerable expert help.
The calculational methods presented in this appendix have been divided into two parts: the
release analysis and the fate analysis. The equations in the release analysis section are used to
estimate contaminant release concentrations and volumetric release rates. The fate analysis deals
with the processes influencing radionuclide transport and how to estimate radionuclide
concentration in ground water.
B.I Release Analysis - Ground Water
(1) Estimating contaminant release concentration. The release concentration of a
radionuclide depends upon characteristics of both the waste and the site. For lagoons or
impoundments, the concentration of the radionuclide in the lagoon or impoundment is
considered to be the concentration of the leachate. For solid-waste disposal sites, the
equilibrium solubility of the solid waste is generally used as the initial concentration, with
the assumption that the waste will have equilibrated with the percolating rainwater. This
may not be the case, however, for all radionuclides. Therefore, it may be necessary to
estimate radionuclide concentrations as a function of the equilibrium partitioning between
the solid and solution, i.e., the distribution coefficient, Kd. The following formulae
provide a means to estimate leachate concentration under equilibrium partitioning
conditions.
B-l
-------
wat~ A'Q'T+pK.
Cwat = concentration in the leachate (Ci/m3-H20)
MCi = amount of nuclide in source (Ci)
A = area of the source (m2)
T = thickness of the source (m)
Kd = distribution coefficient (crrrVgr)
p = bulk density (gr/cm2)
0 = volumetric water content
where
0=0 * R re T\
sat sat \D~*-)
QSat = total porosity
RSat = saturation ratio
Under saturated conditions Rsat = 1. Under unsaturated conditions, the saturation ratio is a
function of the infiltration rate, the saturated hydraulic conductivity, and the texture of the
soil. The saturation ratio can be estimated using the following equation (CLA78):
D _ / -i v 2b+3
J\ , l )
sat ^ if '
sat
where
/ = infiltration rate (m/yr)
Ksat = saturated hydraulic conductivity (m/yr), and
b = soil-specific exponential parameter (dimensionless)
Representative values ofKsat, Qsat and b for various soil textures are listed in Table C.2-
C.4 (Appendix C).
(2) Estimating volumetric release rate. The volume of leachate is calculated in two ways, one
for solid wastes and one for liquid wastes. For solid wastes, percolating water (from
direct precipitation and/or stormwater runoff onto the site) is frequently the primary
source of liquid. In some cases the waste may be buried below the water table so that
B-2
-------
direct contact with groundwater is the principal leaching mechanism. The release rate to
groundwater for radionuclides leaching from percolating precipitation through a buried
source, can thus be calculated using the following equation:
L =
c
(B-3)
where
Lc = contaminant release rate (Ci/day)
/ = infiltration rate (M/day)
A = area of contributing source (M2)
The release rate to ground water for leaching of wastes that are disposed of below the
water table can be calculated using the following equation:
L. = KiA
c
(B-4)
where
K = hydraulic conductivity (M/day)
i = hydraulic gradient (M/M)
For liquid wastes (i.e., lagoons or surface impoundments), precipitation has a minimal
effect, since the liquid wastes will percolate to ground water under the influence of
gravity. In this case, the rate of percolation depends on the permeability of the liner or the
underlying or surrounding soil at the disposal site. The volumetric release rate for liquid
wastes can be estimated using the following equation (BOW79):
Q = Ks i A (B-5)
where
Q = volumetric flow rate (mVsec)
Ks = Darcys coefficient; for unlined lagoons use native soil hydraulic
conductivity (m/sec)
A = area of lagoon (m2).
i =
The hydraulic gradient, i, is determined as follows:
Hydraulic head
(B-6)
liner thickness or underlying soil
The hydraulic head is the sum of the pressure and gravitational heads. In this case, it is
B-3
-------
approximately equal to the depth from the top of the free liquid in the lagoon to the base
of the liner or soil layer from which the parameter Ks in Equation B-5 is obtained; the
liner thickness in Equation B-6 refers to the thickness of a liner when present, or for an
unlined system, the depth of the soil to the water table. When the depth to the water table
is large relative to the depth of free liquid in the lagoon, the hydraulic gradient will be
approximately equal to 1 .0. Alternatively, for systems with a thin liner and an
appreciable depth of free liquid, the hydraulic gradient can be much greater.
The Q, in Equation B-5, is then used to estimate activity release with the following
equation:
L = M 2 (B-7)
c c
where
Lc = contaminant release rate (Ci/day)
Mc = contaminant concentration in lagoon fluid (Ci/m3)
Q = volume release rate (mVday).
Equations B-5 and B-7 model the release rate from a lagoon regardless of whether the
flow is saturated or unsaturated. For unlined active lagoons, the flow is typically
saturated all the way to the water table. For clay-lined lagoons, the flow is saturated
through the liner and unsaturated between the liner and the water table. Equations
B-5 and B-7 are appropriate when lagoon releases are analyzed but should not be used for
spills or other conditions where there is no ponding of the radionuclides on the surface for
a long period of time. Under these conditions, the assumption of saturated flow (through
the liner or soil) may be violated.
Equations B-4 and B-6 apply to liquids that are mostly water. The hydraulic conductivity
is defined in terms of the fluid properties density and viscosity. For liquids with a density
or viscosity that differs from water, Ks can be corrected for this viscosity and density by
calculating the term IQ, using the following:
K =
c
(B-8)
where
Kc = corrected Kg term = hydraulic conductivity of liquid contaminant
(m/sec)
B-4
-------
Kw = hydraulic conductivity for water (m/sec)
p = density of liquids; c = non-aqueous; w = water (mg/liter)
\i = dynamic viscosity of liquids; c = non-aqueous; w = water;
(mg/m/sec).
and then substituting Kc for Ks in Equation B-5. For waste sites that are lined with
flexible membrane liners (FML), the release rate depends on the characteristics of the
contaminant as well as the liner (STE78). Liners that have been in place for long periods
of time or otherwise subjected to significant chemical, radiological, physical, or
geological degradation processes may have significantly greater permeation properties
than in their original undergraded state.
B.2 Fate Analysis - Ground Water
The nature of the ground-water environment restricts the number of processes that control the
fate of radionuclides as they are transported from their source to the receptor area. These
processes fall into two categories: radioactive decay and transport processes. Transport-related
processes (i.e., sorption, ion exchange, and precipitation of solids) can facilitate or retard the
movement of ground-water contaminants, but radioactive decay always results in a loss of
activity (disintegrations or decays per second) of the original radionuclide. However, radioactive
decay can result in an increase in radioactive or chemically toxic daughter products as the parent
radionuclide disintegrates.
B.2.1 Estimations of Ground-water Concentration
There are several different approaches to estimating the concentration of a radionuclide at the
receptor if radioactive decay is the only process affecting concentrations (i.e., no dilution). One
approach is based on the proportionality of volume and concentration of the waste versus those
of ground water.
If limited ground-water monitoring data at the release point are available and sufficient
environmental fate data are available to calculate an overall dilution rate (see Section B.3), the
concentration at the receptor well can be calculated.
The concentration of a decaying substance at the selected point downgradient from the release
point is given by the following equation:
B-5
-------
C =C -e < '
well wat e
where
Cwe// = concentration at downgradient distance x (Ci/m3)
Cwat = leachate concentration (Ci/m3-H20)
e = 2.71828
x = distance downgradient from point of introduction (m)
Vc = contaminant velocity (m/yr)
A = the radioactive decay constant which is equal to Hr\2 /half-life of the
isotope (yr"1 for A).
In the absence of ground-water monitoring data, mathematical models are often used to estimate
concentrations of contaminants in ground water at receptor wells or discharge points. Two
general classes of models can be used for this purpose: numerical and analytical. Numerical
models use various numerical analysis methods to solve the partial differential equations of flow
and transport. Analytical models generally consist of algebraic equations which approximate the
true solution of the differential equations. Both approaches have advantages, disadvantages, and
limitations. For the purpose of scoping calculations, only analytical models will be considered
(Section B.4).
B.3 Quantitative Fate Estimation
Radioactive releases may travel in the unsaturated zone before entering the zone of saturation.
However, the release can also be directly into the zone of saturation. The predominant direction
of the unsaturated zone flow is downward until the flow reaches the zone of saturation. Within
the zone of saturation, the flow is predominantly lateral. Ground-water velocity can be
determined for both the vadose (unsaturated) and saturated zones and is examined in the next
section.
(1) Equations for ground-water flow and radioactivity transport. The movement of
radionuclides in ground water can be described by two equations: one for the movement
of the carrier fluid (water) and one for the mass transport of the dissolved constituents
(radionuclides). In using these equations, the movement of the water in the region under
consideration must be known before the transport equation can be solved.
B-6
-------
(a) Unsaturated flow. The most significant nongaseous contaminant movement in
soils is a function of liquid movement. Vapor phase movement can be significant
for certain volatile contaminants (e.g., tritium and carbon-14). Soluble solid
radionuclides dissolved in rainwater, surface run-off onto the site, or water
applied through human activity will percolate into the soil. After rainwater
infiltrates the surface of the ground, it generally travels vertically downward
through the unsaturated zone (vadose zone) under the influence of gravity and
capillary forces until it reaches the water table.
The movement of water through partially saturated soil is described by Richards
equation, which can be written for an incompressible soil medium as
= V(Kkr (Q))_yh + Vz (B-10)
where
0 = volumetric water content
h = pressure head (length)
z = elevation above datum (length)
K = saturated hydraulic conductivity tensor (length/time)
kr = relative permeability
t = time (time)
V = divergence operator.
Solving Equation B-10 to predict rates of water flow and changes in water content
in a soil, is best accomplished by using a numerical computer model. A variety of
such models exist for simulating one, two, and three dimensional unsaturated flow
problems.
While computer modeling is often an integral part of exposure assessment
analysis, utilization of complex computer models can require a very skilled and
experienced modeler. Another consideration is that the prediction accuracy of
models is often compromised since key parameters (e.g., soil hydraulic
characteristics, heterogeneity), are imperfectly known, and values may have to be
estimated from literature references in the absence of actual site-specific
B-7
-------
measurements.
For these reasons, reliance on relatively simple approaches for determining travel
times through the unsaturated (vadose) zone may be justified. The interstitial pore
water velocity for transport through the unsaturated zone can be calculated from
the average percolation or recharge rate as follows (ENF82):
VPW = -f (B-ll)
where
Vpw = interstitial ground-water (pore water) velocity (length per
unit time)
/ = average infiltration or recharge rate which is the volumetric
flow rate per unit area per unit time
0 = average volumetric water content of the unsaturated zone
(decimal fraction, representing volume of water per volume
of soil).
In general, the flow rate q varies with time, for instance, in response to
atmospheric conditions (rainfall) or man-induced hydraulic loadings. The soil
water content 0 will vary with time and also with depth. However, for the
purposes of estimating travel times, the use of time-averaged flow rates and water
contents may be justified. Under steady-state flow conditions, the volumetric
water content, 0, of the soil will approach a constant and spatially uniform value,
that the soil can support the imposed flow rate, I; i.e., the hydraulic conductivity
of the soil under the unsaturated conditions will be equal to the flow rate, I:
I = K_k
where
(B-12)
Ks = vertical hydraulic conductivity of soil under saturated
conductivity (length/time)
The relative hydraulic conductivity, kr, is generally obtained from an equation that
describes the dependence of hydraulic conductivity or soil water content. One
commonly used expression is the Brooks-Corey relation:
2b+ 3
(B-13)
B-S
-------
where
Qs = volumetric water content of soil under saturated conditions
(porosity) (volume/volume)
b = pore size index (dimensionless)
Equations B-12 and B-13 can be combined to yield the following equation for the
average soil water content (CLA78):
e = e
Representative values of 0, b, and the term l/(2b+3) are listed in Tables C-l - C-
4 (Appendix C).
The saturated volumetric water content, Qs, saturated hydraulic conductivity, Ks,
and the exponential coefficient, b, are all related to soil properties and are usually
different in different soil types. The most reliable values for these parameters are
empirical values (if available) measured at the site. Where empirical values are
unavailable, the values in Tables C-l and C-2 provide guides for the rough
estimation of Qs and the term l/(2b+3). Representative values of Ks from two
different sources are presented in Tables C-3 and C-4. These tables demonstrate
the variability in estimates for these values.
Theoretically, the value of 0 cannot exceed Qs, which is the saturated soil moisture
content. Note also that the percolation rate, /, cannot exceed the saturated
hydraulic conductivity, K^ for the site soil. Whenever / > Ks (and therefore 0 as
calculated by Equation B-14 > Qs) for the duration of the study period, it must be
assumed that saturated conditions exist and that saturated flow prevails.
Equations B-17 and B-18 in the next subsection provide a means of estimating
saturated flow velocities. Records of estimated percolation rates, /, for the site
locality during the time period in question (or annual average recharge rate
estimates) are often available from local hydrology, climate, or soil authorities,
including regional U.S. Geological Survey (USGS) and U.S. Soil Conservation
Service offices. The following equation can be used to estimate the term /
(ENF82):
I = HL + Pr - ET - Qr (B-15)
B-9
-------
where
/ = average infiltration or recharge rate
HL = hydraulic loading from manmade sources (length per unit
time), which is the flow rate divided by the contaminant
source area.
Pr = precipitation (length per unit time)
ET = evapotranspiration (length per unit time)
Qr = runoff (length per unit time).
This estimation procedure can be used to evaluate infiltration rates, 7, at sites
where the information sources listed above cannot provide them directly. This
estimation procedure requires data for precipitation, evapotranspiration, and
runoff rates. In addition to the above sources, the National Weather Service,
Forest Service offices, National Oceanic and Atmospheric Administration
(NOAA) gauging stations, or other first order weather stations (e.g., at local
airports) are possible sources for these three types of data. The average
precipitation rate per unit time, Pr, for the study period can be obtained from
various local weather authorities such as those listed.
A value of ET for substitution into Equation B-15 can be estimated by using
measured Class A pan evaporation rates (a measure of local evaporation rates
under standardized conditions, available from the nearest NOAA gauging station)
in the equation:
ET = EVAP x cefc x cveg (B-16)
where:
EVAP = region-specific or site-specific measured evaporation rates
(length per unit time)
Cet = correction factor for converting measured pan evaporation
rates to evapotranspiration rates from turf grass (unitless)
Cveg = correction factor for converting evapotranspiration from
turf grass to evapotranspiration from other vegetative cover
types (unitless).
Values for Cet are taken from Table C-7, which requires climatological and pan
descriptive information.
B-10
-------
The term Cveg is used mainly for agricultural crops (Table C-8) and varies with the
thickness, depth, and characteristics of vegetative cover. Typical values are 0.87
for shorter broadleaf plants (alfalfa) to 0.6 for taller broadleaf plants (potatoes,
sugar beets) and 0.6 for taller grains and grasses. Where crop-specific data are
unavailable, a conservative default value for this term is the smallest reasonable
value, or 0.6. ET rates can vary significantly, of course, if vegetative cover varies.
A value of Qr, or the average runoff over the study period, for input into Equation
B-15, however, can generally be obtained from local USGS gauging stations. For
relatively level sites, a reasonable conservative default value is Qr = 0, where site-
specific data are unavailable or cannot be estimated.
The above method for predicting the average velocity of water migrating through
the unsaturated zone will in many cases yield reasonable approximations;
however, heterogeneities, such as root holes and macropores, can result in faster
velocities than predicted. The analyst is not expected to correct for this, yet it is
important to be aware of the limitations of the method.
(b) Saturated flow. Darcy's law may be used to describe the volumetric flow of
water through a porous medium under saturated conditions. The volumetric flow
(or discharge) is proportional to the product of the driving force, the soil's ability
to transmit water, and the cross-sectional area perpendicular to the flow direction.
The driving force is the difference in the energy (hydraulic head) between two
points in the aquifer divided by the distance between the two points. This driving
force is called the hydraulic gradient. The ability of soil or rock to transmit water
is represented by an empirically determined coefficient of hydraulic conductivity.
The hydraulic conductivity is determined by the properties of the liquid (water or
contaminant) and the permeability of the porous medium. Typical examples of
hydraulic conductivity for different porous materials are presented in Table C-6.
The soil has an intrinsic of permeability, which is determined by the size,
orientation, and connectedness of the pore spaces.
Estimating radionuclide transport velocity is based on estimating the velocity of
water. For those contaminants that flow with the water, contaminant velocity
equals water velocity (vertical and/or horizontal). For those that flow at rates that
differ from water, the estimated water velocity must be multiplied by a factor to
B-ll
-------
approximate the contaminant velocity.
Ground-water flux per unit cross-sectional area in the saturated zone is calculated
using Darcy's law, which is as follows (BOU78):
v = Ks i (B-17)
where
V = Darcy flux of water, also termed the specific discharge
(length/time)
Ks = hydraulic conductivity of soil or aquifer material
(length/time)
i = hydraulic gradient (length/length).
Although Fhas the units of velocity (length per time), it is a specific discharge
rate or flux (volume of water flowing per unit cross-sectional area of geologic
material per unit time). However, V, is the Darcy flux, rather than the
macroscopic velocity of the water. The actual ground-water velocity is calculated
from the Darcy flux, by dividing it by soil porosity, or, for precise modeling, by
effective porosity. (This approach takes into account the fact that the entire cross-
section of the pore is not flowing because of boundary layer effects, dead end
pores, and unconnected pores.) For clay soils, the effective porosity also corrects
for the effect of electro-osmotic counterfiow and the development of
electrokinetic streaming potentials (BOU78). The equation for calculating
ground-water velocity from Darcy velocity using effective porosity is as follows
(BOU78):
VPW = v/ne (B-18)
where
Vpw = ground-water (pore water) velocity (length/time)
V = Darcy velocity (superficial velocity, specific discharge
(length/time)
ne = effective porosity (dimensionless fraction).
The above terms should be determined for the site being studied. If this is not
possible for all parameters, then literature values can be used when site-specific
data are not available (Tables C-9 and C-10).
The hydraulic gradient (the change in the hydraulic head or elevation of the water
B-12
-------
table over a measured distance) should also be taken from field data obtained
during the site investigation. Water levels in existing nearby wells screened at
appropriate intervals can indicate hydraulic gradient.
Effective porosity, ne, can be approximated by the difference between the moisture
content at saturation and the "wilting point" (-15 bar)3. The equation is as follows
(RAW86):
^e = nsat ~ ^(-15) (B-19)
where
ne = effective porosity (fraction, dimensionless)
nsat = water content when the pores are fully saturated (fraction,
dimensionless)
n(_15) = wilting point moisture content (fraction, dimensionless).
This estimation procedure addresses the fraction of the pore spaces that contributes to flow, but
does not address the effect of electro-osmotic counterfiow and the development of electrokinetic
streaming potentials. For clays, this can be a significant difference. Literature values listed in
Table C-10 should be used for clay solids (these values incorporate the effects of the clay's ionic
double layer (RAW82); either technique Equation B-19 or the tables, can be used for sand or
loam soil.
The above method for predicting the average velocity of groundwater is the most widely accepted
approximation; however, it is only an approximation, and further refinement can be made to this
approach to improve its accuracy. Corrections for the path length difference between the straight
line distance versus the tortuous path that groundwater flows through can improve the precision
(FRE79). However, this correction factor is difficult to estimate.
Example B.I. For saturated ground-water flow, calculate the pore or seepage velocity in an
"average" sandstone under a gradient of 0.01 cm/cm. Use arithmetic mean values in tables
(Appendix C).
Equation B-18 applies. The arithmetic mean hydraulic conductivity, K, is 3.31 x 10"4 cm/s from
The wilting point is determined by drawing a suction of -15 bar to draw water out of the soil in a manner similar
to the section of a plant root; the bar is a measure of pressure (dynes/cm2).
B-13
-------
Table C-6. The arithmetic mean effective porosity, ne, is 0.21 from Table C-10. Therefore,
Vpw = V/ne = ~Ki/ne = 3.31 x 10"4 cm/ s x 0.01/0.21
= 1.58 x lO'5 cm/s.
[End of Example B.I]
(c) Mass transport. Cationic radionuclides that are migrating as a dilute solute may
be subject to retardation effects. Concentrated plumes are not as susceptible to
this phenomenon. Algorithms describing retardation are based on the assumption
that adsorption of radionuclides is primary due to sorption on mineral surfaces.
The mass transport equation uses the retardation coefficient to estimate the rate of
movement of the radionuclide. The most general form of the mass transport
equation is for transport in saturated-unsaturated media. If local equilibrium of
mass transfer and first-order chemical reactions are assumed, sorption can be
represented as a linear relationship, and the general mass transport equation can be
written as:
-V- (9 D-Vc) + V-(Vc) + [(e^F) + XQeRFl c = °,
-------
ne = the effective porosity,
pb = the bulk density (g/cm3),
Kd = the distribution coefficient (mL/g).
By assuming n = ne, Rp can be more conservatively estimated as
R - 1 + P"K
KF ~ L + Kd
(B-22)
An equivalent retardation factor may be defined for fracture flow where the
exposed area of the fracture is used rather than the porosity (FRE79).
Example B.2. Calculate the retardation factor, Rp, for strontium in an "average"
fine sandstone with a bulk density, pb, of 2.8 g/cm3 and a distribution coefficient
of 20 mL/g.
The arithmetic mean values of n, ne are found from Tables C-l 1 and C-12 to be
0.34 and 0.21, respectively. The retardation coefficient, Rd, calculated from
Equation B-21 is therefore
RF= ^!!+^x20 = 268.3.
F 0.21 0.21
Equation B-22 gives
R = l + _L_§_ x 20 = 267.7.
F 0.21
[End of Example B.2]
The approximate rate of movement of the radionuclide is VpJRp (Vpw is equal to
the pore velocity which is defined in Equation B-18), which maybe used to
estimate travel time.
(2) Chain decay of radionuclides. Radionuclides decay to stable products or to other
radioactive species called daughters. In some species, several daughter products maybe
produced before the parent species decays to a stable element. For some radionuclides,
the daughter(s) may present a potentially greater adverse health risk than the parent.
B-15
-------
Accounting for the chain-decay process is particularly important for predicting potential
impacts of actinide and transuranic migration. In considering this process over the
transport path of radionuclides, one transport equation must be written for each original
species and each daughter product to yield the concentration of each radionuclide
(original species and daughter products) at points of interest along the flow path in order
to estimate total radiological exposures. In a constant one-dimensional velocity field, the
general equations can be written as (BUR80):
dc, dc, d2 c,
dc d2
dc . dc . d2c.
V - = D - - - R^.X.c.
. - - - - ^... r -,
Fl dt pw dx x dx2 1 * '
where
RFi = the retardation factor for species i
Vpw = the pore velocity = V/ne
Ct = the concentration of species i
Dx = the dispersion coefficient
A; = the decay coefficient for species i.
Equation B-23 describes the material balances of the ith member of a decay chain and all
preceding chain members.
(3) Net Dilution. Dilution (mixing) processes in ground water are generally less significant
than dilution processes in air and in surface water. In both air and surface water,
dispersive dilution is often a major phenomenon because the flow can be turbulent.
Turbulent flow means that all the flow paths are not essentially parallel to the gross
direction of motion. Flow components that are perpendicular to the bulk fluid motion
cause the plume to spread laterally not longitudinally, thus reducing the concentration in
the plume, while the volume of contaminated air or surface water increases.
However, in ground water, the magnitude of dilution is usually much smaller, partly
because turbulent flow rarely exists. The slow speed of ground water, coupled with the
B-16
-------
effects of small channels in the intergranular pore space, tends to keep the flow smooth
and laminar. In an idealized conceptual model, the interconnecting pore spaces can be
thought of as forming flow channels or tubes; any tendency for the flow to eddy is
resisted by the sides of the flow channel. However, since the interconnecting pore spaces
do not make a continuous flow channel, in real soil there is some lateral mixing due to
branching of flow channels and spatial variation in flow velocity. Dispersion (neglecting
molecular diffusion) is not significantly affected by laminar eddy currents. If molecular
diffusion is momentarily disregarded, dispersion in porous or fractured media is caused
by six principal phenomena: (1) varying permeability, (2) varying pore sizes, (3) varying
path length, (4) variation in the velocity gradient across pore space, (5) anisotropic
permeability and (6) flow splitting around soil particles with mixing within the pore
space. These six phenomena contribute to longitudinal dispersion; the first and last two
phenomena can cause lateral dispersion. In nearly all ground-water systems, longitudinal
dispersion effects are much larger than lateral dispersion effects. Researchers have
reported longitudinal dispersivity values ranging from about 1 to 25 times higher than
transverse dispersivity values (GEL85).
Molecular diffusion is a relatively slow process but also contributes to the overall
dispersion process in two ways: micro-scale mixing within an individual pore or fracture
channels that lead to large-scale bulk dilution and spreading in a slow-moving ground-
water system. For short-term releases (i.e., spills), longitudinal mixing and resulting
dilution of plume concentrations can be significant. In this case, the plume can
effectively mix with the uncontaminated water in front of and behind the slug of
contamination, whereas continuous-release sources can result in plumes of sufficient
length that the middle section cannot effectively mix with clean water in front or behind
it.
A very simplistic approximation of the net dilution may be made by using a form of
Darcys law in conjunction with Equation B-9. The results of Equation B-9 were used in
Section B.3 to estimate the concentration of a radionuclide reaching a receptor if
radioactive decay were the only means by which the concentration was diminished. That
is, there were no dilution effects along the radionuclides' travel path to the receptor. In an
actual system, leachate concentrations would be diluted by mixing with the ambient water
along the travel path. As a first approximation, the degree of mixing may be estimated by
the following:
B-17
-------
Vol=KiTeWQT (B-24)
where
Vol = aquifer volumetric flow rate (mVy)
K = hydraulic conductivity of the aquifer (m/y)
i = hydraulic gradient (m/m)
W = width of the Source Area (m)
Te = effective mixing thickness (m)
0r = total porosity of aquifer (dimensionless)
The effective mixing thickness (Te) of the aquifer may be estimated from the following formula:
Te=<->« (B-25)
where
/= infiltration rate (m/y)
V= specific aquifer (m/y) discharge (B-17)
Q. = distance from the contaminant source to the receptor.
Equation B-25 represents the vertical distance traveled by the contaminant over length H,
assuming the vertical velocity is /, and the horizontal velocity is the aquifer flow rate.
The maximum value for the effective mixing thickness would be the thickness of the aquifer.
The concentration of a radionuclide in a downgradient receptor may subsequently be adjusted for
dilution using the Cwell concentration obtained from formula B-10, in conjunction with Equation
B-26.
_x_
M = C e ~c I A (B-26>
1 well wat
where
Cwat = leachate concentration (Ci/m3)
/ = infiltration rate (m/yr)
A = source area
M
D
wel1 Vol
where
Dwell = concentration of radionuclide in downgradient well corrected for dilution
B-18
-------
(Ci/m3)
Mwell = Mass of radionuclide corrected only for radioactive decay (Ci)
B.4 Analytical Methods for Aquifer Flow and Transport
An excellent discussion of simple mathematical methods to compute radionuclide travel times
and dilution rates is included as Chapter 4 in NUREG/CR-3332 (COD83). The relevant portions
of that discussion, as well as the example problems, figures, and tables, have been excerpted to
form the basis of Sections B.4 and B.5.
Analytical ground-water transport models can be used for certain types of analyses where
available data do not warrant a more complicated numerical analysis. Such models are useful for
scoping the transport problem and may frequently be adequate for regulatory needs if model and
corresponding input data are chosen conservatively.
In this section and Section B.5, a series of simple analytical models used at the U.S. Nuclear
Regulatory Commission (NRC) is presented. Many of these models have been computerized and
are available from the NRC (COD 82). In their simplest forms, however, most can be used with
the aid of only a calculator.
The models are developed for the limiting case of unidirectional saturated advective transport of
a single dissolved substance with three-dimensional dispersion in an isotropic homogeneous
aquifer as discussed in Section B.2.1. Equation B-27 is the governing differential equation of
solute transport for that set of conditions.
8c Vpw 8c Dx d2c D 92C Dz
-^- + ^ -=- = + + -
dt R 8X R x2 R 2 R
where
c = the concentration in the liquid phase (Ci/cm3)
Dx, Dy, D= the dispersion coefficients in the x, y, and z directions respectively (cm2/s)
A = the decay coefficient, (1/s)
Vpw = the x component ground-water pore velocity (cm/s)
Rp = the retardation factor (dimensionless).
The dispersion coefficient can be approximated from Equation B-28. For unidirectional
B-19
-------
flow, V2 = V3 = 0, V] = V, and 0 can be approximated for saturated flow by the effective
porosity, ne. Also, since Vpw = V/ne,
Dx = aLVpw (B-27a)
Dy = aTVpw (B-27b)
DZ = ^TVpw, (B-27c)
where oc£ and ccr are the longitudinal and transverse dispersivities respectively.
9D.. = arV5.. + (aL - og l^/V, (B-28)
..
where
8!y = 1 for i =j, by = 0 for i * j (Kronecker delta function)
0 = the volumetric water content
ccr = the transverse dispersivity (cm)
oc£ = the longitudinal dispersivity (cm)
V = the magnitude of the flux (cm/s)
VpVj = the components of the flux (cm/s).
B.4.1 Point Concentration Model
The first model presented can be used for calculating the concentration in the aquifer at some
point downgradient of a release (e.g., water supply well).
Equation B-27 is solved in terms of Green's functions:
c± = ^X(x,t) Y(y,t) Z(z,t], (B_29)
neRF
where Ci is the concentration at any point in space for an instantaneous one-curie release, ne is the
effective porosity of the medium, andX, Y, Z are the Green's functions in the x, y, z coordinate
directions, respectively. Equation B-29 has been developed for a variety of boundary and source
configurations:
B-20
-------
(1) For the case of a point source at (0, 0, zs) in an aquifer of infinite lateral (x, y) extent and
depth b, as illustrated in Figure B.I,
ec. =
(B-30)
where
exp
(B-31)
exp
(B-32)
d )
cos m n - cosmn r. (B-33)
b b\
(2) For the vertically averaged concentration in case 1 above (equivalent to a vertical line
source of length b),
neRF
(B-34)
where
(B-35)
(3) For a horizontal line source of length w centered at (0, 0, zs), as illustrated in Figure B.2,
neRF
(B-36)
where
B-21
-------
Y -
2 2w
(w/2-y)
(B-37)
and erf is the error function. Tables of the error function are available in standard
mathematical texts (ABR70).
QRNL-OWG 82-14406
POINT SOURCE AT
x = 0, Y =
-------
] IQRIZQNTAL LINE SOURCE
CENTERED AT
x = 0, y = 0, z = z.
TO?'Or Aouirrn OR
BOTTOM Of AQUIFER
Figure B.2. Idealized ground-water system for point concentration model, horizontal line
source (Codell and Duguid, 1983).
B-23
-------
(4)
For the vertically averaged concentration in case 3 above (equivalent to an area source of
width w and depth b),
neRF
(B-38)
(5) For a point source at (0, 0, zs) in an aquifer of infinite lateral extent and depth,
(B-39)
(7)
exp
-u-
exp
(B-40)
where
(6) For a horizontal line source of width w centered at (0, 0, zs) in an aquifer of infinite lateral
extent and depth,
neRF
(B-41)
For a horizontal area source of length / and width w centered at (0, 0, 0) in an aquifer of
constant depth b, as illustrated in Figure B.3, the solution to Equation B-29 becomes:
" "2Z2' (B-42)
Cl " neRF
where
erf
- erf-
exp (-Xt) .
(B-43)
B-24
-------
97-14403
SOURCE AREA
x
BOTTOM BOUNDARY
Figure B.3 Vertically averaged ground-water dispersion model (Codell and Duguid, 1983).
Example B.3. Concentration in an aquifer of limited thickness.
One curie of a radioactive pollutant leaks quickly into a water table aquifer through a
highly permeable ground cover over a square surface area 50 m on each side. The
pollutant has a half-life of 30 y. A well tracer test indicates that the ground water is
moving in the direction of two wells at a speed, Vpw, of 1.5 m/d and that the longitudinal
and transverse dispersivities, oc£ and ccr, are 20 to 10 m, respectively.
The saturated thickness of the water table aquifer, b, is 50 m and has an effective
porosity, ne, of 0.2. The pollutant has been determined to have a retardation factor, RF, of
20 in the aquifer.
Calculate the concentration of the pollutant in wells whose downgradient coordinates
with respect to the center of the source area are
(a) x = 200m, y = 0m
B-25
-------
(b) x = 400m, y = 50m
The wells are open to the entire depth of the aquifer.
Case 7 in the preceding section applies to this example, since the source is a horizontal
area type and the wells are screened over the total depth, which would vertically average
the concentration (Figure B.4).
Equation B-42 is therefore evaluated with Green's function:
X2 determined by Eq. B-43,
Y2 determined by Eq. B-37, and
Z2 determined by Eq. B-35.
The dispersion coefficients are calculated by Equations B-27a and B-27b.
Dx = ccLVpw = 20 x 1.5 = 30m2
Dy = ccTVpw= 10 x 1.5= 15m2
Figure B.4 shows the concentration as a function of time calculated for the two wells.
[End of Example B.3]
B.4.2 Flux Models
The flux model is used to calculate the discharge rate of a radionuclide entering a surface water
body that has intercepted the aquifer containing the transported material as depicted in Figure
B.5. It is assumed that all material entering the aquifer eventually enters the surface water except
for that which has been lost through radioactive decay. The assumptions that apply to the point
concentration model also apply to this model. The model provides only the rate of input to the
surface water at an average distance x downgradient from the surface. Actually, the contaminant
would enter the surface water as a diffuse patch, but the model described here gives no
information about the spatial distribution of this patch.
B-26
-------
ORM.-DWS BZC-207SI
TO
O
10
TIME OOQO
Figure B.4 Concentration in downgradient wells for Example B.3.
QRNL-DWG
APPROXIMATE O.ISTANCE
TO RIVER
X
SOURCE AT x = 0
Figure B.5. Ground-water/surface-water interface, flux model.
B-27
-------
In the unidirectional flow field assumed, the flux F (Ci/s) of material crossing an area dA = dy dz
perpendicular to the x axis is described by the equation
dF ( Bc\
= VDWC ~ Dx -5- ne ' (B-44)
i pw x :j_, i e v /
where c is the concentration in the dissolved phase. The total flux across the plane would be
/ -, \
t> < I Tr
F = n
dydz .
(B-45)
B.4.3 Source Released from a Vertical Plane (x = 0)
If Q is the concentration from an instantaneous release of 1 Ci at x = 0 and time t = 0, as
described by Equation B-29, then the resulting flux at distance x downgradient would be
x+ V
pw R
exp
x- V
pw R
- At
(B-46)
BAA Horizontal Area Source
For conditions expressed by Equation B-43, the corresponding flux would be
Fi "
pw
R
?1) - erf(z2)]
D
- -2 [exp(-zO - exp(-z^)]
R
exp(-Xt) ,
(B-47)
where
z =
x -
^t+±
D O
X\,_, ^
x
-5~t-±
and
B-28
-------
Example B.4. For the same conditions in the previous example, calculate the flux of the
pollutant into a river intercepting the ground-water flow, which is a distance x of 2000 m
downgradient from the center of the source.
Equation B-47 applies in this case. Figure B.6 shows the flux into the river as a function
of time.
[End of Example B.4]
24
20
16
^*
_>
o
.5 12
x
ZJ
I
J_^J'
I
4 a 12 16 tO 2< 28 32 36 40
tliul (1000 -a}
Figure B.6. Flux of pollutant into river for Example B.4.
B.4.5 Generalization of Instantaneous Models
Equations B-5 and B-27 are formulated only in terms of instantaneous releases. They can be
generalized for arbitrary releases by use of the convolution integral:
6 = htf(T)Q.(t-T)dT
(B-48)
where 0 is the solution at time t for the arbitrary release, 0;(t -1) is the solution at time
(t - t) for an instantaneous release at (t - t) = 0, and^t) is the source release rate at T in curies/s.
B-29
-------
Certain analytical solutions can be found to Equation B-48 for simple source release rate
functions. For example, Wilson and Miller (WIL78) develop the solution to Equation B-48 for a
continuous release in terms of the "well function." Most useful solutions to Equation B-48 use
numerical integration, generally involving a digital computer.
Several special precautions must be taken, however, to preserve computational accuracy, because
the terms within the integral of Equation B-48 can be very nearly zero over part of the integration
range. Computer programs for solving the equations in this section are described by Codell et.
al. (COD82). Program listings in BASIC and FORTRAN are given in this reference. An
alternative method for simulating a continuous source function is to present the continuous
source as a series of instantaneous ones. The analytical solutions are then linearly summed.
Complicated areal source terms can also be solved in an analogous fashion by representing the
source area by a series of point sources and linearly summing the solutions.
B.5 Simplified Analytical Methods for Minimum Dilutions
Simplified forms of the equations of Section B.4 have been developed for calculating the
minimum dilutions (i.e., maximum concentration) of volume Vr of a substance instantaneously
released from a point source into an aquifer.
B.5.1 Dilution at Downgradient Wells in Confined Aquifers for an Instantaneous Point Source
at the Surface
At some distance downgradient from a release at the surface of a confined aquifer, the
concentration can be considered to be mixed in the vertical direction. Close to the point of
release, or in an unconfined aquifer, the vertical dispersion will not be influenced by the vertical
boundaries of the aquifer. Between these regions, there is a region where the concentration
cannot be considered mixed, but the boundaries (top and bottom) affect the dispersion. The
degree of vertical mixing can be characterized in a confined aquifer of constant thickness and
uniform transport properties by the factor
' (B-49)
where
the vertical (transverse) dispersivity (ft)
the thickness of the aquifer (ft)
B-30
-------
x = the distance downgradient of the release (ft).
The factor (J) can be used to characterize the aquifer in three approximate regions:
(a) If (J) < 3.3, the release may be considered to be within 10% of being
vertically mixed in the aquifer;
(b) If (J) > 12, the release maybe considered to be within 10% of the
aquifer;
(c) If 3.3 < (J) < 12, the release is neither completely mixed nor
unaffected by the boundaries.
Different methods apply to each of the three regions.
Vertically Mixed Region ((J) < 3.3). For an instantaneous release at x = 0, the minimum dilution
corrected for decay directly downgradient of a source would be
DL = RF4nnexb±_exp(xt) , (B-50)
where
DL = minimum dilution = c0/c
Rf = retardation factor
ne = effective porosity
VT = volume of liquid source term (cm3)
aL,aT = dispersivities (cm) in the indicated direction
x = distance downgradient (cm)
b = aquifer thickness (cm)
t = travel time (y)
A = decay constant = In 2/t1/2(l/y).
The travel time, t, can be approximated as
V
pw
where V is the pore velocity defined by Equation B-18.
B-31
-------
Unmixed Region ((J) > 12). For an instantaneous release at x = 0 on the surface of the aquifer,
the minimum dilution of the surface of the aquifer directly downgradient from the source would
be determined from Equation B-52,
exp (Xt) , (B-52)
where CCL, CCT are dispersivities in the indicated direction and the other terms are as previously
defined.
Intermediate Region (3.3 < (J) < 12). For an instantaneous release at x = 0 on the surface of an
aquifer, the minimum dilution on the surface of the aquifer directly downgradient from the
source would be
£ 4nrz x£>A/a a exp (Xt)
(B-53)
where
and the other terms are as previously defined.
The function F((J>) is conveniently plotted in Figure B.7. It can be easily seen that for small
values of (J), F approaches the value of 1 .0, which yields the vertically mixed case. For large
values of (J), the slope of F is 1/2, and the unmixed case prevails. This method may be used for
any value of (J) that can be read on Figure B.5.
B-32
-------
Figure B.7. Mixing factor for confined aquifers.
B.5.2 Ground-Water/Surface Water Interface-Instantaneous Source
For an instantaneous release to the ground-water at x = 0, the minimum dilution in an
intercepting river, corrected for decay, can be determined from:
DL =
2RQnuTx
exp(Xt) ,
(B-55)
where
fi
V,
pw
flow rate of river (cmVs)
the longitudinal dispersivity of the aquifer (cm)
the volume of release (e.g., tank volume) (cm3)
pore velocity of ground water (cm/s).
Relatively simple equations can be used for estimating average concentration in ground water or
in surface water supplies contaminated by ground water (Equations B.56 and B.57).
B.5.3 Quantity of Released Radioactivity Crossing a Vertical Plane
In the case of ground-water flow to an intercepting river, the total quantity M (curies) of the
dissolved substance entering the river would be
B-33
-------
M = /; Fdt ' (B-56)
where F is the flux defined for either an instantaneous point or vertical plane source by Equation
B-45 or a horizontal area source by Equation B-47. Equation B-56 can be integrated graphically
or numerically and in some cases may have an analytical solution.
If dispersion is relatively small (e.g., ccx « C), the following approximation may be used:
M= MQe~xt curies, (B-57)
where M0 is the quantity of radioactivity released instantaneously from the source (curies), t is the
travel time (y), and A is the decay coefficient (1/y).
If the substance is being released from the source at a rate proportional to the quantity remaining
(e.g., an exponentially decaying source term),
M= M° e"Xt ' (B'58)
where A' is the release rate from the source (1/y), andM0 is the initial quantity of material in the
source term (curies).
B.5.4 Direct Ground- Water Usage
The U.S. Nuclear Regulatory Commission developed a model for calculating the quantity of a
radionuclide ingested by a population using the contaminated ground water (NRC78). Ground-
water usage was considered to be spatially continuous instead of being from discrete well points.
The total amount of the released radionuclide ingested by the population is
I = °° °° °° cQ dxdydtf
JD j-~ j-~ g
where
B-34
-------
I
c
Q*
the ultimate number of curies ingested from the release
the ground-water concentration (Ci/L)
the ground-water withdrawal rate for drinking water purposes
(m3/d/m2).
If all usage is restricted to a downgradient distance C and beyond from the release point, Equation
B-59 may be integrated in closed form to give
I =
MQQg ^
2neRFb °"P
«%,
2Dx
/ \
RFP (X+y)
( D* J
1/2
(B-60)
where
Y =
pw'
M0 is the total quantity of the radionuclide discharged to the point source, and the other terms are
as previously defined.
If usage of the ground water is restricted between two downgradient distances, ^ and £2, the
curies ingested would be defined as:
where IftJ and/(1>^) are evaluations of Equation B-60 for fj and H2 respectively.
Example B.5. The use of several of the simpler analytical models in Section B.4 will be
demonstrated by way of a hypothetical example:
Leakage into the ground water rapidly empties a 1000-ft3 tank containing 4000 |j,Ci/mL of
3H, 2000 jiCi/mL of 90Sr, and 3000 jiCi/mL of 137Cs at a radioactive waste site. The site is
50 ft above the mean level and 3000 ft upgradient from a river that has representative low
flow of 5000 ft3/s and is the sink for all surficial ground water in the area. Two shallow
wells are located 400 and 2500 ft directly downgradient from the site of the spill. Ground
water exists in a homogeneous alluvial sand layer 100 ft thick under water table
B-35
-------
conditions. Dispersivities for the sand have been determined in the near field from
single-well tracer tests to be 0.5 ft for CCT and 1.0 ft for OCL. The bulk density pb of the sand
is 2.6 g/cm3. Its total porosity n and effective porosity ne are 0.4 and 0.25, respectively.
The permeability K is 0.02 cm/s. Distribution coefficients Kd for the sand have been
determined to be 0, 2.0, and 20.0 mL/g for dilute solutions of 3H, 90Sr and 137Cs,
respectively. Using this information, calculate the following:
(a) the maximum concentrations of the radioactive components in the river,
(b) the maximum concentrations of the components in the near well,
(c) the maximum concentrations of the components in the far well, and
(d) the total quantity of each radionuclide escaping to the river.
Solution
(a) If it is assumed that the source is released over a short period, Equation B-55 for
instantaneous releases may be used to calculate the maximum river concentrations
of 3H, 90Sr, and 137Cs. First determine the pore velocity Vpw from Equation B-18
and the effective porosity ne:
K
v - - - -
= - 0.0167 ;
ne ne
The gradient
_AH = - 50 ft
Ax 3000 ft
therefore,
- 2 x 10"2 cm/sx - 0.0167 86,000s/d _ ->nof.,,
v x J./D ft/a
pw 0.25 30.48 cm/ft
The retardation factors for 3H, 90Sr and 137Cs can be determined from
Equation B-22:
B-36
-------
3U D 1 j. ^ ' U
H ^ " X ^4
*>Sr R = i + 111 x 2.0 - 14 ,
F
0.4
137Cs £P = 1 + l x 20 = 131 .
F 0.4
The travel times for the three components are calculated by Equation B-47:
. **_, . 3000 ft x 1 x _^_ _
Vp 3.78 ft/d 365 d
oc *. 3000 ft x 14 y _- .
Sr t = x i = 30.4 y,
3.78 ft d 365 d
-Cs t - 3000 ft x 131 x _^_ _ 284_
3.78 ft/d 365 d
The half-lives of 3H, 90Sr, and 137Cs are 12.3 y, 29 y, and 30.1 y, respectively. The
decay-corrected minimum dilutions in the river are found by applying Equation B-
55:
5000 ft3
3H
2 x 1.0 x _^^_^^_ x^nx l.O ft x 3000 ft
DL = £
3.78 ft/d x 1000 ft3 x d
86,400s
f In 2 . . ]
x exp - x 2.17y
P y
= 2.51 x 107 ,
B-37
-------
5000 ft3
9°Sr
DL
2 x 14 x ^^_^^_ xJnx 1.0 ft x 3000 ft
3.78 ft/d x 1000 ft3
86,400s
I In 2 on /, I
x exp - x 30. 4y
y
= 6.42 x 10
5000 ft3
137Cs
2 x 131 x _^^_^^_ x^nx l.Oftx 3000ft
DL =
3.78 ft/d x 1000 ft3 d
86,400s
x exp I ln2 x 284. 8y|
P Y
= 2.05 x 1012 .
The peak concentrations in the river are determined by dividing the tank
concentrations by the dilution factors:
c(3H) = 4000 uCi/mL /2.51 xlO7 = 1 . 59 xlQ'4 yCi / mL ,
c(90Sr) = 2000uCi/mL/6.42 xlO8 = 3 . 12 xlQ'5 yd / mL ,
c(137Cs) = 3000uCi/mL/2.05 xlO12 = 1 . 46 xlQ'9 yCi / mL .
(b) Minimum dilution in well (400 ft downgradient).
B-38
-------
First determine whether or not the thickness of the aquifer would affect the results
by calculating the factor (J) from Equation B-50:
* (100 ^)2 = 50.
0.5 ft* 400 ft
Therefore, in this region the release will be relatively unaffected by the thickness
of the aquifer, and Equation B-53 applies.
The travel times are estimated using the retardation factors and pore velocity
calculated above:
3H = 400 ft x 1 x _^_ _ Q_
3.78 ft/d 365 d
oc *. 4°0 ft x 14 y ......
Sr t = x i = 4.06y,
3.78 ft d 365 d
-Cs t - 400 ft x 131 x _^_ __
3.78 ft/d 365 d
B-39
-------
Applying equation B.52:
3H 0.25 x l x (4nx400 ft)3/2Jl ft* 0.5 ft* 0.5 ft
L
2 xiooo ft3
x exp [ ln2 x 0 . 2 9 y I
Y
= 22.6 ,
= 0.25 x 14 x (4nx400 ft)372/I f t x 0 . 5 f t x 0 . 5 ft
L 2 xlOOO ft3
9°Sr ( in 2
x exp x 4.06y
= 343.6 ,
= 0.25 x 131 x (4nx400 ft)32l f t x Q . 5 f t x Q . 5 ft
2 xlOOO ft3
I In 2 ,. )
x exp x 38y
I 30.ly )
= 6999.9 .
B-40
-------
The peak well concentrations are therefore 177 |j,Ci/mL for 3H, 5.8 |j,Ci/mL for 90Sr, and 0.43
for 137Cs.
(c) Well 2500 ft downgradient. Calculate (J) for this region from Equation B-49:
(100 ft)2
0.5 ft x 2000 ft
.O.
Therefore, this well is in the intermediate region, and Equation B-53 applies. The
factor F ((J>) can be read from Figure B.6 to be 1.6. Travel times for each
component calculated from Equation B-46 are
= 2500 ft x 1 x _^_ __
3.78 ft/d 365 d
» 2500 ft x 14
Sr t - x __ __ 25
3.78 ft/d 365 d
?Cs = 2500 ft x 131 x _^_ __ 237.4y.
3.78 ft/d 365 d
Applying Equation B-53:
3H = 1 x 4nx Q.25^/0.5 ft x l.Q ft x 2500 ft x 100 ft
x 1000ft3xl.6
( ln2 i on 1
x exp x 1. 81 y
Pll2-3y yj
= 271.8 ,
B-41
-------
9°
= 14 x 4nx 0.25^/0.5 ft x l.Q ft x 2500 f t x 100 ft
L 1000ft3xl.6
Sr / in 2 ^
I 29y
= 8874 ,
137,
Cs
131 x 4nx 0.25^0.5 ft x l.Q ft x 2500 f t x 100
1000 ft3 x 1.6
x exp x 237.4 y
= 1.07 x 107 .
The peak well concentrations are therefore 14.7 |iCi/mL for 3H, 0.32 |j,Ci/mL for
90Sr, and 2.8 x 10'4 nCi/mL for 137Cs.
(d) Quantity M of each radionuclide eventually reaching river.
Equation B-57 applies to this case because CCL «C (i.e., 1 ft vs 1000 ft). Travel
times are estimated in part (a) above. The quantity of each radionuclide initially
in the tank is the concentration multiplied by the volume. Therefore,
3H M = 4000 uCi/mL x 1000 ft3 x 28,300 mi/ ft3
xexp
12.3 y
= 1.002 x 105 Ci ,
B-42
-------
90c
sr M = 2000 ]iCi/mL x 1000 ft3 x 28,300 mL/ff
xexp ix30.4y| x 10"6Ci/uCi
I 29 y I
= 27,370 Ci ,
Cs M = 3000 uCi/mL x 1000 ft3 x 28,300 mL/ft3
xexpf ~ ln 2 x 284. 7y| xlO~6Ci/iaCi
^ 30.1 y )
= 120.7 Ci .
B-43
-------
B.6 References
ABR70 Abramowitz, M., and I. Stegun, eds., November 1970. Handbook of
Mathematical Functions, Applied Mathematics Series 55, Nat. Bur. Standards.
BOU78 Bouwer, H. 1978. Groundwater Hydrology. New York: McGraw-Hill Pub. Co.
BOW79 Bowers, J.F., et al. 1979. Industrial source complex (ISC) dispersion model
user's guide, volumes I and II. Washington, B.C.: U.S. Environmental Protection
Agency. EPA 450/4-79-030. As reviewed in: Bonazountas, M., Fiksel, J., et al.
1982. Environmental mathematical pollutant fate modeling handbook/catalog.
Draft. Washington, B.C.: U.S. Environmental Protection Agency, Office of
Policy and Resource Management. Contract No. 68-01-5146. As cited in
EPA88b.
BUR80 Burkholder, H.C., and E.L.J. Rosinger, June 1980. "A Model for the Transport of
Radionuclides and Their Decay Products Through Geologic Media," Nucl.
Tecnol. 49. 150-58.
CLA78 Clapp, R.B., and G.M. Hornberger. 1978. "Empercial Equations for Some Soil
Hydraulic Properties." Water Resour. Res. 14(4): 601-604.
COD82 Codell, R.B., K.T. Key, G. Whelan, 1982. A collection of mathematical models
for dispersion in surface water and ground water. Washington, B.C.: U.S.
Nuclear Regulatory Commission. NUREG/CR-0868. As cited in EPA88b.
COD83 Codell, R.B., and J. Duguid, 1983. Radiological Assessment; A Textbook on
Environmental Dose Analysis, NUREG/CR-3332, U.S. Nuclear Regulatory
Commission.
ENF82 Enfield, C.G., R.F. Carsel, S.Z. Cohen, T. Phan, D.M. Walters, 1982.
Approximating pollutant transport to ground water. Ground Water 20(6): 711-
722. As cited in EPA88b.
FRE79 Freeze, R.A. and J. Cherry, 1979. Groundwater. Prentice-Hall, Englewood Cliffs,
N.J.
GEL85 Gelhar, L.W., A. Montoglou, C. Welty, and K.R. Rehfeldt, 1985. A review of
field-scale physical solute transport processes in saturated and unsaturated porous
media. Palo Alto, CA: Electric Power Research Institute. EPRIEA-190. As
cited in EPA88b.
RAW82
Rawls, W.J., D.L. Brakensiek, K.E. Saxton, 1982. Estimation of soil water
B-44
-------
properties. Trans. Am. Soc. Agri. Eng. 25(5):1316-1320 and 1328. As cited in
EPA88b.
RAW86 Rawls, W.J., 1986. Computer printouts from the soils data base, dated August 28,
1986. From WJ Rawls, Beltsville, MD: Beltsville Agricultural Research Center.
As cited in EPA88b.
STE78 Steingiser, S., S.P. Nemphos, M. Salame, 1978. Barrier polymers. In: Kirk-
Othmer encyclopedia of chemical technology, 3rd ed. New York: John Wiley
and Sons. As cited in EPA88b.
WIL78 Wilson, J.L. and P.J. Miller, April 1978. "Two Dimensional Plume in Uniform
Ground-Water Flow." J. Hydraulics Division, ASCE 104 (HY4), 503-14.
B-45
-------
APPENDIX C
DEFAULT PARAMETER VALUES
-------
Contents - Appendix C
Tables
C-l. Representative Values for Saturated Moisture Contents and Field Capacities of Various
Soil Types C-l
C-2. Representative Values of Hydraulic Parameters C-2
C-3. Representative Values of Saturated Hydraulic Conductivity C-3
C-4. Saturated Hydraulic Conductivity Ranges for Selected Rock and Soil Types C-4
C-5. Distribution Coefficient (Kd) of Selected Radionuclides Sorbed by Clays and Cation
Exchange Capacity for Selected Clay Minerals C-5
C-6. Typical Values of Hydraulic Conductivity of Porous Materials C-6
C-7. Suggested Value for Cet Relating Evaporation from a U.S. Class A Pan to
Evapotranspiration from 8 to 15-cm Tall, Well-Watered Grass Turf C-7
C-8. Crop Coefficients for Estimating Evapotranspiration C-8
C-9. Typical Values of Porosity of Aquifer Materials C-9
C-10. Typical Values of Effective Porosity (or Specific Yield) of Aquifer Materials C-10
C-11. Dispersivity Values aL and ccr Obtained Directly through Measurements of Tracer
Breakthrough Curves in Groundwater Solute Transport C-l 1
C-i
-------
Table C-l. Representative Values for Saturated Moisture Contents and Field
Capacities of Various Soil Types
Number of soils
Sand
Loamy sand
Sandy loam
Loam
Silt loam
Sandy clay loam
Clay loam
Silty clay loam
Sandy clay
Silty clay
Clay
762
338
666
383
1,206
498
366
689
45
127
291
Saturated moisture content
(9s)a
Mean ± 1 standard deviation
0.437
0.437
0.453
0.463
0.501
0.398
0.464
0.471
0.430
0.479
0.475
0.347 - 0.500
0.368-0.506
0.351-0.555
0.375-0.551
0.420 - 0.582
0.332 - 0.464
0.409-0.519
0.418-0.524
0.370 - 0.490
0.425 - 0.533
0.427 - 0.523
a From total soil porosity measurements compiled by Rawls et al. (1982) from numerous
sources.
Source: Rawls, W.J.,
Trans. Am. Soc. Asri.
D.L. Brakensiek, K.E.
. Eng. 25(5): 13 16-1320
Saxton, 1982.
and 1328. As
Estimation of soil water props
cited in EPA88b.
C-l
-------
Table C-2. Representative Values of Hydraulic Parameters
Soil texture
Sand
Loamy sand
Sandy loam
Silt loam
Loam
Sandy clay loam
Silt clay loam
Clay loam
Sandy clay
Silt clay
Clay
No. of
soils3
13
30
204
384
125
80
147
262
19
441
140
bb
4.05(1.78)d
4.38(1.47)
4.90(1.75)
5.30(1.87)
5.39(1.87)
7.12(2.43)
7.75 (2.77)
8.52 (3.44)
10.40(1.64)
10.40 (4.45)
11.40(3.70)
1
2b+3
0.090
0.085
0.080
0.074
0.073
0.058
0.054
0.050
0.042
0.042
0.039
e;
0.395 (0.056)
0.410 (0.068)
0.435 (0.086)
0.485 (0.059)
0.451 (0.078)
0.420 (0.059)
0.477 (0.057)
0.476 (0.053)
0.426 (0.057)
0.492 (0.064)
0.482 (0.050)
a Number of individual soil samples included in data compiled by Clapp and Hornberger
(1978).
b Empirical parameter relating soil matrix potential and moisture content; shown to be strongly
dependent on soil texture.
c Volumetric soil moisture content (volume of water per volume of soil).
d Standard deviation in parentheses.
Source: Adapted from: Clapp, R.B., and G.M. Hornberger. 1978. "Empercial Equations for
Some Soil Hydraulic Properties." Water Resour. Res. 14(4): 601-604.
C-2
-------
Table C-3. Representative Values of Saturated Hydraulic Conductivity
Soil texture
Sand
Loamy sand
Sandy loam
Loam
Silt loam
Sandy clay loam
Silt clay loam
Clay loam
Sandy clay
Silt clay
Clay
Number of soils"
762
338
666
383
1,206
498
366
689
45
127
291
Hydraulic i
(Ks; cm
5.8x
1.7 x
7.2 x
3.7 x
1.9 x
1.2 x
4.2 x
6.4 x
3.3 x
2.5 x
1.7 x
cone
/sec"
io-3
io-3
io-4
io-4
io-4
io-4
io-5
io-5
io-5
io-5
io-5
a Number of individual soil samples included in data compiled by Rawls et al. (1982).
b Predicted values based on compiled soil properties.
Source: Adapted from: Rawls, W.J., 1986. Computer printouts from the soils data base, dated
August 28, 1986. From WJ Rawls, Beltsville, MD: Beltsville Agricultural Research Center. As
cited in EPA88b.
C-3
-------
Table C-4. Saturated Hydraulic Conductivity Ranges for Selected Rock and Soil Types
Saturated hydraulic
conductivity (cm/sec)
Soils
Unweathered marine clay
Glacial till
Silt, loess
Silty sand
Clean sand
Gravel
Rocks
Unfractured metamorphic and igneous rock
Shale
Sandstone
Limestone and dolomite
Fractured igneous and metamorphic rock
Permeable basalt
Karst limestone
Source: Adapted from: Freeze, R.A. and J. Cherry, 1979.
5 x ID'11
io-7
1Q-io
io-7
io-5
io-4
io-1
io-8
5 x IQ-12
io-8
5 x 1Q-8
ID'6
io-5
io-4
Groundwater.
-
- io-4
- io-3
- io-1
- 1
- IO2
- io-2
- io-7
- 5 x 1Q-4
- 5 x 1Q-4
- io-2
- 1
- 1
Prentice-Hall,
Englewood Cliffs, N.J.
C-4
-------
Table C-5. Distribution Coefficient (Kd) of Selected Radionuclides Sorbed by Clays and Cation
Exchange Capacity for Selected Clay Minerals
Material
Illite
Kaolinite
Montmorillonite
Vermiculite
Average
Cesium
98
68
50
100
Percent Activity Sorbed(a)
Cobalt
86
61
62
99
Strontium-85
27
66
67
97
Zirconium-
Niobium-85
94
94
35
Cesium
180,000
2,200
1,000
12,000
Average K^
Cobalt
6,400
3,100
1,700
4,700
Strontium-85
370
4,000
2,100
1,800
Zirconium-
Niobium- 8 5
47,500
56,000
540
(a) Average percent and average K^ of radionuclides sorbed by clays in distilled water at pH 6 over a period of 7 days (vermiculite, 8
days).
(b)
Reported in millequivalents per lOOg of soil [Source: GRI68].
Source: Derived from:
Webster, G.B. et al., 1976. "Radionuclide Migration from Low-Level Wastes: A Generic Overview," in M.W. Carter et al. (editors),
Management of Low-Level Radioactive Waste. Pergamon Press, New York, NY, pp. 1041-1072.
Grim, 1968.
-------
Table C-6. Typical Values of Hydraulic Conductivity of Porous Materials
Material
Number of
analyses
Igneous rocks
Weathered granite 7
Weathered gabbro 4
Basalt 93
Sedimentary materials
Sandstone (fine) 20
Siltstone 8
Sand (fine) 159
Sand (medium) 255
Sand (coarse) 158
Gravel 40
Silt 39
Clay 19
Metamorphic rocks
Schist 17
Arithmetic
Range
(cm/s)
(3.3-52X10'4
(0.5-3.8) X10-4
(0.2-4250) X1Q-8
(0.5-2270) X1Q-6
(0.1-142) X10-8
(0.2-189)X1Q-4
(0.9-567) X10-4
(0.3-6610)X1Q-4
(0.3-31.2) X10-1
(0.09-7090) X10-7
(0.1-47)X1Q-8
(0.002-1130) X 10
1-6
mean
(cm/s)
1.65X10'3
1.89X10-4
9.45 X1Q-6
3.31 X1Q-4
1.9X10-7
2.88 X1Q-3
1.42X10-2
5.20X10'2
4.03 X ID'1
2.83 X ID'5
9X10'8
1.9X10"
Source: McWhorter, D.B., and D.K. Sunada, 1977. Ground-Water Hydrology and
Hydraulics, Water Resources Publications, Fort Collins, Colo.
C-6
-------
Table C-7. Suggested Value for Cet Relating Evaporation from a U.S. Class A Pan to Evapotranspiration
from 8 to 15-cm Tall, Well-Watered Grass Turf
Pan surrounded by a
short green crop
Wind
Light
< 170 km/day
Moderate
170-425 km/day
Strong
425-700 km/day
Very strong
>700 km/day
Pan surrounded bv a dry surface ground
Upwind
fetch of
crop
(m from pan)
0.75
10
100
1000
0
10
100
1000
0
10
100
1000
0
10
100
1000
Average regional
relative humiditv. %*
20-40
0
0.65
0.7
0.7
0.5
0.6
0.65
0.7
0.45
0.55
0.6
0.65
0.4
0.45
0.5
0.55
* Mean of maximum and minimum relative
Source: Jenson,
Civil Engineers.
M.E., ed., 1973.
As presented by
40-70
0.7
0.75
0.8
0.85
0.6
0.7
0.75
0.8
0.5
0.6
0.65
0.7
0.45
0.55
0.6
0.6
>70
0.8
0.85
0.85
0.85
0.65
0.75
0.8
0.8
0.6
0.65
0.7
0.75
0.5
0.6
0.65
0.65
Upwind
fetch of dry Average regional
fallow relative humiditv. %*
(m from pan) 20-40
0.85
10
100
1000
0
10
100
1000
0
10
100
1000
0
10
100
1000
0.6
0.55
0.5
0.65
0.55
0.5
0.45
0.6
0.5
0.45
0.4
0.5
0.45
0.4
0.3
40-70 >70
0 0.55 (
0.7
0.65
0.6
0.75
0.65
0.6
0.55
0.65
0.55
0.5
0.45
0.6
0.5
0.45
0.4
0.8
0.75
0.7
0.8
0.7
0.65
0.6
0.7
0.65
0.6
0.55
0.65
0.55
0.5
0.45
humidities.
Consumptive use of water and irrigation water requirements. New York
Enfield et al. 1982. Approximating pollutant transport to ground water.
, NY: American Soci
Ground Water 20(6):
0.65
722. As cited in EPA88b.
Enfield, C.G., R.F. Carsel, S.Z. Cohen, T. Phan, D.M. Walters, 1982. Approximating pollutant transport to ground water. Ground
Water 20(6): 711-722. As cited in EPA88b.
-------
Table C-8. Crop Coefficients for Estimating Evapotranspiration
Crop
Alfalfa
Potatoes
Small grains
Sugar beets
Period
April 1 - October 10
May 10 - September 15
April 1 - July 20
April 10 - October 15
Coefficient
(Cveg)
0.87
0.65
0.6
0.6
Source: Jenson, M.E., ed., 1973. Consumptive use of water and irrigation water requirements.
New York, NY: American Society of Civil Engineers. As presented by Enfield et al. 1982.
Approximating pollutant transport to ground water. Ground Water 20(6): 711-722. As cited in
EPA88b.
Enfield, C.G., R.F. Carsel, S.Z. Cohen, T. Phan, D.M. Walters, 1982. Approximating pollutant
transport to ground water. Ground Water 20(6): 711-722. As cited in EPA88b.
-------
Table C-9. Typical Values of Porosity of Aquifer Materials
Aquifer material
Number of
analyses
Range
Igneous Rocks
Weathered granite
Weathered gabbro
Basalt
4
94
0.34-0.57
0.42-0.45
0.03-0.35
Arithmetic
mean
0.45
0.43
0.43
0.17
Sedimentary Materials
Sandstone 65
Siltstone 7
Sand (fine) 245
Sand (coarse) 26
Gravel (fine) 0.2380.38
Gravel (coarse) 15
Silt 281
Clay 74
Limestone 74
0.14-0.49
0.21-0.41
0.25-0.53
0.31-0.46
0.24-0.36
0.34-0.51
0.34-0.57
0.07-0.56
0.34
0.35
0.43
0.39
0.34
0.28
0.45
0.42
0.30
Metamorphic Rocks
Schist
18
0.04-0.49
0.38
Source: McWhorter, D.B., and D.K. Sunada, 1977. Ground-Water Hydrology and
Hydraulics, Water Resources Publications, Fort Collins, Colo. Reprinted with
permission.
C-9
-------
Table C-10. Typical Values of Effective Porosity (or Specific Yield) of Aquifer Materials
Aquifer material
Sedimentary Materials
Sandstone (fine)
Sandstone (medium)
Siltstone
Sand (fine)
Sand (medium)
Sand (coarse)
Gravel (fine) 33
Gravel (medium)
Gravel (coarse)
Silt
Clay
Limestone
Wind-Laid Materials
Loess
Eolian Sand
Tuff
Metamorphic Rock
Schist
Number of
analyses
47
10
13
287
297
143
0.13-0.40
13
9
299
27
32
5
14
90
11
Range
0.02-040
0.12-0.41
0.01-0.33
0.01-0.46
0.16-0.46
0.18-0.43
0.28
0.17-0.44
0.13-0.25
0.01-0.39
0.01-0.18
~0-0.36
0.14-0.22
0.32-0.47
0.02-0.47
0.22-0.33
Arithmetic
mean
0.21
0.27
0.12
0.33
0.32
0.30
0.24
0.21
0.20
0.06
0.14
0.18
0.38
0.21
Source: McWhorter, D.B., and D.K. Sunada, 1977. Ground-Water Hydrology and
Hydraulics, Water Resources Publications, Fort Collins, Colo. Reprinted with
permission.
0.26
C-10
-------
Table C-l 1. Dispersivity Values oc£ and ccr Obtained Directly through Measurements of Tracer Breakthrough Curves in
Groundwater Solute Transport
Setting
Chalk River, Ontario
alluvial aquifer
Chalk River, strata of
high velocity
Alluvial aquifer
Alluvial, strata of
high velocity
Lyons, France
alluvial aquifer
Lyons (full aquifer)
Lyons (full aquifer)
Lyons (full aquifer)
Lyons (full aquifer)
Lyons (full aquifer)
Alsace, France
alluvial sediments
Carlsbad, N. Mex.
fractured dolomite
Savannah River, S.C.
fractured schistgneiss
a£ ccr Axa Ub
(m) (m) (m) (m/d)
0.034
0.034-0.1
0.5
0.1
0.1-0.5
5
12.0 31.1-14 7.2
8 0.015-1 9.6
5 0.145-14.5 13
7 0.009-1 9
12 4
38.1 38.1 0.15
134.1 538 0.4
Method
Single-well tracer
test
Single-well
Two -well
Two -well
Single-well
Single-well
Single-well test
with resistivity
Single-well test
with resistivity
Single-well test
with resistivity
Single-well test
with resistivity
Environmental tracer
Two-well tracer
Two-well
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Table C-11 (Continued)
Setting
Barstow, Calif.
alluvial sediments
Dorset, England
chalk (fractured)
(intact)
Berkeley, Calif.
sand/gravel
Mississippi limestone
NTS, carbonate
aquifer
Pensacola, Fla.
limestone
«£
(m)
15.2
3.1
1.0
2-3
11.6
15
10
ccr Axa
(m) (m)
6.4
8
8
8
312
Ub
(m/d)
311-1382
0.6
Method
Two-well
Two-well
Two-well
Multiwell trace test
Single-well
Two-well tracer
Two-well
aAx = distance between wells in two-well test.
bU = groundwater seepage velocity.
Source: Evenson, D.E., and M.D. Dettinger, 1980. Dispersive Processes in Models of Regional Radionuclide Migration,
University of California, Lawrence Livermore Laboratory, Livermore.
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