Modeling and Low-Level
Waste Management:
An Interagency Workshop
December 1-4, ' D
Denver, Color?
Sponsored by
U.S. Department of Energy
U.S. Environmental Protection Agency
U.S. Geological Survey
U.S. Nuclear Regulatory Commission
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Modeling and Low-Level
Waste Management:
An Interagency Workshop
December 1-4. 1980
Denver. Colorado
Compiled by
Craig A. Little
Health and Safety Research Division
and
Leroy E. Stratton
Environmental Sciences Division
Oak Ridge National Laboratory
Oak Ridge, Tennessee 37830
for the
INTERAGENCY LOW-LEVEL WASTE MODELING COMMITTEE:
Robert S. Lowrie (Chairman), Program Manager, Low-Level Waste Management Program,
Oak Ridge National Laboratory, Oak Ridge, Tennessee
Edward F. Hawkins, Low-Level Waste Licensing Branch, Division of Waste Management,
U.S. Nuclear Regulatory Commission, Washington, D.C.
Dewey E. Large, Manager, ORO Low-Level Radioactive Waste Management Program,
U.S. Department of Energy, Oak Ridge, Tennessee
G. Lewis Meyer, Criterion and Standards Division, Office of Radiation Programs,
U.S. Environmental Protection Agency, Washington, D.C.
John B. Robertson, Office of Radiohydrology,
U.S. Geological Survey, Reston, Virginia
Craig A. Little (Scientific Secretary), Health and Safety Research Division,
Oak Ridge National Laboratory, Oak Ridge, Tennessee
Workshop sponsored jointly by U.S. Environmental Protection Agency, U.S. Nuclear
Regulatory Commission, U.S. Geological Survey, and the Oak Ridge Operations Low-Level
Radioactive Waste Management Program, U.S. Department of Energy, under contract W-7405-
eng-26 with the Union Carbide Corporation.
Date Published: June 1981
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Preface
This workshop was organized by the Interagency Low-Level Waste Modeling
Committee and was held December 1-4,1980 in Denver, Colorado. Committee membership
is listed on the title page. Each of the four federal agencies represented on the committee
contributed equal funding to support the meeting.
The paramount objective was to foster cooperation between the agencies having
various responsibilities and involvements in modeling for low-level waste applications.
Another aim of the workshop was to begin a dialogue between model users, such as
regulators and assessors, and modelers. These Proceedings are testimony that progress has
been made toward attainment of these goals.
The meeting had three distinct sessions. First, each agency presented its point of view
concerning modeling and the need for models in low-level radioactive waste applications.
Second, a larger group of more technical papers was presented by persons actively involved
in model development or applications. Each of these papers came from one of the
sponsoring agencies or from an agency contractor. Last, four workshops—two running
concurrently—were held to attempt to reach a consensus among participants regarding
numerous waste modeling topics. Each of the three-sections is identified in the Contents.
We are grateful to many people for their assistance in organizing the meeting and
compiling the Proceedings: speakers and authors who promptly delivered their
manuscripts; workshop chairmen and secretaries who recorded and summarized their
sessions; Bonnie Reesor of the ORNL Conference Office; and particularly Thelma Patton
and Brenda Baber of the ORNL Low-Level Waste Management Programs Office. Thanks.
Craig A. Little
Secretary for Interagency Low-Level
Waste Modeling Committee
June 1981
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Contents
Page
Preface iii
Agency Papers
USDOE Needs in Modeling for Low-Level Radioactive
Waste Management 3
Dewey E. Large
Modeling and Analyses to Support EPA's Low-Level
Radioactive Waste Standards 9
G. Lewis Meyer
Modeling in Low-Level Radioactive Waste Management
from the U.S. Geological Survey Perspective 21
John B. Robertson
Low-Level Radioactive Waste Modeling Needs from the
USNRC Perspective 31
R. Dale Smith and Edward F. Hawkins
Technical Papers
Modeling Contaminant Transport in Porous Media in
Relation to Nuclear-Waste Disposal: A Review 43
David B. Grove and Kenneth L. Kipp
An Optimum Model to Predict Radionuclide Transport in
an Aquifer for the Application to Health Effects Evaluations 65
Cheng Y. Hung
Modeling of Water and Solute Transport Under Variably
Saturated Conditions: State of the Art 81
E. G. Lappala
Verification of Partially Saturated Zone Moisture
Migration: Model and Field Facilities 139
S. J. Phillips and T. L. Jones
Radionuclide Transport Under Changing Soil
Salinity Conditions 155
T. L. Jones, G. W. Gee, and P. J. Wierenga
On the Computation of the Velocity Field and Mass
Balance in the Finite-Element Modeling of Groundwater Flow 165
Gour-Tsyh Yeh
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Fracture Flow Modeling for Low-Level Nuclear Waste Disposal 187
Brian Y. Kanehiro, Varuthamadhina Guvanasen,
Charles R. Wilson, and Paul A. Witherspoon
A Conceptual Analysis of the Groundwater Flow System
at the Maxey Flats Radioactive Waste Burial Site,
Fleming County, Kentucky 197
D. W. Pollock and H. H. Zehner
Computer Simulation of Groundwater Flow at a Commercial
Radioactive-Waste Landfill Near West Valley, Cattaraugus
County, New York 215
David E. Prudic
State-of-the-Art in Modeling Solute and Sediment
Transport in Rivers 249
William W. Sayre
An Overview of Resuspension Models: Application to
Low-Level Waste Management 273
J. W. Healy
An Overview of EPA's Health-Risk Assessment Model for
the Shallow Land Disposal of LLW 283
G. Lewis Meyer and Cheng Y. Hung
Risk Assessment and Radioactive Waste Management 303
Jerry J. Cohen and Craig F. Smith
Modeling in the Determination of Use Conditions for a
Decommissioned Shallow-Land Burial Site 315
E. S. Murphy
A Systems Model for Evaluating Population Dose from
Low-Level Waste Activities 327
J. A. Stoddard and D. H. Lester
Models and Criteria for LLW Disposal Performance 333
Craig F. Smith and Jerry J. Cohen
Workshop Summaries
Release Mechanisms Workshop 343
Les Dole and David E. Fields
Overall Systems Modeling Workshop 351
James Steger and Leroy E. Stratton
Environmental Transport Parameters and Processes Workshop 357
Edward L. Albenesius and Craig A. Little
Model Verification/Validation Workshop 373
T. L. Jones and Donald Jacobs
List of Attendees 379
VI
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Agency Papers
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USDOE NEEDS IN MODELING FOR LOW-LEVEL RADIOACTIVE WASTE MANAGEMENT
Dewey E. Large
National Low-Level Waste Management Program
U.S. Department of Energy
Oak Ridge, Tennessee 37830
Greetings from John Peel who is Manager of the National
Low-Level Waste Management Program (NLLWMP) for the Department of
Energy Idaho Operations. Speaking for him and as Associate Program
Manager of the NLLWMP for Technology Development for Oak Ridge
Operations, we will give you this Program's perspective of the DOE's
requirements for modeling in Low-Level Radioactive Waste Management.
On the agenda, this presentation is entitled, "USDOE Needs in
Modeling for Low-Level Waste Management." At the outset, we want to
emphasize that the Department of Energy does not need additional
models as entities. We do need reliable and credible tools for
decision making which involve identification, selection, adaptation,
and implementation of existing models. This is an important
distinction and is the basis for the Low-Level Waste Management
Program's efforts in environmental modeling. Recently, the Program
issued a strategy document which endorses states being allowed to
assume and implement their responsibilities for managing their
respective low-level wastes; it identifies areas of assistance that
federal agencies can provide to states. In summary, the states must
decide upon a course of action regarding siting and selection of
disposal sites; making that decision often requires and is expected
to use environmental models as decision-making tools.
We are not being trite when we emphasize that an environmental
model is only a tool that can be used in a decision making process
and should not be used as a replacement for management judgement.
The value of a model is to focus complex technical considerations to
provide management with the ability to examine various options,
compare impacts, and to explore the mitigative effects of
alternatives. If and when it becomes necessary, models are used to
predict events and to aid in resolving the "What if?" questions.
The reliability for prediction is, of course, a function of the
model itself, but as importantly, the accuracy of the input data.
Because of the inherent inaccuracies and uncertainties of data, or
simplifying assumptions used when data do not exist, the reliability
of model prediction will always be subject to question. For this
reason, decision making cannot and should not be based solely on
model prediction.
In July, many of you attended a meeting on Low-Level Waste
Modeling held in Bethesda, Maryland. At that meeting, several key
issues common to all modeling groups were identified. Tonight, let
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us examine some conclusions reached at that meeting and how the
Program intends to proceed in its modeling efforts. We will also
present some issues which we believe can only be resolved through
interagency cooperation.
An important conclusion at our July meeting was that many
models lack credibility. Consequently, a "Credibility Gap" does
exist. This is partly due to ineffective use of models; but, as
importantly, it is due to a lack of understanding by model
developers of what is required for decision making. Are models
really telling us what we need to know? Models are used to predict,
but results are often presented as absolute numbers. When looking
at those numbers, how often have you seen model results that attempt
questionable accuracy and predict one "FEMTO-REM" per billion
population? These numbers do not deal with a real world situation;
and, based on cause-effect relationships, such a degree of precision
is neither necessary nor desirable.
Clearly, there is a need for model developers to state what
their models can do and provide some bounds or limitations on model
results. Concurrently, the user must be educated concerning the
real function of that model and not attempt to push it beyond the
limits of credibility. This brings up another point concerning
credibility. How often have we seen "worst-case analysis" end up as
"absolute truth"? Decision making is an interactive process between
government and public. A report analyzing some worst case scenario
may be so prohibitive that reasonable alternatives may be ignored.
As our colleague Jerry Cohen says, "Give me a worst case and I'll
develop yet a worse one." We must reemphasize the use of
engineering and management judgement in defining model scenarios.
Our principal short term goal is to provide a siting model by
September 1981. Until this task is completed, the program will not
be able to identify and support other areas of model development
work. We have neither the time nor the funds to conduct extensive
new model development, and we do not believe that such effort and
expense is necessary since many adequate models already exist.
The siting model is needed by decision makers to examine the
adequacy of a site, explore various alternatives, and develop
preliminary operating and design criteria.
The basic guidance for our model development has been that the
model should:
1. Estimate health effects and maximum dose to any individual for
the population at risk;
2. Estimate the health effects with reasonable accuracy;
3. Estimate the health effects within a reasonable cost of
computer execution;
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4. Be as simple as possible while giving adequate detail for
standards development; and
5. Be comprised from existing sub-models; when adequate
sub-models are available, thereby reducing development time
and costs.
Our approach to model development has been to:
1. Review existing models;
2. Select viable models;
3. Utilize viable sub-models from selected models; and
4. Fill in gaps with new sub-models.
Our consideration and treatment of models include:
1. Verification and validation
The credibility of a model is dependent upon its ability to
simulate and predict. A verified model reasonably simulates
physical transports and pathways; whereas, a validated model
predicts (within limitations) actual transport of a substance.
There are many models that cannot be validated because of
simplifying assumptions used in their development. But, any
model to be used in decision making must, as a minimum,
provide a reasonable simulation of the real world. The
majority of our laboratory and field efforts will be directed
toward validation and verification.
2. Improved characterization
Unreliable or inadequate data will always reduce the
credibility of model results. Field measurements and uniform
site characterization procedures are necessary to improve
input data.
3. Bounding the credibility of model results
Sensitivity analyses, error bars, and probabilities are
important. Errors in input data should be propagated through
the model to provide estimates of result error. Sensitivity
analyses identify limitations of model boundary conditions.
Analysis of probability allows the user to identify the
dominant pathways and events. All of these serve to bound the
credibility of the results and provide the user with an
understanding of the model's limitations in decision making.
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At this time, let us turn our attention to a discussion of
criteria. A model should be used to determine how effectively a site
will meet criteria. At the present time, we do not have recognized and
approved criteria for reference of implementation. The federal
agencies must come to an agreement on what we are expecting models to
provide; until we have such criteria, it may be pointless to continue
modeling.
The most basic criteria should be expressed in terms of the
maximum allowable doses to the public from routine and accidental
releases. In determining these criteria, we must take into
consideration the economic impact on industry and the principles of as
low as reasonably achievable (ALARA).
The 500 mrem maximum individual exposure limit is based on sound
scientific evidence and reasoning. It is a regulation and industry
standard. However, do we select and operate sites based solely on the
500 mrem limit or do we establish some lower limit? If we establish a
lower limit, is it scientifically defensible, and can we justify the
economic impact on industry? For purposes of discussion, let us
examine the following "straw man" criteria:
1. The maximum allowable dose from any accident or non-routine
release is 500 mrem under normal conditions.
2. The physical properties of that site should not permit
exposure of individuals to more than 50 mrem under normal
conditions. This does not take into account any properties
for waste form or special retardation barriers but is based
solely on waste inventory and the site characteristics.
3. The dose from a release event should be inversely proportional
to the probability of the event. This, of course, means the
higher the probability, the lower the dose.
4. Sites are to be selected, designed, and operated so that
during operational periods the maximum dose will not exceed
500 mrem; for the 25 years of post-operational monitoring, the
maximum dose does not exceed 100 mrem; and after 25 years the
maximum dose does not exceed 50 mrem. This last "straw man"
criteria is our obligation not to pass on to future
generations a burden for maintaining these sites.
One way for a decision maker to determine if his site meets these
"straw-man" criteria is through the use of a model. Based upon our
experience, it would seem that such criteria would not only be easy to
meet but also difficult to violate under any reasonable management
system. Given the guidelines we have suggested, it might be that
sufficient modeling capabilities already exist to provide reasonable
assurance of compliance.
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The following are a few major items we believe can only be
resolved through the effective interaction of the federal agencies:
1. Reference sets of source term and dose conversion factors
There is a need to establish a set of reference source terms
to describe a low-level radioactive waste site. One of the
major complaints of model users and developers is that too
much time and effort is spent in developing source terms, and
there is no guarantee that those terms will be considered
reasonable. There already exist several excellent detailed
source term documents; all that is required is for the
interagency group to agree on terms and compile these into a
single reference document.
2. Standard documentation format
A model to be transferred must be in a form easily
understandable to the user. There is a need to develop
procedures and formats for documenting and transferring models
from developers to users.
3. Standard site characterization procedures
Collection of data for the characterization of a site are not
uniform procedures and result in large variances in data. A
set of reference characterization procedures should be
developed to identify data to be collected and acceptable
collection methods.
4. Validation experiments
This was an idea proposed at our July meeting. In essence, we
would develop experiments that would test various scale model
shallow land disposal facilities to points of failure. The
data obtained would be used to define the conservativeness of
our models.
5. Reference validation data sets
Data from field tests should be developed into a reference
document which would allow model users to test their models
against real world data.
6. Model categorization system
Model users often have little to guide them in selecting a
model. By providing a categorization system, models would be
grouped according to function, application, and boundary
condition. This categorization system should be developed
concurrently with the model documentation procedures.
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In conclusion, we believe that by working together with common
objectives the several agencies and supporting organizations can
successfully find adequate tools and use them to make appropriate
decisions for managing the Nation's low-level radioactive waste.
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MODELING AND ANALYSES TO SUPPORT EPA's
LOW-LEVEL RADIOACTIVE WASTE STANDARDS
G. Lewis Meyer
Criteria and Standards Division
Office of Radiation Programs
U. S. Environmental Protection Agency
Washington, D.C. 20460
ABSTRACT
EPA is developing a generally applicable standard for the
management and disposal of LLW as part of the U.S. National
Radioactive Waste Management Program. It will issue a proposed
standard in September, 1981, and a final standard in early 1984.
EPA will use predetermined models to estimate the health risks and
costs/benefits from disposing of LLW by a number of land and sea
disposal alternatives. These estimates will be used to support
EPA's decision on the proposed standard and for the supporting EIS
and Regulatory Analysis. This paper describes why EPA is using a
generic "systems" approach to modeling the impact of LLW disposal,
the types of disposal alternatives which will be evaluated, its
approach to cost/benefit analysis, the current status of model
development, general input data requirements for the model, and a
general schedule for development of the models and the standard.
THE NEED FOR A STANDARD AND MODELING
Before 1975, there was no demonstrated need for an
environmental standard for the disposal of low-level radioactive
waste (LLW) because it was believed that there would be no release
of radioactive from the disposal sites.d) Since then, studies at
the Maxey Flats (KY), West Valley (NY), and Oak Ridge (TN) disposal
sites have shown that unplanned releases to the uncontrolled
environment can occur if LLW are not properly disposed and that
the impact of these releases could be substantial. For example,
more than 24.6 million liters of radioactive leachate containing
appreciable concentrations of H-3, Co-60, Sr-90, Cs-137, Pu-238,
and Pu-239/240 have been pumped from the trenches at Maxey Flats and
treated.(2) Pumping operations are continuing. If stopped, these
radioactive leachates would be released by overflow directly at land
surface or to the groundwater. The experience at West Valley has
been similar.(3)
The public has become increasingly concerned about the safe
disposal of all radioactive wastes including LLW. Generators of LLW
have become very concerned about the lack of adequate disposal space
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in the future since three of six operating commercial LLW disposal
facilities have closed. State regulators have become concerned
about the quality and the quantity of LLW being shipped to the
disposal sites in their states. These occurances established the
need for EPA to act pursuant to its charge to identify radiation
hazards to the general environment and to set generally applicable
environmental standards where needed.(4,5) On February 12, 1980,
the President directed EPA to develop the necessary radioactive
waste standards on an accelerated basis.(6)
Consequently, EPA is developing a generally applicable
environmental standard for the management and disposal of LLW.
It has guidance on how to develop this standard from a number of
sources. These include NEPA (7), EPA's "Improving Government
Regulations" (8), BEIR II (9), "Decision Making in EPA" (10), and
ICRP-26 (11). This guidance, which clearly establishes EPA's needs
for modeling and models, has been summarized quite well in EPA's
proposed "Criteria for Radioactive Wastes" (12) as follows,
"...Radiation protection requirements for radioactive wastes
should be based primarily on an assessment of risk to individuals
and populations; such assessments should be based on predetermined
models..." and
"...Any risks due to radioactive waste management or disposal
activities should be deemed unacceptable unless it has been
justified that the further reduction in risk that could be achieved
by more complete isolation is impracticable on the basis of
technical and social considerationd..."
Before going on to discuss EPA's general modeling needs, the
role of modeling in EPA's standards development process should be
put into perspective. MODELING IS ONLY A TOOL! Developing and
setting environmental standards is less than an exact science.
In most cases, arrival at the standard involves a combination of
public acceptance, political accord, good judgment stemming from
long experience in public health protection, and expert technical
judgement. The models will support our judgement and standard.
The results from our modeling efforts will not be the basis for
EPA's standard.
EPA'S APPROACH - A GENERIC "SYSTEMS" APPROACH
The primary purpose of EPA's modeling efforts is to assess the
potential impacts from managing and disposing of LLW by a number of
realistic alternatives. This includes estimating the potential
releases to the environment, doses, health risks, and costs.
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A number of factors have influenced EPA's approach to doing this.
These factors, which include EPA's authorizing legislation and
mandates, the character of the wastes, themselves, and the realistic
means by which they may be managed or disposed, are discussed
briefly below.
The guidance given to EPA for developing the standard is broad,
general, and sometimes contradictory. The environment should not
be degraded.(7) We should assess the impact of all reasonable
alternatives.(8) Even the smallest amount of radiation is
harmful.(9) The risk should be reduced as low as reasonably
achievable (ALARA) but we should take social and economic factors
into account.(9,11,12)
The radioactive wastes for which this standard is being
developed are an extremely heterogeneous group of wastes.
They include all wastes which have been categorized as "low-level"
radioactive wastes and, in addition, those transuranic-contaminated
wastes (TRU) which do not fall under the High-Level Radioactive
Waste (HLW) Standard.(13) The term "low-level" does not adequately
reflect the potential hazard of wastes currently classified as LLW.
LLW are, in fact, a large diverse class of wastes in which some
types may contain tens of thousands of curies per cubic meter
(14,15) and remain radioactive for millions of years. Other LLW may
contain only traces of radioactive material. They may be solid,
mushy, liquid, or gaseous; may range from insoluble to completely
soluble; and may be animal, vegetable, or mineral.
Disposal in the ground or in the ocean are the disposal methods
which are most readily available at this time. Any land or ocean
disposal facility is a complex "system" which includes the waste,
the site geology, hydrology or oceanography, and meteorology, the
emplacement method, site engineering, and all of the pathways from
it and the performance of the system will change if changes are
made to any of its parts. Analysis of a single pathway would not
provide an adequate analysis of the impact of a "disposal system."
Thus, the disposal facility and the wastes in it must be regarded as
a disposal system and analyzed as a whole if we are to assess its
overall impact.
Because of the above factors, our model(s) will be used to
evaluate the effectiveness of each disposal alternative using a
systems approach — to examine each alternative as a disposal
system and to compare it with alternative disposal systems.
Sufficient detail will be included in the models to permit
evaluation of the effectiveness of proposed changes in the system
design for improving the retention capability of the disposal system
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(i.e., change in waste form or packaging, engineering barrier,
etc.). Changes in the design of the disposal system may also affect
the cost of disposal. These costs will be assessed or modeled
separately.
There are a number of land and sea disposal methods which are
suitable for the disposal of some types of LLW. However, no one
disposal method is suitable for all types of LLW. Therefore, the
model(s) should be applicable to a variety of disposal methods,
types of wastes, and hydrogeological settings; the results of which
should be useful in comparing the potential impacts and costs of the
different disposal alternatives. Thus, it is also clear that a
generic systems model(s) is needed.
The natural characteristics of a disposal site or "disposal
system" (i.e., geology, hydrology, and meteorology) are determined
and relatively fixed once a site is selected. Normally, they can be
changed by selecting another site. However, emplacing wastes in the
site, the interaction of the wastes with the site, and engineering
changes to the site can significantly alter the functioning of the
hydrogeologic portion of the "disposal system" and directly affect
its performance. The wastes, their form, the method of their
emplacement, and site engineering also affect the performance of the
"disposal system" but are mainly determined by man. The point being
made is — the natural characteristics of the "disposal system" are
mainly determined in the site selection process; man can control the
performance and costs of the rest of the system.
TYPES OF ENVIRONMENTAL ASSESSMENT MODELS
We have reviewed a number of alternative methods for disposing
of LLW to identify those which could be implemented quickly without
delay for technical hardware or development. So far, we have
identified eight land and two ocean disposal methods which could be
used for the disposal of one or more types of LLW. Our review has
shown, however, that not one of these methods is suitable for the
disposal of all types of LLW.
In five land disposal methods, the wastes would generally be
disposed of at or near land surface and generally not below 50
meters depth. These shallow land disposal (SLD) methods include
engineered surface storage (SLD-SS), sanitary landfill (SLD-LF),
shallow land disposal as presently practiced (SLD-P), improved
shallow land disposal (SLS-I), and intermediate depth disposal
(SLD-I). In three land disposal methods, the wastes would generally
be disposed of at depths greater than 50 meters. The deep land
disposal (DLD) methods include deep geological disposal in a mined
cavity (DLD-MC), hydrofracturing (DLD-HF), and deep-well injection
(DLD-DWI). The two ocean disposal methods include disposal on the
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sea floor (OD-OSF) and beneath the sea floor (OD-BSF). We are,
therefore, developing environmental assessment models for shallow
land disposal (SLD), deep land disposal (DLD), and ocean disposal
(OD).
The SLD assessment model will be used to analyze the potential
health risk from disposing of LLW at or near land surface to a depth
of 50 meters but generally above the water table. Experience has
shown that near surface geology, hydrology, and meteorology affect
the performance and retention capability of a SLD site very much.
Therefore, the SLD model is being designed with sufficient
flexibility to cover at least three types of wide-spread
hydrogeologic/climatic settings which we believe will cover the most
sensitive parameters that affect radionuclide retention and site
performance. These settings include: (1) an arid zone site for
which no particular regard is given to the permeability of the
disposal medium (because of a lack of water as a driving force);
(2) a humid zone site with a low-permeability disposal medium (which
would fill like a "bathtub" in the event the trench cap leaked); and
(3) a humid zone site with a moderately permeable disposal medium
(which would allow water infiltrating through the trench cap to leak
out the bottom of the trench like a "sieve").
Additional settings will be developed as necessary by altering
in the input parameters. The water pathways would certainly be
less important for conventional SLD and improved SLD in the arid
climates, whereas, all of the pathways would be important for these
methods in the two humid climate scenarios. For intermediate depth
SLD (10-l8m deep), the groundwater pathway would be considered most
important.
The DLD assessment model will closely parallel the SLD
assessment model in design, pathways, etc. However, it will
basically be used to analyze the impact of disposing of LLW at
depths below 50 meters and generally below the water table, although
there are some areas in the western United States where water tables
are as deep as 700 meters. Near surface hydrogeology and meteor
ology normally will not have much affect on the performance of the
DLD alternatives. More emphasis will be given to hydrodynamics and
groundwater transport.
There is not sufficient information on the ocean disposal
assessment model to discuss it at this time.
BENEFIT/COST MODELING
The benefit/cost analysis program to support the LLW waste
standard will be different and more extensive than the one conducted
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for the HLW standard. To begin with, we have extensive experience
and impact and cost data from operating SLD facilities. These data
can be used to develop a base case against which other disposal
methods can be compared. For HLW, there was no such experience or
data. There are also at least ten alternatives for disposing of LLW
for which the benefits and costs can realistically be estimated and
compared. No comparison was made for HLW. Finally, the length of
time which LLW must be retained is generally much less than for HLW;
in most cases, only several hundred years or less rather than
thousands of years. Thus, the period of analysis can be much
shorter for LLW.
The benefit gained in waste disposal, as used herein, is the
reduction in health effects gained from taking a control action.
The greater the reduction in health effects, the greater is the
benefit. The benefits, or reduction in health effects, which may
be gained from each disposal alternative will be calculated using
one of the SLD, DLD, or OD assessment models discussed earlier.
To help the user, the benefits will be expressed, so far as
possible, in several forms including radionuclides released to
the environment and doses, health effects and health risks to
individuals and populations.
On the cost side of the analysis, one, and possibly two,
analyses will be made. The projected direct costs for controlling
or disposing of a unit of waste, whether it be in specific activity
or unit volume, will be estimated for each disposal alternative
considered. These costs will be useful for the tradiational
"cost-effectiveness" types of analysis. They will also furnish
guidance on how much is being spent on the control of LLW.
A model(s) or method(s) will be developed to calculate and compare
these direct costs for the different disposal methods.
A second, broader type of economic analysis may be required,
depending upon whether EPA internal triggering criteria relating
to total dollars for costs of compliance, increases in energy
consumption, etc., are met.(8) The purpose of these economic
analyses would be to fully identify, on a national basis, the
overall benefits and costs of the regulations; not just the direct
costs. They would include preparation of a profile of the affected
sectors of the economy including producers, consumers, and
industry. The following impacts would be analyzed: price effects;
production effects; industry growth; profitability and capital
availability effects; employment effects; balance of trade effects;
and energy effects.
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15
CURRENT STATUS OF MODEL DEVELOPMENT PROGRAM
As noted earlier, our general modeling program is directed at
making realistic technical analyses of the health risks and costs of
disposing of LLW by a number of practical disposal alternatives.
The major steps and flow of achieving this goal includes: defining
our modeling needs and approach; developing the models; compiling
data on the wastes, disposal alternatives and actual disposal sites
to use in the models; and conducting the analyses.
Definition of our modeling needs and approaches to environ-
mental assessment modeling is being done in-house. The concept and
design of our SLD model is complete.(16) Design is in progress on
our DLD and cost models. Design of the OD model has not begun yet.
Specific guidance for development of the infiltration, leaching,
and groundwater transport submodels for the main SLD assessment
model release was developed in-house by Dr. C. Y. Hung.(16,17,18,19)•
The submodels for the atmospheric and terrestrial pathways submodels
and dose and health risk which will be used in the SLD model are
models which were or are being developed for EPA for use in the
Clean Air Act.(20,21) They were developed by Oak Ridge National
Laboratory (ORNL). ORNL is currently developing our SLD assessment
model under an interagency agreement with DOE. This includes
development of several additional submodels and, as necessary,
modifying and coding all of the submodels into an overall main
assessment model.
The overall design and submodels of the DLD assessment model
will closely parallel the SLD model. The major difference is that
the DLD model will be used to analyze the impact of disposing of LLW
at depths below 50 meters and generally below the water table.
The DLD model will probably include the modification and combination
of submodels from our HLW model and the SLD model currently being
developed. However, the dose/health risk model used will be the one
from the SLD model, which is different from the one currently used
in our HLW model. Work on the DLD model has not begun yet, nor has
work on the cost model.
Disposal of LLW on and beneath the ocean floor, which includes
two of ten disposal options currently considered practical, is
regulated under different authorities than land disposal. The ocean
disposal program is moving at a different pace than the LLW
program. At present, evaluations of former ocean disposal sites,
development of siting criteria, and evaluations of ocean disposal
technology are under way. However, development of an environmental
assessment model for ocean disposal appears to be several years in
the future.
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INPUT DATA FOR THE MODELS
With an estimated 20 million dollars or more per year being
spent by other Federal Agencies on research and development for LLW
management, we believe that most of the data needed to support EPA's
models is either already available or is being collected by others.
Therefore, most of EPA's efforts will be directed at collecting,
collating and critically evaluating data from others. We are
presently carefully reviewing the information available on
radioactive and hazardous waste-related projects funded by other
Offices within EPA, other Federal agencies, the States, and
industry, to facilitate rapid collection of the needed data.
Gaps in our data needs will be filled as necessary.
Field data will be collected from three existing LLW disposal
sites to provide actual or, at least, realistic data for input into
the SLD assessment model and for making the cost/risk analysis for
disposing of LLW by this method. The sites are Barnwell (SC), West
Valley (NY), and Beatty (NV) which represent the three general
regional hydrogeological and climatic settings we intend to model.
Two categories of site data will be collected; one from
existing data, including published and unpublished data, and the
other from field observations. Existing data will be used whenever
possible. ORNL is currently preparing a list of data needs for the
SLD model and is compiling some of the existing data. Collection of
additional data to fill in the gaps between the model data needs and
the existing site data will start shortly.
We currently have an in-house project for compiling and
evaluating information on waste characteristics and on improved
methods for treating, packaging, and managing LLW. This information
will be used to estimate the potential costs and reductions in
health effects which can be achieved through waste management
alternatives which already exist. Information is being collected
on: the character and quantities of primary wastes being produced
by typical waste generating facilities before treatment and
packaging; what kinds of segregation and elimlination of wastes
may be done within each facility; the treatment, volume reduction,
solidification, and packaging options that are available for the
primary wastes and the characteristics and quantities of the
resulting wastes; the hazard potential of the various wastes; and
projections of future wastes based on several optimum management
and treatment modes.
We will soon begin to collect information and develop the
general characteristics of the natural (geological), engineering,
and institutional barriers which are currently being used, or could
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be used in the near future to retain LLW when disposed of by shallow
land disposal and other selected land disposal alternatives. The
types of land disposal methods to be considered was discussed
earlier. Information to be collected includes; technical and cost
data on SLD as presently practiced, improved SLD, and the other land
disposal alternatives; site selection criteria for all land disposal
alternatives; and post-operational controls for all land disposal
alternatives.
TIMING FOR DEVELOPMENT OF LLW MODELS AND STANDARD
The schedule for development of the LLW standard, development
of the models, technical, environmental and cost assessments, and
collection of data to support the assessments is shown in Figure 1.
PROPOSED
FINAL
SLD MODEL
OLD MODEL
1980 I 1981 I 1982 j 1983 | 1984
I
K
h_
STANDARD . n
I A y
OD MODEL I No Schedule at This Tin
COST/BENEFIT ANALYSES , A_
OF LAND DISPOSAL
Figure 1. General schedule for the development of EPA's LLW assessment
models, analysis of the risks, costs and benefits of different disposal
alternatives, and development of the LLW standard.
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18
REFERENCES
1. U.S. Atomic Energy Commission, Draft Generic Environmental
Statement for Mixed Oxide Fuel (GESMO), WASH-1337 (1974).
2. Dames and Moore, Inc., "Review of Low-Level Radioactive Waste
Disposal History," an unpublished report for USNRC on LLW
Disposal Technology (1980).
3. U.S. Environmental Protection Agency, Summary Report on the
Low-Levej^Radioactive Waste Burial Site, West Valley, New York
(1963-1975), EPA-902/4-77-010 (Feb. 1977).
4. U.S. Code of Federal Regulations, Reorganization Plan No. 3, of
1970, (1970).
5. U.S. Code of Federal Regulations, Atomic Energy Act of 1954, as
amended, (1970).
6. Office of the U.S. President, Comprehensive Radioactive Waste
Management Program - Message from the President of the United
States, (Feb. 12, 1980).
7. U.S. Code of Federal Regulations, National Environmental Policy
Act of 1969, (1969).
8. U.S. Environmental Protection Agency, Improving Government
Regulations, (Jul. 1978).
9. U.S. Environmental Protection Agency, Considerations of Health
Benefit - Cost Analysis for Activities Involving Ionizing
Radiation Exposure and Alternatives, EPA 520/4-77-003, (1977).
10. U.S. National Research Council, Volume II, Decision Making in
the Enviromental Protection Agency, A National Research
Council Report to EPA on Environmental Decision Making, (1977).
11. International Commission on Radiation Protection, Publication
26: Recommendations of the International Commission on
Radiation Protection, (Jan. 1977).
12. U.S. Enviornmental Protection Agency, Criteria for Radioactive
Wastes, Recommendations for Federal Radiation Guidance,
(Nov. 15, 1978).
13. U.S. Environmental Protection Agency, Generally Applicable
Standard for the Management and Disposal of High-Level
Radioactive Wastes, unpublished draft (1980).
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19
14. U.S. Nuclear Regulatory Commission, Draft Programmatic
Environmental Impact Statement Related to Decontamination and
Disposal of Radioactive Wastes Resulting from March 28, 1979,'
accident. Three Mile Island Nuclear Station, Unit 2,
NUREG-0683, (Jul. 1980).
15. Atomic Industrial Forum, A Survey and Evaluation of Handling
and Disposing of Solid Low-Level Nuclear Fuel Cycle Wastes,
AIF/NESP-008ES, (1977).
16. U.S. Environmental Protection Agency, Technical Specifications
for Environmental Assessment Model Development, unpublished
TAD/ORP document, (1979).
17. G.L. Meyer and C.Y. Hung, "An Overview of EPA's Health Risk
Models for Shallow Land Disposal of Low-Level Radioactive
Waste," Proceedings of DOE/EPA/NRC/USGS Interagency Workshop on
Modeling and Low-Level Waste Management, Denver, Colorado,
(Dec. 1980).
18. C.Y. Hung, "Prediction of Long-Term Leachability of a
Solidified Radioactive Waste from a Short-Term Test by a
Similitude Law for Leaching Systems," Proceedings of ORNL
Conference on the Leachability of Radioactive Solids,
Gatlingburg, Tennessee, (Dec. 1980).
19. C.Y. Hung, "An Optimum Model to Predict Radionuclide Transport
in an Aquifer for Application to Health Risk Evaluations,"
Proceedings of DOE/EPA/NRC/USGS Interagency Workshop on
Modeling and Low-Level Waste Management, Denver, Colorado,
(Dec. 1980).
20. U.S. Environmental Protection Agency, NEMOS (Atmospheric) and
TERRA (Terrestrial) Transport Codes, under development by ORNL,
(1980).
21. U.S. Environmental Protection Agency, ANDROS, DARTAB and
RADRISK (Dose and Risk Factor) Codes, under development by
ORNL, (1979).
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MODELING IN LOW-LEVEL RADIOACTIVE WASTE MANAGEMENT
FROM THE U.S. GEOLOGICAL SURVEY PERSPECTIVE*
John B. Robertson
U.S. Geological Survey
Reston, Virginia
ABSTRACT
The United States Geological Survey (USGS) is a long-
standing proponent of using models as tools in geohydrologic
investigations. These models vary from maps and core samples
to elaborate digital computer algorithms, depending on the
needed application and resources available. Being a non-
regulatory scientific agency, the USGS uses models primarily
for: improving modeling technology, testing hypotheses,
management of water resources, providing technical advice to
other agencies, parameter sensitivity analysis, and deter-
mination of parameter values (inverse problems). At low-level
radioactive waste disposal sites, we are most interested in
developing better capabilities for understanding the ground-
water flow regime within and away from burial trenches, geo-
chemical factors affecting nuclide concentration and mobility
in groundwater, and the effects that various changes in the
geohydrologic conditions have on groundwater flow and nuclide
migration. Although the Geological Survey has modeling
capabilities in a variety of complex problems, significant
deficiencies and limitations remain in certain areas, such as
fracture flow conditions and solute transport in the unsatu-
rated zone. However, even more serious are the deficiencies
in measuring or estimating adequate input data for models and
verification of model utility on real problems. Flow and
transport models are being used by the USGS in several low-
level disposal site studies, with varying degrees of success.
INTRODUCTION
A model generally means a simulator of a thing or process that
cannot easily be observed directly. This broad definition is applicable
to use of the term in the U.S. Geological Survey (USGS) and in this
presentation. Everyone uses models, whether they be analogue, mathemati-
cal, or conceptual. Our favorite photographs of people and scenes are
*
This paper is prepared for inclusion in the program of the Interagency
Workshop on Modeling and Low-Level Waste Management, December 1-4, 1980,
in Denver, Colorado, sponsored by the DOE-NRC-EPA-USGS Interagency Low-
Level Radioactive Waste Committee.
21
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simplified models of more illusive and complex real-world conditions.
What driver hasn't used a road map as a scale model of the real highway
system? (That example alone quickly teaches us lessons in the problems
of model inaccuracies.) Nearly all of us use our car radio or home
stereo to simulate our favorite live-music performances.
Scientific models are often thought of as mysterious, electro-
mathematical devices fully understood by no one and worthy of consider-
able distrust. However, the most commonly and confidently used models
in science and technology are similar to those used by most everyone
else — photographs, drawings, diagrams, and simple equations. The types
of models chosen, and the ways in which they are used, simply depend on
the desired output and available resources. Types and applications of
models used by the USGS are those appropriate to the Agency's mission
and its resources. The same is generally true of other agencies and
organizations. Before elaborating on the USGS's use of models in low-
level waste management technology, it would seem appropriate first to
describe the mission and philosophy of the Survey as it relates to
radioactive waste disposal. This should help clarify the differences in
modeling philosophies and approaches between the Survey and various
other Federal or State agencies. These differences often appear confus-
ing, conflicting, or even contradictory, though in reality they are
usually complimentary.
WHAT IS THE USGS?
The Geological Survey is an old-line Federal agency established by
an act of Congress in 1879 and charged with the responsibility for
"classification of the public lands and examination of geological struc-
ture, mineral resources, and products of the national domain." It is the
Nation's principal source of information about its physical resources -
the configuration and character of the land surface; the composition and
structure of the underlying rocks; the quantity, extent, and distribution
of water and mineral resources. In contrast to many agencies, the
Survey's mission has changed very little in more than 100 years; it
remains principally a scientific research and fact-finding agency,
rather than an action or regulatory organization.
Most of the USGS activities in waste disposal stem from its respon-
sibility to provide data on surface and groundwater quality essential to
the development and conservation of water resources. Fundamental in
this mission is the development of appropriate technology and tools to
better assess physical and chemical behavior of hydro!ogic systems and
their response to stress.
The Water Resources Division of the Survey has a line item in its
appropriation to conduct investigations and research aimed at establish-
ing a technical basis upon which earth science criteria can be developed,
tested, and enforced by other agencies for the selecting and operating
of low-level waste disposal sites. This program includes several field
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investigations of existing commercial disposal sites and more fundamental
research into such fields as groundwater geochemistry and modeling
techniques.
As the Federal Government's principal earth-science research agency,
a fundamental part of the USGS role is that of technical advisor and
consultant to other Federal agencies and states. The Survey is often
mandated by the President or Congress to provide specific technical
services to Federal and state programs. A recent example is our Presi-
dentially specified participation in the National high-level waste
management program. Our role as technical consultant to other agencies
includes critical review of proposed regulations and criteria, investiga-
tions of Federal waste disposal sites, review of proposed and ongoing
programs, cooperative investigation at the requests of states, and other
related investigations, generally of nonregulatory nature. It is
generally the policy of the USGS to avoid specific disposal site selec-
tion and characterization studies, preferring instead to provide technical
advice and tools for other groups to conduct such investigations properly.
Specific states, for example, have come to us for help in searching
for a low-level waste burial site. Rather than suggest specific sites,
we recommend approaches based on our knowledge of positive and negative
factors that appear important in site performance. In addition, we
offer any regional, areal, or local data which we have gathered that
could be pertinent to their considerations. Our role is restricted
solely to earth science considerations, whereas other agencies such as
EPA and NRC must consider much broader social, political, legal, and
technical issues. Therein lies the fundamental difference between USGS
and the other, more action and regulation-orientated, agencies.
MODELING APPLICATIONS IN THE USGS
The Geological Survey has long held a keen interest in development
and application of models as tools for quantitative analysis of complex
hydrologic problems. Models have always been regarded by the Survey as
means, rather than ends, to problem solving. The principal use of
models is to aid in gaining a more quantitative understanding of complex
hydrologic systems involving many interdependent physical and chemical
processes and boundary conditions. Waste disposal and contamination
problems represent some of the most complex problems investigated by the
Survey. For many of these problems, modeling is the only means we have
of examining all major components of a system together.
Another key application of models is to test and evaluate our
concepts of system dynamics. Interpretations of field conditions gener-
ally must be based on very limited field data. The integrated picture
developed from a few point-observations can often turn out like the
blind men and the elephant. If one set of data indicates the elephant
is like a snake, a model can be constructed to simulate the snake-like
attributes. If the-model is good, it would soon indicate that a snake's
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24
characteristics and behavior are different from those of the elephant's
trunk, suggesting that our initial impression was wrong and that more
data are required. Eventually we might obtain data from enough parts of
the elephant to construct a model that confirms our blind interpretation
of the beast.
Models have been used extensively to help interpret hydrology and
decipher performance of existing low-level radioactive waste burial
sites. These sites represent long-term complex experiments whose results
we are currently observing and recording. The unknowns in these experi-
ments are the experimental conditions. The challenge then, is to deter-
mine the experimental conditions that produced the results currently
observable. Models are one of the few tools that can allow us to test
various combinations of conjectured conditions. One of the most important
outputs of this type of modeling is a sensitivity analysis - a ranking
of processes and parameters according to the magnitude of their influence
on the result of interest (perhaps the concentration distribution of a
particular nuclide). These results then serve as guidelines for more
detailed studies. Thus, the model output (combined with all of our
other tools) can be invaluable in selecting which segments of a system
deserve more attention.
Our philosophy is to engage the use of modeling early in any complex
study as a feedback tool — repeatedly adjusting the model, in an iterative
procedure, as new information is gathered, accompanied by corresponding
adjustments in data-gathering and testing priorities, according to
modeling results. This application of modeling is immensely helpful in
establishing earth-science guidelines for disposal site selection and
operation.
Another example of USGS model employment is in resource and problem
management. We are often requested to assess impacts of various develop-
ment options on an aquifer so that a management agency can make sound
development decisions. This may involve geochemical and hydraulic
impacts of a potentially degrading activity such as a waste disposal
facility; or it could involve assessment of various remedial approaches
to a contamination problem.
Finally, USGS modeling capabilities can be used as an independent
test and verification of modeling results of other agencies used for
judging the conformance of a particular site to regulatory criteria.
This can be done directly or by providing the appropriate models to
other agencies for their use. We view this as one of our most important
roles — developing more accurate, more versatile, and more efficient
models for use by other government organizations. We maintain several
research projects dedicated to advancing model technology and teach
courses on modeling open to some non-USGS personnel.
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WHAT KIND OF MODELS DOES THE USGS USE?
As mentioned previously, the Geological Survey uses many types of
models, ranging from core samples to elaborate room-sized physical
models. Although the types of models generally depend on the need and
available resources, USGS modeling can be classified into two general
approaches.
The first approach is to use a small sample or portion of the real
field system to represent the total system. A core sample is an example.
In some studies, a core sample might be used for measuring hydraulic
properties which might then be extrapolated to represent those properties
over a much larger field area. Water samples similarly are often used
as models to represent chemical characteristics of a larger zone of an
aquifer. These models generally have very limited use and, in fact,
have been commonly misused. Hydraulic conductivity measurements on core
samples, for instance, are often orders of magnitude different from
average areal conductivities in many formations.
The second approach is use of models that simulate the entire
system of interest, generally in some simplified way. One of the first,
and still the most commonly used model of this type in groundwater
hydrology, is Darcy's Law:
where
q is the water flux rate, L/T
K = hydraulic conductivity, L/T
g— = hydraulic head gradient, dimensionless
This empirically based model is the basis for nearly all other more
sophisticated groundwater flow and solute transport models. One of the
more elaborate of these (but still simple) is the Theis equation (Theis,
1935), which is perhaps the most widely used model in the world for
interpreting the hydraulic characteristics of aquifers. Although very
few actual field conditions meet the strict qualifying conditions for
application of the Theis equation, it has nonetheless enjoyed widespread
success as an approximation.
Models receiving most attention in recent years are, of course,
ones which approximate solutions to complex partial differential equations
of flow, solute, and energy transport. These models range from elaborate
physical and electrical analogues to ingenious mathematical schemes to
solve huge matrices of simultaneous equation. These mathematical
approaches include a variety of finite difference schemes, such as
iterative-alternating-direction-implicit procedure, and various finite
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26
element algorithms such as the Galerkin procedure. All of these methods
and a number of others have advantages and disadvantages for different
applications. It is not the purpose of this paper to describe or compare
these models. Anderson (1979) made an excellent review of model use for
simulating contaminant movement in groundwater.
One of the few examples of the development of application and
verification of a numerical model to a low-level waste problem was by
Robertson (1974). In that case, the field-observed distributions of
chloride, tritium, and strontium-90 in a large aquifer were successfully
simulated by a two-dimension transient flow and transport model which
included effects of dispersion, radioactive decay, and cation exchange.
Figure 1 indicates one of the results of that modeling effort for tritium.
We have recently attempted to model groundwater flow (and in some
cases, solute transport) at four of the commercial low-level radioactive
waste disposal sites with varying degrees of success. Two of these
investigations are described in other presentations of this meeting.
USGS MODELING CAPABILITIES
Models and modelers developed in the Geological Survey are capable
of modeling groundwater flow in stimulated, three-dimensional, hetero-
geneous, anisotropic, transient conditions, on a fairly routine basis.
Non-reactive solute transport, including dispersion, can be modeled
under similar conditions. Transport of reactive contaminants can be
modeled in two dimensions for simple linear sorption and first-order
rate reactions. Multi-component reactive solutes can generally be
modeled only in one dimension. Multi-component chemical equilibrium
models, such as WATEQ, are used routinely for complex solutions in a
non-transport mode. Flow in the unsaturated zone can be simulated for
fairly simple transient non-uniform conditions with no hysteresis;
however, solute transport modeling in the unsaturated zone is in its
infancy. Other papers in these proceedings review the state-of-the-art in
the USGS as well as other models in saturated and unsaturated ground-
water conditions and in streams.
We do not yet have satisfactory modeling capabilities in the fol-
lowing areas:
• fractured rock systems, such as Maxey Flats, Kentucky, except
for very high and/or very uniform fractures systems;
• flow in very dry, heterogeneous unsaturated zones, such as Beatty,
Nevada;
• unsaturated flow with hysteresis or reactive solute transport;
• flow in any media involving complex multicomponent reactions or
non-first order rate reactions;
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27
Solid waste
burial
Explanation
Map adapted from Robertson (1974)
Equal Tritium Concentration in pico curies per milliliter for 1968
^-50 Well samples
'"—50 Digital model
INEL = Idaho National Engineering Laboratory
ICPP = Idaho Chemical Processing Plant (nuclear fuel reprocessing)
TRA = Test Reactor Area
EBR-1 = Experimental Breeder Reactor 1 site (inactive)
CFA = Central Facilities Area
INEL boundary
Figure 1. Comparison of ICPP-TRA waste tritium plumes in the Snake River
Plain aquifer for 1968 based on well sample data and computer
model.
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• flow involving transport and biologically-controlled reactions.
Although these limitations are severe, we have research proceeding on
all these subjects.
For most situations amenable to modeling, our models are better
than our field data. Field data on dispersivity, chemical reaction
parameters, and other inputs are grossly inadequate at most sites of
concern. Therefore, in order to build new and better models or even to
apply existing models, development of field and laboratory testing
technology must proceed at comparable or faster rates. Deficiencies in
our ability to measure and estimate model input parameters are actually
more serious than model limitations and currently are deserving of more
research attention. Related to this is a continuing need to enhance the
credibility of models with many more applications and demonstrations on
well-documented field problems.
CONCLUSIONS
The USGS is a strong proponent of the use of appropriate models in
all hydro!ogic problems. The more complex the problems such as those
involving waste disposal and groundwater contamination, the greater the
need for modeling and the more complex are the models needed. Models
are only tools to be used in conjunction with, not a substitute for,
investigative analytical tools.
In waste disposal studies, the USGS needs and uses models for the
following main purposes:
• to learn how to build better models;
• to test and confirm theories and concepts;
• to aid in managing water resources and stresses upon those
resources;
• for intellectual stimulation;
• to gain competence and experience in order to advise other groups
on model application;
• sensitivity analysis to screen or rank interacting processes and
parameters;
• to "back-out" values of parameters such as dispersivity, distribu-
tion coefficients, and hydraulic conductivity (inverse problems);
• to gain better general insight into complex multi-component
systems.
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Although significant deficiencies exist in current models, the
greater deficiency is in our ability to provide adequate input informa-
tion to models and to demonstrate model accuracy and reliability on real
field problems.
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30
REFERENCES
Anderson, Mary P., 1979, Using models to simulate the movement of contam-
inants through groundwater flow systems: CRC Critical Reviews in
Environmental Control, November 1979, pp. 97-155.
Robertson, J. B., 1974, Digital modeling of radioactive and chemical waste
transport in the Snake River Plain Aquifer of the National Reactor
Testing Station, Idaho: U.S. Geological Survey open-file rept.
ID022054.
Theis, C. V., 1935, The relation between the lowering of the piezometric
surface and the rate and duration of discharge of a well usins
groundwater storage: Trans. Amer. Geophysical Union, vol. 16,
pp. 519-524.
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LOW-LEVEL RADIOACTIVE WASTE MODELING NEEDS
FROM THE USNRC PERSPECTIVE
R. Dale Smith
Edward F. Hawkins
Low-Level Waste Licensing Branch
Division of Waste Management
U.S. Nuclear Regulatory Commission
Washington, D.C. 20555
ABSTRACT
The NRC is developing a low-level waste regulation to be issued
for public comment by April 30, 1981. Publication of the final rule
and final EIS is expected in 1982. This regulation, 10 CFR Part 61
will apply only to near surface disposal. Two basic approaches have
been considered in developing specific regulations: a prescriptive
approach and a performance objective approach.
To accomplish the licensing and regulation of low-level waste
disposal facilities, the information that will be needed and the
methodologies to be used to assess performance are being established.
Regulatory guides will cover site selection and characterization,
facility design and operations, waste classification, waste form
performance, site closure, post-operational surveillance, funding
and monitoring. In developing licensing review procedures, initial
efforts are to establish analytical modeling capabilities. We are
establishing our modeling needs, determining models availability,
obtaining potentially useful models and adopting those that meet
our needs. As a test case, the Barnwell facility will be modeled
and its performance assessed. Our analyses will then be compared
with those of our contractor, ORNL and the Licensee.
The NRC schedule for promulgation of a low-level waste regulation
calls for issuance of the notice of proposed rulemaking along with a
supporting draft EIS by April 30, 1981. Publication of a final rule and
final EIS is expected in 1982 and will depend on the nature and extent
of comments received and whether a formal hearing is held.
31
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Part 61 will apply to the near surface disposal of waste by such
means as shallow-burial, engineered structures, and deeper burial.
The disposal of waste by other methods (e.g., deep mined cavity or
other deep burial) will not be dealt with in this rulemaking action
and will be considered at a later time through a separate rulemaking.
In developing specific regulations, we have considered two basic
approaches: a prescriptive approach and a performance objective
approach. In the former approach, specific detailed requirements for
design and operation of a waste disposal facility would be set out in
the regulations. Prescriptive standards would specify the particular
practices, designs, or methods which are to be employed—for example,
the thickness of the cover material (the cap) over a shallow land
burial disposal trench, or the maximum slope of the trench walls.
Setting of prescriptive standards requires a considerable amount
of detailed knowledge about potential designs, techniques, and proce-
dures for disposing of waste in order to prescribe which designs,
techniques, and procedures are best and also assumes that the state-
of-art in waste disposal is developed to the point where there are
clear choices to be made among all the potential approaches.
A regulation oriented toward performance objectives, on the other
hand, would establish the overall objectives to be achieved in waste
disposal and would leave flexibility in how the objectives would be
achieved. The performance standards would specify levels of impact
which should not be exceeded at a disposal facility in order to
provide protection against radiological hazards.
NRC staff believes that neither a purely prescriptive nor a
performance objective approach will be satisfactory in Part 61. A
combination of both approaches is needed such that performance ob-
jectives are established to define the overall performance expected
in disposal while at the same time, certain specific minimum tech-
nical criteria would be established.
NRC's overall goal is to assure protection of the public health
and safety. In considering radioactive waste disposal, this goal
would appear to fall into two time frames. That is, we must be
concerned with protection of workers and the public during the short-
term operational phase and protection of the public over the long
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33
term after operations cease. Thus, the near-term performance objective
will be to assure that the disposal facility will be operated in com-
pliance with existing standards as set out in Part 20 of our regula-
tions. Although routine operational releases are not expected at a
disposal facility, they might occur in those cases where waste pro-
cessing or volume reduction operations are conducted at a disposal
site. In such cases we would propose to apply the same limits that
would apply if these operations were conducted at fuel cycle facili-
ties (40 CFR Part 190).
Assuring safety over the long term involves two considerations:
1) protection of an individual who might unknowingly contact the
waste at some point in the future; and 2) protection of the general
public from potential releases to the environment. We believe that
intentional intrusion into the waste—e.g., an archaeologist
reclaiming artifacts-cannot reasonably be protected against. After
active institutional controls cease, however, one or a few individuals
could inadvertently disturb the waste through such activities as
construction or farming. It is important to note that consideration
of intrusion involves hypothetical exposure pathways which are assumed
to occur. Actual intrusion into the waste may never occur. But,
for purposes of Part 61, intrusion is considered such that if it
should occur the one or few individuals contacting the waste should
not receive an unacceptable exposure. With respect to the unknowing
intruder, we are presently considering use of the Part 20 maximum
individual dose limit of 500 mrem/yr to assure protection of such an
intruder.
With respect to migration and protection of the ground water,
we are evaluating a range of exposure guidelines from 1 to 25 mrem/yr
that would be applied at the site boundary. We expect our analyses
will result in a limit around 4-5 mrem/yr.
I'd like to review what these performance objectives mean
in terms establishing minimum technical requirements; that is, the
prescriptive part of the regulation. These requirements would in-
volve placing controls on the various parts or barriers of the
overall disposal "system" and will determine which wastes are ac-
ceptable for near-surface disposal and which wastes are not and must
therefore be disposed of by other methods providing much greater
confinement. The principal parts or barriers of an overall LLW
disposal system that are readily identifiable and will be addressed
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34
in the prescriptive requirements are:
— the characteristics of the waste and its packaging;
— the characteristics of the site into which the waste is placed;
— the methods by which the site is utilized, the waste emplaced and
the site closed; and
— the degree and length of institutional control, surveillance,
and monitoring of the site after closure.
It is instructive to start with the intruder performance objec-
tive since although it deals with a hypothetical pathway, it provides
a basis for several limiting requirements and establishes two
"classes" of waste suitable for near-surface disposal.
There are two principal means of controlling potential exposures
to an intruder—use of institutional controls and use of natural or
engineered barriers which would make it more difficult for a potential
intruder to contact the waste. Institutional controls can be further
separated into two types: (1) "active" controls, which require per-
formance of some action by a person or agency-e.g., controlled access
to the site, including barries to entry (fences) and periodic inspec-
tion and monitoring of the disposal site by a regulatory or other
governmental agency; and (2) "passive" controls, which are not
specifically dependent upon human actions-e.g., government ownership
of land, redundant records and restrictions on land use. Natural
or engineered barriers could include deeper burial or use of caissons
backfilled with concrete.
Neither institutional controls nor engineered barriers can be
relied upon to completely prevent human intrusion. Although we have
not completed our final analyses we expect to allow reliance on
active institutional controls to prevent intrusion for about 100
years. This will mean that certain wastes (e.g., those with short
half-lives) will not present an undue risk to an intruder when
active institutional controls are removed. The active controls will
be followed by several hundred additional years of passive controls.
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35
Passive controls, although certainly minimizing potential for inad-
vertent intrusion may not be completely effective in preventing it.
A second class of waste would then be that which may present an undue
risk to an intruder when active controls are removed. For this waste
additional measures must be taken to reduce this risk. This can be
done by introducing an additional barrier to intrusion, such as
deeper burial or engineered barriers. These additional measures
will not be relied on for more than 500 years; thus waste which
could potentially result in an exposure to an intruder greater than
500 mrem after 500 years is not acceptable for near surface disposal.
A similar type of analysis for migration cannot be performed in
as direct a manner. While intruder exposures are relatively non-site
specific, potential ground-water releases are very site specific and
are more difficult to treat in as generic a fashion as the intruder.
In order to carry out a site specific analysis, the variables beyond
the control of the site operator need to be fixed to some extent.
Of principal importance is the long-term stability of the waste and
disposal facility. Unless the waste and the disposal site are stable
over time, there is no way to predict the long-term radiological
impacts of disposal, or the activities (maintenance, monitoring, etc.)
and associated costs required to maintain potential impacts to low
levels. This would suggest that all waste should be placed in a
solid, non-compressible form for disposal in the near surface.
However, many wastes—particularly trash waste streams—contain very
low levels of activity and present little or no concern from the
standpoint of long-term migration. Thus, a more reasonable require-
ment would be that compressible waste containing low-levels of activity
should be segregated and disposed of separately. All other waste
would be placed in a solid, non-compressible form (e.g., solidifica-
tion in the case of resins and filter sludges) and the disposal
facility should be designed and operated to enhance stability over the
long-term. Certain nuclides, concentrations of nuclides or waste
forms may also require individual consideration from the standpoint
of migration on a site specific basis. This might include, for example,
a large one-time shipment of a very mobile radionuclide or a large
quantity of chelating agent in a single shipment. Thus, migration
considerations establish "classes" of wastes that require special
disposal considerations.
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36
We expect to establish a common sense base of siting requirements
that could be consistently applied throughout the country. The require-
ments would essentially eliminate certain limited areas from consider-
ation due to undesirable characteristics leaving large areas in each
region where acceptable sites could be found. The kinds of require-
ments we would expect to establish include:
1) Avoid areas of fractured bedrock;
2) Avoid siting in wetlands, high hazard coastal areas, flood-
plains, swamps or other types of very wet or potentially
very wet terrain;
3) The land surface should be relatively stable structurally
and geomorphically;
4) The hydrogeologic conditions of the site should be simple
enough for reliable residence time prediction to be made; and
5) The site should provide long travel times for radionuclides
from disposal trenches to the biosphere.
The completion of Part 61 does not, however, mean that our job
is done. To accomplish the licensing and regulation of low-level
waste disposal facilities, we are also establishing the information
needed and the methodologies we will use to assess the performance of
low-level waste disposal facilities. These efforts are concentrated in
two activities—the development of regulatory guides to support the
regulation and the development of licensing review procedures. The
regulatory guides are being initiated more or less in parallel with
our efforts on licensing review procedures. As we now envision it,
regulatory guides will be developed in the areas of site selection,
site characterization, facility design and operations, waste classi-
fication, waste form performance, site closure, post-operational
surveillance, funding and monitoring. As with the regulation, these
guides will be directed at disposal by means of shallow and inter-
mediate depth land burial. Future guides, as needed, will be
directed at other methods of disposal.
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37
To develop our licensing review procedures, that is, the way
we will review applications for low-level waste disposal, we are
initially concentrating on methodologies we will use to assess per-
formance. The first step in this development is to establish our
capability to perform analytical modeling. Although NRC has extensive
experience in modeling and some modeling has been done of low-level
waste disposal in the past, a systematic approach is now being taken
to establish a program to assure that our analytical capabilities
are in place as soon as possible. To accomplish this, we have em-
barked on an effort to establish our modeling needs, determine the
availability of models to meet these needs, obtain prospective models,
develop our operational capability of the selected models, and test
the models on existing or proposed sites. Let me make it clear that
it is not our intention to develop a new set of models. Rather, we
will investigate models that are already in use and adapt, or adopt
them for our use. If our investigations reveal that there are no
acceptable models to meet a specific need, we will then initiate a
program of model development.
Our first task will be to develop screening criteria to be used
to select potential models that can be used to simulate low-level
waste sites and the effects of construction, operation and ultimate
closure. Models will need to be able to simulate the geological
environment, ground and surface water pathways, air pathways, biotic
and vegetation pathways, radionuclide transport and the assessment
of potential radiological and non-radiological impacts. Models
will also be screened in terms of their level of detail and our need
for such detail. In some cases site screening and gross evaluation
type of models may be all that are required; whereas for other cases
very detailed, highly accurate models may be needed to fully assess
impacts. Perhaps in a few cases a model may need to be developed
that is applicable to a specific site. The screening criteria must
also take into account the need for models in a wide variety of
geologic and climatic settings, e.g., the obvious differences be-
tween arid and humid sites. This screening will also be done in the
context of the time and computer systems available to us.
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38
Through our own efforts and those of our contractors and consul-
tants, our search for available and acceptable models will begin with
a review of those that NRC already has. We will also investigate what
other agencies are doing and what they have available. Very specifical-
ly, this is one of the primary reasons we are here at this workshop. We
are very anxious to find out what you are doing. Our efforts will
also include the work going on at the National Laboratories, the
universities by private industry and any other source that we can find.
As we accumulate models for potential use, we will evaluate and
categorize them for our particular needs in terms of their usefulness
to us for making licensing decisions. Key items in our evaluations
will be:
— their applicability (humid vs. arid sites, saturated and unsaturated
flow, single and multiple aquifers, solute transport, dose assess-
ment, etc.);
— the details of the model (data requirements; simple, conservative
vs. detailed, more realistic; difficulty of use, model options,
computer requirements, etc.);
— availability of documentation (users and programmers manuals,
sample runs, examples, etc.); and
— validation and verification of models (theoretical proofs, pre-
vious applications to field conditions, reproducibility, re-
peatability, etc.).
It is obvious that this is a major undertaking both in terms of
time and staff effort. However, it is necessary if we are to be able
to make knowledgeable licensing decisions in a timely manner. To enable
us to focus on a specific goal, we have elected to concentrate our
initial efforts on performing an assessment of the Barnwell Low-Level
Waste Disposal Facility in South Carolina as a test case. There are
several reasons why we made this choice. First, we have agreed to
assist the state of South Carolina this next year in the environmen-
tal assessment of the Barnwell site as a part of their review for the
renewal of the state license for the faciltiy. Second, we have con-
tracted with Oak Ridge National Laboratory to assist us in this ef-
fort, including performance assessment of the facility. Theirs and the
licensees analyses should provide a good comparison to our efforts.
Last, since the Barnwell facility has been operating for some time
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39
and there are other nuclear facilities nearby that have been studied
extensively, we feel that data requirements for the selected models
should probably not be a limiting factor. This should provide us more
flexbility in model selection.
Once we have completed one test case (Barnwell) we will then turn
over efforts to expanding our capabilities to other types of sites
and situations. Hopefully, we will not have to go back to the
beginning of the exercise since we should have learned a great deal
the first time through and should have accumulated some applicable
information.
In conclusion, I would like to summarize the modeling needs from
our perspective:
1) We have identified the need for models that are directed at
enabling us to make licensing decisions.
2) We have determined that the ability to perform assessments
must be contained within our own staff.
3) Our modeling capabilities must be able, eventually, to
evaluate the full range of potential sites, facilities,
designs and operations that may occur. We have, however,
elected to first concentrate on models that are poten-
tially applicable to the Barnwell facility, which will
be used as a test case.
4) We have determined that the best course for us is to
adopt, or adapt, existing models to our use to the fullest
extent possible.
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Technical Papers
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MODELING CONTAMINANT TRANSPORT IN POROUS MEDIA
IN RELATION TO NUCLEAR-WASTE DISPOSAL: A REVIEW
David B. Grove and Kenneth L. Kipp
U.S. Geological Survey
P. 0. Box 25046, Mail Stop 413
Federal Center
Denver, Colorado 80225
ABSTRACT
The modeling of solute transport in saturated porous
media is reviewed as it is applied to the movement of radio-
active waste in the subsurface. Those processes, both
physical and chemical, that affect radionuclide movement are
discussed and the references that best illustrate these pro-
cesses listed. Movement is separated into convection,
convection-dispersion, and convection-dispersion and chemical
reactions. Solutions of equations describing such movement
are divided into one-, two-, and three-dimensional analytical
and numerical examples. Discussions of recent work in the
area of stochastic modeling are followed by discussions of
applications of the models to selected field sites.
INTRODUCTION
An important consideration in the evaluation of existing or pro-
posed nuclear waste-disposal proposals is the impact on the environment
and the exposure to man of waste material that may escape containment
and be transported out of the engineered repository. For the purpose
of this paper, presently available solute-transport models will be
reviewed with respect to the transport of radionuclides by ground-water
flow under saturated conditions from such a repository.
Mathematical modeling of radionuclide transport in porous media is
the only way to quantify transport rates and discharges. The steps in
modeling involve: (1) Formulation of mathematical expressions that
represent the primary physical and chemical mechanisms involved;
(2) determination of the parameters that appear in these expressions
using measured laboratory and field data; (3) finding an acceptable
method of solving these mathematical expressions; (4) evaluation of the
acceptability of the formulated mathematical model in reproducing
observed physical and chemical phenomena; and (5) the application of
the model to the prediction of the rates of radionuclide transport and
to evaluate various waste-disposal proposals. Examples and references
have been selected that the authors feel best illustrate the various
factors that affect such transport. The interested reader is referred
to a comprehensive review by Anderson (1979) for more detailed infor-
mation.
43
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THE GENERAL EQUATIONS
Because the primary mechanism of solute transport in ground water
depends on the direction and magnitude of the ground-water flow, the
equation describing such flow must be formulated and solved with the
appropriate boundary and initial conditions.
One form of the flow equation for a compressible fluid in a satu-
rated nonhomogeneous anisotropic porous medium is that given by
Konikow and Grove (1977) as:
4. no
P« at + pone at
s 3m. + +
v5- E gT+Wp (1)
vo *=1 3t
where
2
k.. is the intrinsic permeability (a second-order tensor), L ;
p is the fluid density, ML"3;
y is the dynamic viscosity, ML T ;
P is the fluid pressure, ML"1! ;
2
g is the gravitational acceleration constant, LT ;
*
z is the elevation of the reference point above a standard
datum, L;
* *
W = W (x,y,z,t) is the volume flux per unit volume (positive sign
for outflow and negative for inflow), T"';
* -3
p is the density of the source/sink fluid, ML ;
a is the vertical compressibility coefficient of the medium,
p is the fluid density at a reference pressure, temperature, and
concentration, ML"3;
n is the total porosity for compressive storage (dimensionless);
e is the effective porosity (dimensionless);
1 2
3 is the compressibility coefficient of the fluid, LM T ;
o
v is the reference volume of the fluid, L ;
m, is the mass of species k in the reference volume v , M;
K . 0
s is the number of species, (dimensionless);
x. are the cartesian coordinates, L; and
t is time, T.
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The equation for transport of a dissolved chemical species can be
obtained by combining the principle of the conservation of mass with
Pick's law of diffusion. Assumptions that result in the following
equation can be found in Konikow and Grove (1977):
8C 3
e "at" ~ "iw~~
dt cX •
T, \ d / £- \
- C'W* + £ Rk (2)
where
C is the concentration of the solute species in the flowing
phase, ML"3;
e is the effective porosity of the medium;
2 -1
D.. is the hydrodynamic dispersion coefficient (a tensor), L T ;
V. is the interstitial velocity, LT"1;
W is the volumetric flow rate/unit volume of porous medium of a
source/sink, T~^;
C1 is the concentration of the species in the source/sink flow,
ML"3; and
R, is the rate of production of solute species in the kth
reaction T~lL~3.
Once the set of reactions described by R, is specified, equation
(2) can be written for each chemical species of a system and the set of
equations solved simultaneously with the flow equation. The boundary
conditions for the solute-transport equation include boundaries along
which the solute concentration is specified, and those along which
continuity of the flux of the solute species is required. Initial
conditions specifying the solute-concentration distributions must be
given for all species.
The primary coupling of the flow equation to the solute-transport
equation is through the interstitial velocity obtained from Darcy's law.
The back-coupling or effect of the transport equation on the flow
equation is through the dependence of fluid density and viscosity on
solute concentration.
For instances where the solute concentrations are very dilute it
commonly is assumed that the fluid density and viscosity are constant.
The flow equation can then be solved independently from the transport
equation. In essence, the presence of the chemical-solute species does
not affect the flow field, but the flow field is still the primary
mechanism for solute transport.
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46
For studies where a two-dimensional areal model is appropriate,
the above equation can be integrated in the vertical direction through-
out the saturated thickness of the aquifer. Such situations occur when
horizontal ground-water movement is much larger than vertical movement.
TRANSPORT MECHANISMS
The primary solute-transport mechanism is the flow of ground water
resulting in convection of the solute species, and is represented by
the second term on the right hand side of equation (2).
Hydrodynamic dispersion, a secondary transport mechanism, des-
cribes the net macroscopic transport effect of movements of the
individual solute particles through the porous medium, Bear (1972).
Hydrodynamic dispersion consists of two processes. One is diffusion
of solute on the molecular scale described by a Fickian diffusion
term with a molecular-diffusion coefficient. The other is mechanical
dispersion that describes transport due to the variations of velocity
across individual pores, and the mixing of fluid at pore junctions.
The two coefficients usually are lumped together and the mechanical
dispersion dominates in all but the slowest of flow systems.
Scheidegger (1961) relates the dispersion coefficient to the
interstitial velocity by the equation:
V V
D.. = a.. -*-JL (3)
^J i3im I y I
where
a.. is the dispersivity or characteristic length of the porous
•z-jroi med-jym (a fourth-order tensor), L;
V and V- are the components of the flow velocity of the fluid in
m the m and « direction, respectively, LT~1; and
|V| is the magnitude of the velocity vector, LT~ .
When the coordinate system is aligned with the average velocity
in a steady, uniform flow field in an isotropic medium, the dispersion
coefficient tensor reduces to two components, a longitudinal and a
transverse coefficient (Scheidegger, 1961, and Bachmat and Bear, 1964,
and Bredehoeft, 1969).
Recent analyses (Smith and Schwartz, 1980) of the dispersive
effects caused by macroscale heterogeneities in the permeability of a
porous medium, have shown that dispersion of solute at this scale is
not described by a Fickian diffusion term.
Transport and the associated dispersion and diffusion through
fractures,where there may be a flowing and a stagnant fluid phase, can
be analyzed using the concept of dead-end pores and is discussed by
De Smedt and Wierenga (1979).
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47
CHEMICAL REACTIONS
The reaction term includes all chemical processes that can occur
during transport (Back and Cherry, 1976). These may include homo-
geneous reactions that occur within the fluid phase and heterogeneous
reactions that take place between the solid and liquid phases.
An example of a common homogeneous reaction is an irreversible
rate reaction such as radioactive decay. In this instance the reaction
rate term R in equation 2 becomes:
R = - x(ec + pb c) (4)
where
X is the radioactive-decay constant (equals In 2/half life), T ;
and
c is the concentration of the species adsorbed on the solid (mass
of solute/mass of sediment), ML"3.
Heterogeneous reactions include sorption ion exchange, diffusion into a
stagnant fluid phase with possible sorption or reaction, precipitation
and dissolution. When heterogeneous reactions are being considered, a
separate mass balance equation for the sorbed or stagnant phase must
be solved simultaneously with equation (2). For the simple hetero-
geneous reaction such as equilibrium-controlled sorption or ion exchange
with a linear adsorption isotherm the reaction rate term for a single
species can be written as:
R = - K. p. T£ (5)
Q D dt
where
K. is the distribution coefficient that defines the relationship
between the dissolved and adsorbed ions.
Grove (1976) and Enfield and others (1976) discuss basic forms of
sorption and ion-exchange mechanisms. The sorption mechanisms that
have been simulated to date include kinetically controlled sorption
and equilibrium sorption.
MATHEMATICAL MODELS OF SOLUTE TRANSPORT
The solutions to the governing equations of flow and solute
transport for a given system geometry, boundary conditions, initial
conditions, and reaction-mechanism procedure form the mathematical
model of the system. The following sections summarize transport
models that have been developed which are relevant to nuclear-waste
disposal. Analytical and numerical solutions to the models are dis-
cussed separately with these being further subdivided under the
headings of (1) convection, (2) convection-dispersion, and (3) convection-
dispersion and chemical reactions. When appropriate, the types of
models are further divided according to dimensionality.
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48
Analytical Models
Convection. The transport model for convection only in one
dimension is trivial as the output matches the input with only a time
delay for travel. For two dimensions, in a horizontal plane, Nelson
(1978a, 1978b) gives several examples for travel times and concentra-
tion changes during solute movement from a source to an outflow boundary.
Convection-dispersion. For the convection-dispersion model in one
dimension the flow field is usually taken as uniform and steady, thus
eliminating the need to solve the flow equation. Bear (1972) presents
some solutions for a variety of boundary conditions and source terms
for this case. Other solutions include those of Brenner (1962),
Lindstrom and Boersma (1971), Gelhar and Collins (1971), and Al-Niami
and Rushton (1977). For two dimensions, solutions are presented by
Bear (1972), Sauty (1980) and Fried and Combarnous (1971). For the
cases of radial transport and equation transformation of equations to
simpler forms, results are available from Bear and Jacobs (1965), Bear
(1972), Hoopes and Harleman (1967), and Phillips and Gelhar (1978).
Convection-dispersion-reaction. Examples of one-dimensional
solutions that involve linear homogeneous reactions are given by
Bear (1972), Marino (1974a), and Parlange and Starr (1978). Two-
dimensional examples with similar reactions have been presented by
Hunt (1973, 1974). Many one-dimensional transport models have been
developed for various heterogeneous reactions. Examples for linear
equilibrium sorption and linear kinetically-controlled sorption are
given by Lapidus and Amundson (1952), Ogata (1964), Girshon and Nir
(1969), Marino (1974a, 1974b) and Enfield and others (1976). An
analytical transport model for a three-member radionuclide decay chain
with linear equilibrium sorption and an impulse or decaying-slug source
terms has been derived by Lester and others (1975). The cases of both
dispersive and nondispersive transport were treated. Nondispersive
transport was examined in a similar manner by Rosinger and Tremaine
(1978). Details of the numerical implementation of the model are
given by Kipp (1979) and DeMier and others (1979).
Examples of analytical solutions for two-dimensional transport
including kinetic sorption and decay are presented by Eldor and Dagan
(1972) and Wilson and Miller (1978).
Numerical Models
Convection. The one-dimensional convective-transport case with
constant interstitial velocity can be solved analytically and is of no
interest numerically. For two dimensions Nelson (1978c, 1978d) gives
several good examples of modeling convective-transport and transient-
flow systems.
Convection-dispersion. The numerical solution of the solute-
transport equation with dispersion can be quite difficult because of
numerical-diffusion errors and solution oscillations caused by the
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49
particular technique. Because of these problems a variety of numerical
techniques have been used. These include finite-difference and finite-
element methods and the method of characteristics (MOC). Examples
illustrating these problems and the use of various methods to minimize
numerical dispersion (Lantz, 1971) and oscillations have been proposed
by Stone and Brian (1963), Chaudari (1971), and Chhatwal and others
(1973). Examples of finite-element methods used to overcome such
difficulties for one-dimensional problems are given by Price and
others (1968), Pinder and Shapiro (1979) and Van Genuchten and Gray
(1978). Huyakorn (1977) proposed using an upwind finite-element method
to overcome the oscillation problem in convection-dominated transport.
Pinder and Cooper (1970) and Reddell and Sunada (1970) describe the
method of characteristics/particle in cell model for transport cal-
culations that overcomes numerical-dispersion and overshoot/undershoot
problems. A description of the U.S. Geological Survey's MOC model is
given by Konikow and Bredehoeft (1978).
Peaceman and Rachford (1955, 1962) presented one of the first
finite-difference numerical techniques for two-dimensional transport.
Several of the finite-difference formulations cited previously have
been extended to two-dimensional problems with convective and dis-
persive transport.
Many two-dimensional finite-element based solute-transport models
have been developed. Guymon (1970) used a variational principle;
Nalluswami and others (1972) included the mixed partial-derivative
terms; Pinder (1973) used the Galerkin technique; and Smith and others
(1973) demonstrated the greater generality of the Galerkin technique
over variational formulations. Lee and Cheng (1974) and Segol and
others (1975) used a finite-element method to solve the density
dependent coupled flow and solute-transport equations.
Huyakorn and Taylor (1977) compared the relative merits of the
velocity-pressure formulation, the stream-function formulation, and
the hydraulic-potential formulation for the coupled flow and solute-
transport problem and found the first method to be the most accurate,
yet the most expensive.
The three-dimensional solute-transport models with dispersion and
convection include Khaleel and Reddell's (1977) and Reddell and Sunada's
(1970) method of characteristics model and Gupta and others (1975)
finite-element model.
Convection-dispersion-reaction. The numerical simulation of
convection, dispersion, and reaction of chemical species through
porous media in most instances simply involves the addition of the
reaction term on the convection-dispersion equation. While most
applications of this are for one-dimensional column experiments the
extension of the principle to two- and three-dimensional systems is
quite straight forward.
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The one-dimensional PERCOL model of Routson and Serne (1972)
treats convective transport with equilibrium reactions of a set of
"macroions" (ions present in high concentrations). It then computes
the transport of microions (ions present in low concentrations), which
are radionuclides present in small concentrations and undergo equilib-
rium sorption. The sorption coefficients are a function of the macroion
concentrations.
The MMT1D radionuclide-transport model of Washburn and others
(1980) computes the transport of N-member radionuclide decay chains and
includes linear equilibrium-sorption with constant sorption coefficients.
The dispersive transport is simulated by a discrete parcel random-walk
technique described by Ahlstrom and others (1977). Hill and Grimwood
(1978) also developed a one-dimensional model of radionuclide-chain
transport with linear equilibrium sorption and radioactive decay.
Finite-difference techniques are illustrated by Lai and Jurinak
(1972) and finite-element methods by Rubin and James (1973). Both
solve the general differential equation with linear or nonlinear
reactions.
An early two-dimensional transport model with chemical reactions
was that of Schwartz and Domenico (1973); it used finite differences,
neglected dispersion, but treated a multiplicity of chemical reactions
including both equilibrium and kinetic. The REFQS model of Shaffer
and Ribbens (1974) also calculated the transport of several chemical
species using the same techniques as Schwartz and Domenico, but
included dispersive transport.
Schwartz (1975) developed a radionuclide transport method-of-
characteristies model which included binary cation exchange and linear
decay. Finite-element techniques were used by Duguid and Reeves (1976)
to simulate transport with equilibrium sorption and linear decay; by
Prakash (1976) for transport in a bounded radial-flow system for a
partially penetrating well; by Cabrera and Marino (1976) for two-
dimensional flow and transport in a stream-aquifer system; by Pickens
and others (1979) for transport to tile drains; and by Pickins and
Lennox (1976) for a steady free surface-flow system with constant
equilibrium sorption.
Ahlstrom and others (1977) presented the Discrete Parcel Random
Walk transport model, MMT-DPRW, which is an extension of the particle-
in-cell method in that the dispersive transport is simulated by a
statistical random-walk computation. The idea of a numerical random-
walk simulation for dispersive transport in porous media also was used
by Todorovic (1975). A cell grid is used and parcels of fluid are
tracked during the simulations. The model can incorporate the chemical
reaction computations performed by Routson and Seme's (1972) PERCOL.
The case of flow in a fractured medium with diffusion into adjacent
porous blocks, including linear equilibrium sorption and linear decay
mechanisms, has been investigated using finite elements by Grisak and
Pickens (1980).
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Three-dimensional flow systems with solute transport are difficult
to model numerically because of the large number of computational nodes
involved, even for systems of moderate size. An approximate method of
treating this problem was given by Ross and Koplik (1979) who perform
the one-dimensional transport calculations with equilibrium sorption
and linear decay along steady-state streamtubes generated by an appro-
priate three-dimensional flow model. A convolution integration with
the analytical Green's function solution is used to obtain the con-
centration field.
The INTERA (Intercomp Resource Development and Engineering, Inc.,
1976, INTERA Environmental Consultants, Inc., 1979) three-dimensional
flow, solute- and energy-transport model is a fully coupled flow and
transport simulation code using finite-difference techniques. Linear
equilibrium sorption and linear decay for single-species transport is
included. This model was extended to handle radionuclide-decay chains
and named SWIFT (Dillon and others, 1978). The radionuclide species
are assumed to be in dilute solution and thus do not affect the flow
field.
COMPUTATIONAL PROBLEMS IN SOLUTE TRANSPORT MODELING
With analytical-solution models, the only difficulties that may
arise are roundoff errors, overflows in certain function evaluations,
and insufficient accuracy in numerical quadrature. The first two
problems can be resolved by combining terms in the proper order. This
is particularly necessary in the radionuclide-chain calculations.
Accurate numerical quadrature can usually be achieved by using an
adaptive method and isolating any significant peaks of the integrand.
With finite-difference and finite-element models there is the
possibility of excessive numerical dispersion (Oster and others, 1970;
Lantz, 1971) oscillation in the solution due to the space- or time-
discretization method, (Price and others, 1966), and overshoot/undershoot
near a steep concentration gradient (Gray and Pinder, 1976). Oscillation
and overshoot/undershoot are probably related if not the same thing. As
mentioned above, various correction terms have been used to minimize
numerical dispersion. The method of characteristics was developed to
avoid this problem.
Oscillation problems can be avoided by proper selection of time-
and space-step sizes (Price and others, 1966; Siemieniuch and Gladwell,
1978) but restrictions may make a given computation quite impractical.
Higher order time-integration methods have been proposed to avoid
oscillation problems, for example, Norsett methods (Smith and others,
1977). Overshoot/undershoot occurs because a finite-difference or
finite-element representation cannot propagate all frequencies contained
in the solution-function representation at the same phase speed or
without attenuation (Gray and Pinder, 1976). One of the advantages
of finite-element methods is that the use of higher-order basis functions
gives better propagation behavior over a wider range of frequencies.
Higher-order basis functions minimize numerical instabilities but cost
more in computer-solution time (Grove, 1977).
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52
Discrete-parcel tracking or Lagrangian-simulation techniques are
oscillation free and nearly free from numerical dispersion and are mass
conservative at the cost of a slower computation and the presence of
some random noise errors from the dispersion calculation (Ahlstrom and
others, 1977).
The main problems with three-dimensional models are the computation-
time and computer-storage requirements. This is particularly true if
transport of multiple species are being simulated. Fast and efficient
equation solvers for the matrix equations generated have been developed
and are being improved. Out-of-core matrix storage is another technique
being used to minimize core-storage requirements. At the present time
however, many of the larger three-dimensional problems can only be run
on large, very fast computers.
STOCHASTIC CONCEPTS IN SOLUTE TRANSPORT MODELING
It has long been realized that there is uncertainty in the des-
cription of the physical and chemical properties of the porous medium.
To collect sufficient data on a real system for a precise deterministic
model is an impossible task. Thus, some effort has been made to attempt
to quantify the uncertainty in transport-model prediction and correlate
it with the uncertainty in the various model parameters. Only a few
results have been achieved to date. A group of studies involved the
modeling of large-scale dispersion by using macroscopically variable
hydraulic-conductivity fields (Warren and Skiba, 1964; Heller, 1972;
Schwartz, 1977; Smith and Schwartz, 1980). The models used tracer-
particle tracking techniques with a one-dimensional mean flow field.
Results of these studies showed that a unique dispersivity cannot be
defined when the low-permeability inclusions are not arranged homo-
geneously. Smith and Schwartz (1980) found that even with statistically
homogeneous conductivity fields, the tracer particles do not take on
a normal distribution and a constant dispersivity does not exist. Also,
a large uncertainty is associated with the predicted solute elution
curve even when the statistical features of the porous medium are known.
Gelhar and others (1979) looked at longitudinal dispersion due to
variations in hydraulic conductivity in a stratified porous medium.
They developed a stochastic differential equation describing the system
and used spectral techniques to solve it. The transient nature of the
dispersion process and non-Fickian dispersion effects were identified.
All of the above studies show that, unless sufficient transport
time has elapsed, dispersion due to large scale heterogeneities of
the porous medium does not obey the deterministic Fickian type
mechanism.
FIELD APPLICATION OF TRANSPORT MODELS
Low-level radioactive waste-disposal sites are being thoroughly
studied by the U.S. Geological Survey as well as other Federal and State
agencies. Although a qualitative knowledge as to what is going on at
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53
these sites is usually known,a quantitative understanding of contaminant
movement, both within and off site, generally is lacking. Such movement
can be studied through the use of the ground-water solute-transport
models discussed earlier. As mentioned, a wide variety of models are
available for use. However, examples of field applications using
such models available in the general literature are somewhat limited.
There are, however, several situations where modeling has been
applied to the movement of chemical species in ground water with good
results. Although those discussed in the following paragraphs are not
exhaustive they are representative of "state-of-the-art" applications.
Discussions of each field site will be brief and interested readers are
referred to the references for more detail.
As was mentioned previously, solution of the solute-transport
equation presents significant mathematical problems that must be over-
come by fairly sophisticated solution methods. For this reason the
first studies of field applications used a particle-tracking technique
that was entitled the method of characteristics (HOC). The technique
was used by Bredehoeft and Finder (1973) who studied the salt-water
contamination of a limestone aquifer at Brunswick, Georgia. The source
of the contamination was underlying brackish water migrating upward
through natural conduits in response to lowered hydraulic heads in the
overlying aquifers. Simulation studies showed interceptor wells to be
the most feasible method to protect the well field. This reference also
provides an extensive theoretical background to the partial differential
equations that describe solute transport.
Konikow and Bredehoeft (1974) investigated salinity increases in
ground water and surface water in an alluvial valley in southeastern
Colorado as related to irrigation practices. The MOC model was used.
Dissolved-solids concentrations calculated by the model were within
10 percent of the measured values for both the aquifer and stream
approximately 80 percent of the time.
Konikow (1976) used this same model to reproduce the 30-year
history of chloride contamination at the Rocky Mountain Arsenal,
Colorado. Here, liquid industrial waste had seeped out of unlined
disposal ponds and spread over many square miles in an alluvial
aquifer.
Robson (1974) used the MOC to model the water quality of a shallow
alluvial aquifer near Barstow, California. The aquifer had been
polluted by percolation of wastes and sewage from industrial and
municipal sources for about 60 years. Several strategies to alleviate
the pollution problem and to protect downgradient domestic wells were
investigated by model simulation. Robson (1978) also modeled the same
system in a vertical section. This model has the advantage of being
able to simulate vertical flow and water quality changes in a single-
or multiple-aquifer system.
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54
Robertson (1974) used a MOC water-quality model to predict the
migration of industrial and radioactive liquid wastes throughout a
15-square-mile area at the Idaho National Engineering Laboratories
(INEL) in Idaho. The wastes were disposed of in the Snake River Plain
aquifer. Chloride and tritium were found to have moved as far as
5 miles downgradient from the injection point. Strontium-90 and
cesium-137 were significantly retarded due to sorption. The strontium
was detected no more than 1.5 miles downgradient and cesium migration
was not detected. All of the above species were adequately simulated
using the MOC model and estimates made of the long-term effects of
these contaminants on aquifer usage. Robertson (1977) also modeled
solute transport from waste seepage ponds at INEL. A three-segment
numerical model, incorporating analytical and multidimensional
numerical solutions, was used in this study.
Grove (1977) used a two-dimensional finite-element solute-
transport code to simulate the same data from Robertson's (1974) study.
Virtually the same results were obtained using this numerical technique.
Finder used the finite-element method to simulate several field
situations of water-quality change. A chromate contamination problem
at Long Island, New York was well simulated by Pinder (1973). He also
simulated the two-dimensional movement of a saltwater front in the
Cutler area of the Biscayne aquifer near Miami, Florida (Pinder, 1975).
Due to the density effect caused by the saltwater concentrations, a set
of non-linear partial differential equations had to be solved simul-
taneously.
Cheng (1975) used a finite-element method to study the movement of
variable-density fluids through a causeway dividing the Great Salt Lake
in Utah. The model did not match concentration variables measured in
the field, but it did give some insight into the hydrology of the system.
Ahlstrom and others (1977) documented the theoretical foundation
and numerical-solution procedure for the Hanford MMT-DPRW model. A
simulation of the tritium plume beneath the Hanford, Washington site
was included to illustrate the use of the model in a typical application.
CONCLUSION
It is clear from the review of available ground-water solute-
transport models for saturated flow systems that many of the basic
mechanisms that apply to real site situations can be simulated with
certain simplifying assumptions. Analytical models are adequate for
parameter-sensitivity analyses and preliminary investigations of
transport phenomena, particularly when there is a gross lack of field
data for model-parameter determination. The simplifying assumptions
necessary for analytical solutions restrict their ability to realisti-
cally model a real system. Numerical models have the ability to treat
the more complicated systems such as heterogeneous porous media.
However, determining the parameter distributions for a real system from
a sparse data set can be fraught with uncertainties that limit the
quality of the model predictions.
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55
Areas where the modeling technology need further development
include flow and transport in fractured/porous media, transport of
multiple chemical species with complex chemical and sorptive inter-
actions, and transport modeling using stochastic principles in order
to handle the uncertainties in real systems.
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medium: Water Resources Research, v. 11, no. 2, p. 348-354.
Van Genuchten, M. Th., and Gray, W. G., 1978, Analysis of some
dispersion corrected numerical schemes for solution of the
transport equation: International Journal of Numerical Methods
in Engineering, v. 12, no. 3, p. 387-404.
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63
Warren, J. E., and Skiba, F. F., 1964, Macroscopic dispersion:
Society Petroleum Engineers Journal, p. 215-230.
Washburn, J. F., Kaszeta, F. E., Simmons, C. S.,
1980, Multicomponent mass transport model:
migration of radionuclides in groundwater:
Laboratory, Rich!and, Washington, PNL-3179,
and Cole, C. R.,
A model for simulating
Pacific Northwest
130 p.
Wilson, J. L., and Miller, P. J., 1978, Two-dimensional plume in
uniform ground-water flow: Proceedings American Society Chemical
Engineers, v. 104, HY4, p. 503-514.
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AN OPTIMUM MODEL TO PREDICT RADIONUCLIDE TRANSPORT IN AN AQUIFER
FOR THE APPLICATION TO HEALTH EFFECTS EVALUATIONS
Cheng Y. Hung
Criteria and Standards Division
Office of Radiation Programs
U.S. Environmental Protection Agency
Washington, D.C. 20460
ABSTRACT
This paper presents an optimum groundwater transport
model for simulating radionuclide transport in an aquifer
using an approximate solution of the basic transport
equation. The model is designed to avoid (1) relatively
high computer simulation costs normally experienced in
numerical models and (2) the large errors which are
sometimes introduced when the physical boundary conditions
are converted to a mathematical form suitable for an
analytical model. The model neglects the effect of
radionuclide transport through dispersion first and then
compensates for this effect later with a health effects
correction factor. This correction factor is found to be
a function of the Peclete number and the "transport
number," which has been defined, and can be determined by
using the parameters of the groundwater transport system.
The model has a low cost of simulation and yet maintain-
ing reasonable accuracy for the peak and cumulative radio-
nuclide discharges, which are of primary interest for
risk assessments. Preliminary analyses indicated that
the model can be integrated into most of the risk assess-
ment models for health effects evaluations.
INTRODUCTION
Comprehensive studies on the evaluation of the optimum method of
disposal of radioactive wastes have been conducted by various government
agencies [1,2] and by an Interagency Review Group [3]. All of these
studies have unanimously concluded that the geological disposal method
is one of the most viable alternatives. Because transport through an
aquifer is a primary pathway of transporting radionuclides to the
biosphere, a groundwater model which simulates the migration of radio-
nuclides in an aquifer is one of the important transport submodels
employed for health effects evaluation.
To date, there are more than one hundred and eight groundwater
transport models [4]. However, some of the models fail to consider the
process of radioactive decay and/or sorption and are, therefore, not
65
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66
suitable for application to health effects evaluations. The rest of the
models which may be suitable for health effects evaluation can be sub-
divided into two groups: the analytical model and the numerical model.
In general, analytical models are limited by specific mathematical bound-
ary conditions which require some approximation of actual physical condi-
tions [5, 6, 7]. As a result, these models suffer considerable error in
simulation due to these approximations.
The application of a numerical model requires, in general, many more
tedious computations than the analytical approach does. Besides, the
accuracy of simulation depends greatly on the adjustments of time and
space increments; severe errors may result if they are improperly adjusted.
On the other hand, a proper adjustment of these increments, in some
cases, may result in excessive computer time required for the simulation.
The cost may become prohibitive when the model is applied to long-term
simulations such as health effects evaluations.
The purpose of this study is to present a groundwater transport
model which could overcome the shortcomings of existing models and to
characterize the model when it is applied in health effects evaluations.
APPLICATION OF EXISTING MODEL IN HEALTH EFFECTS EVALUATIONS
The basic equations for a groundwater transport system include the
momentum, the energy and the continuity equations for the hydrodynamic
system, and the continuity equation for the solute and the constituent
radionuclide in a decay chain. Using tensor notations, the equations
take the form: [12]
Y. = -(k/y)(vp - pgvz) (1)
V-[(pk/y)H(vp - pgvz)] + V-j^-VT - q, - q'H
= (3/3t)[npU + (1 -nMpCpTj] (2)
7Py_ + q- = -(a/3t)(np) (3)
v-[pC(k/u)(vp - pgvz)] + nV-pD^-vC - q'C
= (3/at)(pnRC) (4)
in which C is the concentration of the radionuclide in the fluid phase;
y^ is the velocity vector; k is the permeability; v is the viscosity of
the fluid; p is the pressure; p is the mass density; g is the gravita-
tional acceleration; D is the dispersivity tensor; T is the temperature;
qL is the rate of heat loss; H is the fluid enthalpy; n is the porosity;
z is the height above reference plane; \A is the decay constant; R is
the retardation factor; U is the internal energy; Cp is the specific
heat; q' is the rate of fluid withdrawal; and subscripts h and c are
the heat energy and component of mass respectively.
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67
The above non-linear equations characterize the transport of radio-
nuclides in the groundwater system. Since each of the above equations
are related through dependent variables, the direct solution of the
system equation for any boundary and initial conditions is extremely
difficult and not practical. A commonly used practice in solving the
system equation is to assume that the flow of groundwater is steady and
that there is no heat energy being generated or absorbed in the system.
The system equation then reduces to a single equation.
R(9C/3t) - v-(jl'VC) + VvC + AdRC = 0 (5)
in which V is the interstitial velocity (V = v/n). Equation (5) has
been further simplified and solved by numerous authors [7, 8, 9] employ-
ing numerical or analytical approaches. The difficulties encountered in
applying these models for health effects assessments are described in the
following sections.
Numerical Models
The existing multidimensional models are solved either by the finite
difference method [8] or by the finite-element method [9]. Although a
model employing the finite-element method is, in general, found to be
more efficient than one employing the finite difference method [10], the
time required to execute a computer model employing the finite element
method is still far beyond the limitations of a normal project budget for
risk assessment when a number of cases must be considered.
The simulation of the vertical migration of tritium from a burial
trench toward the groundwater table conducted for the West Valley, New
York burial site [11] used finite element model developed by Duguid and
Reeves [9]. Approximately seven minutes of computer process time was
required on UNIVAC-110 computer for simulating 200 years of real time.
The analysis also indicated that the half strength concentration
(C/Co = 0.5) wave front moved 0.6 m in 20 years. Assuming that 1000
years of real time is required to simulate the system and that the time
required for the simulation of the radionuclide migration in a ground-
water stream is the same as that for the vertical migration, then the
total time required for simulating the West Valley or similar groundwater
system would be 700 min for completing simulation of ten critical radio-
nuclides. This is equivalent to $10,500 per run, a cost that is unques-
tionably excessive.
A one-dimensional INTERA model [12] developed by INTERA Environ-
mental Consultants, Inc., employs a finite difference method. It has a
computer processing cost of $200 for simulating the migration of a single
radionuclide in a one mile long aquifer for a 100,000 years real time
simulation. This is also too costly when many other radionuclides and
pathways are to be considered for a complete health effects evaluation.
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68
Analytical Models
Lester, Jansen, and Buckholder developed an analytical groundwater
transport model for a one-dimensional, semi -infinite aquifer system
which has impulse release and decay step release boundary conditions [6].
These boundary conditions at x = 0 are expressed mathematically as
C = C0fi(t) (6)
and
C = (C0/td)exp(-Xdt) (7)
for the impulse release and the decaying step release, respectively. In
the above equation, x is the space coordinate, t is the time, td is the
duration of radionuclide leaching, and CQ is the concentration of radio-
nuclides at t = 0.
Ford, Bacon, and Davis, Inc., developed an analytical model which
assumed that the rate of radionuclide release at any time is propor-
tional to the inventory of radionuclides remaining at the source point.
Mathematically, it is expressed as
C = (\yQw)exPL"-Ad + XL)t] (8)
in which AL is the leaching constant, Im is the initial inventory of
the radionuclides in the waste depository, and Qw is the rate of ground-
water flow.
These analytical models have made considerable contributions in
groundwater transport modeling by simplifying the procedures of simula-
tion. However, the application of these models to health effects
assessments is limited because it requires the approximation of con-
verting the physical boundary condition to a form meeting the require-
ments of an analytical model. This may result in considerable error
of simulation in some cases.
The above discussion implies that existing numerical and analytical
groundwater models may either be too costly to compute or may introduce
large errors when applied to health effects assessments. Therefore, a
more accurate and more economic groundwater transport model has been
developed for health risk assessment and is presented herein.
THEORETICAL BACKGROUND OF THE OPTIMUM GROUNDWATER TRANSPORT MODEL
FOR HEALTH EFFECTS EVALUATIONS
Derivation of Basic Equation
This transport model is intended to simulate a uniform, one-
dimensional flow in an aquifer. It receives radionuclides released from
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69
the waste disposal site and transports them to the biosphere for bio-
logical uptake. This includes wells, springs, streams, and other use
and discharge points. It is assumed that the dissolved raionuclides
are in sorption equilibrium with the aquifer formation and that decay
is in progress for both dissolved and sorbed radionuclides.
The basic one-dimensional groundwater transport equation for the
model simulating radionuclide migration in an aquifer may be reduced
from Eq. (4) to
D(32C/3x2) - V(9C/3x) - R(3C/3t) - AdRc = 0 (9)
and should be solved by using the following initial and boundary
conditions:
C = 0, at all x, when t = 0, (10)
C = Cn(t), at all x, when t > 0,
° (ID
C = finite, at x = °° when t > 0
For the convenience of health effects analysis, one may convert the
radionuclide concentration, C, into the rate of radionuclide transport
by multiplying Eqs. (9), (10), and (11) by the rate of groundwater flow.
When this transformation is completed, Eqs. (10) and (11) become:
D(32Q/3x2) - V(3Q/3x) - R(3Q/3t) - RXdQ = 0 (12)
Q = 0, at all x, when t = 0,
Q = QQ(t), at x = 0, when t > 0, and (13)
Q finite, CyH at x = °°, when t > 0
•
where Q denotes the rate of radionuclide transport.
Since Eq. (13) is an undefined boundary condition, the analytical
solution for Eq. (12) cannot be obtained. However, its solution can
be expressed in an integral form by employing the method of Green's
function, that is:
Q(t) = \(t - T)ll(T)dT (14)
0
in which u denotes the radionuclide release rate at the discharge end,
x = L, which responds to the unit release of a radionuclide at x = 0
and T is a dummy variable of time.
The response of unit release in Eq. (14), U(T), has been thoroughly
studied by Burkholder et al. [6] with the following results:
U(T) = (V/2L)V(RP/"03)Exp{-Nd0 - (P0/4R)[(R/0) - I]2} (15)
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70
in which P is the Peclet number (= VL/D); 0 is the dimension! ess time
(= TV/L); Nd is the decay number (= xrfL/V). By substituting Eq. (15)
into Eq. (14) one obtains.
Q(t) = Q/oU-r) (V/2L)V(RP/*e3)
Exp{-Nde-(Pe/4R)[(R/e) - l]
Equation (16) can not be integrated because Q0(t-r) is undefined.
However, if the dispersion term, D(92Q/3x2), in Eq. (12) is neglected,
then this response of unit release, U(T) can be simplified and becomes [6]
U'(T) = Exp(-RLXd/V)6(T - RL/V) (17)
As a consequence, Eq. (14) can be integrated and the result is
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71
If this is the case, there is a linear relationship between the cumulative
radionuclides released and the expected total health effects which is
governed by the health effects conversion factors [13]. Thus, Eq. (21)
can be rewritten as:
n = [HECFo/°Q(t)dt]/[HECF /°0/(t)dt] (22)
where HECF represents the health effects conversion factor. Subsequent
substitution of Eq. (14) into Eq. (22) yields
00 t« oo t
U U U
Since no radionuclides are being released at x = 0, when t<0, the upper
limit of the integration in Eq. (23) may be converted from time t to
infinity without changing the results of the integration. Thus, Eq. (23)
may be rewritten as:
00 „). m ^
^ = '-n ^ Qfp^~T'^*TjdTdt]/[ / / Qn(t-t)u"(T)dTdt1 (?4^
Now, since the time integral of the functions Qo(t--r) and U(T) converge
to finite numbers when t is infinite, Eq. (24) can be transformed to
Tl = [ /"U(T)dT /\(t-T)dt]/[ /Y(T)dTn/"Qn(t-T)dt]
co (25)
Equation (25) implies that the health effects correction factor n is
independent of environmental time and, hence, can be characterized
independent of time. Once the health effects correction factor is deter-
mined, the rate of radionuclide release at the discharge end may be
approximated by:
Q(t) = nExp(-RLXd/V)Q0(t - RL/V) (26)
Equation (26) is the basic equation of this groundwater transport model.
Characterization of Health Effects Correction Factor
To determine the characteristics of the health effects correction
factor, one may substitute Eqs. (15) and (17) into Eq. (25) and then
complete the integration of the denominator. This yields
y°0/2)J(RP/pe3)Exp[-N.e - (Pe/4R)(R/e - l)2]de
n = 0 1 d (27)
Exp(-RLxd/V)
A series of computations were made for the numerator and the denominator
of Eq. (27), using various values for P, Nd, and R. The error between
the results of analysis obtained for the denominator and for the numera-
tor were also computed and extended to evaluate the relative error.
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72
Careful examination of the results reveal that the relative error
is a primary function of the Peclet number and the parameter expressed
by RxdL/V. This parameter represents the ratio of the radioactive decay
constant, x^, and the "transport constant," V/RL, and is designated as
"transport number" for this study. The above results were plotted on a
Peclet number vs. transport number plane as shown in Fig. 1. This figure
indicates that the relative error, e, increases with the increase in
transport number and decreases with the increase in Peclet number.
Based on the definition of relative error, the health effects
correction factor can be computed by
n = 1 + e (28)
Since the relative error is always greater than 0, the health effects
correction factor is always greater than 1.
Proposed Groundwater Transport Model for Health Effects Assessment
Based on the previous discussions a groundwater system with initial
and boundary conditions as shown in Eq. (13) may be simulated by Eq. (26),
which is duplicated as;
Q(t) = nExp(-RLx /V)6 (t - RL/V) (29)
d 0
Equation (29) represents an algebraic equation, in which, constant n is
determined from Fig. 1 and Eq. (28); and the terms, RL/V and xd are also
known constants determined from the characteristics of the groundwater
system. Therefore, the proposed model described by Eq. (26) is the
simplest and most economical to process when it is integrated into a
health effects assessment model.
It should be noted that when this model is employed in a health
effects assessment model, there may be some error in the health effects
for each generation. The size of this error depends on the characteris-
tics of the system, the nature of the radionuclide being considered, and
the boundary conditions. The nature of this error is characterized in the
following section. Nevertheless, the cumulative health effects evaluated
for each generation, based on this model, is expected to be a insignifi-
cant error relative to the true solution obtained from the one dimensional
model expressed in Eqs. (9), (10), and (11). This fact is also discussed
in the following section.
CHARACTERIZATION OF THE POTENTIAL ERROR OF THE PROPOSED MODEL
Hypothetical Groundwater System
In order to characterize the nature of error incurred by the pro-
posed model, a hypothetical groundwater transport system was established.
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73
In the hypothetical groundwater system, it was assumed that: the veloc-
ity of groundwater flow is 100 m/yr; the location of radionuclide dis-
charge is 500 m down stream from the source point where the radionuclide
is released; the coefficient of dispersion is 300 m2/yr; the rate of
radionuclide release at the source point, x = 0, varies with time and
is represented by
Q0(t) = S1n[(ir/td)t] for 0 < t < td
Qo(t) = 0 for t > td (30)
where td is the duration of radionuclide release.
Three cases are assumed for the analysis. Case I represents a
fast release, t^j = 200 yr, and long radionuclide half-life, Xd =
0.000693. Case II represents a fast release, td = 200 yr, and short
radionuclide half-life xd = 0.0693. Case III represents a slower
release, td = 400 yr, and long radionuclide half-life, Xd = 0.000693.
Each case is also subdivided into three subcases with the retardation
factors equal to 1, 10, and 100, respectively.
Method and Results of Analysis
Two models were developed for the analysis. One of them was
developed based on a numerical integration of Eq. (16).and was desig-
nated the "Exact Model." The other model was developed based on
Eq. (29) and was designated as the "Optimum Model." Each of these
models were designated to compute the radionuclide discharge rate at
the downstream end of the aquifer and the cumulative radionuclides
released at the discharging point over infinite time. The results of
computer analyses using these two models are presented in Figs. 2,
3, and 4.
Discussion of Results
In case of long radionuclide half-life, Cases I and III, the rate
of radionuclide release and the cumulative radionuclide release com-
puted from the exact model and the optimum model agree very weTT for
all three subcases assumed. However, the results of the Case II
analysis, where the radionuclide half-life is short, indicate, that
the rate of radionuclide discharge obtained from the optimum model
deviates more and more from that obtained from the exact model as
the retardation factor increases. Nevertheless, the major deviations
are merely in the time-lag of radionuclide discharge, whereas the peak
discharge showed little deviation (Fig. 3). The results of this latter
analysis also indicated that the above deviation will be mitigated for
those radionuclides with longer half-life, i.e., with smaller transport
number (Fig. 2). Fortunately, a system with larger transport numbers
results in a limited quantity of radionuclide discharge at its dis-
charging point as can be seen by comparing case II-3 versus cases II-2
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74
or II-1. This result implies that the discharge of radionuclides under
a high transport number, such as Case II-3, would not be a major concern
from the view point of health effects assessment. Besides, this
deviation should further decrease with the increase in the duration of
radionuclide release at the radionuclide source point, x = 0, as shown
in Figs. 2 and 4.
Nevertheless, the cumulative radioactive discharge, which is the
primary parameter in determining the total health effects from a radio-
active waste disposal site, remains the same for values obtained from
the exact model and from the optimum model. This agreement implies
that the proposed optimum groundwater transport model can be integrated
into the health effects assessment model without introducing any signifi-
cant error in the assessment of health effects.
CONCLUSION
An optimum groundwater transport model for health effects assess-
ment has been derived and characterized. The processing time required
to simulate this model is significantly less than for other comparable
models because it only requires algebraic calculations.
The error from simulating the peak discharge of radionuclides
using the "optimum" model, as compared with the error from use of the
"exact" model [i.e., the numerical model represented by Eqs. (16) and
(30)], may be noticeable when the groundwater transport number is greater
than 5. However, the cumulative radionuclides released under the same
conditions is normally only a small fraction of the cumulative radio-
nuclides released at the source point. Therefore, the effect of this
error is mitigated.
The error in assessment of health effects from the proposed optimum
model, as compared to that obtained from an exact model, has been
found to be is insignificant in practical applications.
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75
REFERENCES
1. The Mitre Corporation, Alternative Disposal Concepts for High-level
and Transuranic Radioactive Waste Disposal, report prepared for
USEPA, ORP/CSD 79-1 (1979).
2. Ford, Bacon, & Davis Utah, Inc., Screening of Alternative Methods
for the Disposal of Low-Level Radioactive Wastes, report prepared
for USNRC, NUREG/CR-0308, UC 209-02 (1978).
3. Interagency Review Group, Report to the President by the Interagenc.y
Review Group on Nuclear Waste Management. TID-29442 (March 1979).
4. Science Applications, Inc., A Compilation of Models and Monitoring
Techniques, report prepared for USDOE, ORNL/SUB-79/13617/2,
SAI/OR-565-2.
5. C. B. Smith, D. J. Egan, W. A. Williams, J. M. Gruhlke, C. Y. Hung,
and B. Serini, Population Risks from Disposal of High-level
Radioactive Wastes in Geological Repositories. USEPA report,
520/3-80-006.
6. D. H. Lester, George Jansen, and H. C. Burkholder, "Migration of
Radionuclide Chains through an Adsorbing Medium," AICHE Symp.
Series 7j_: 202 (1975).
7. Ford, Bacon, & Davis Utah, Inc., A Classification System for
Radioactive Waste Disposal —What Waste Goes Where, report prepared
for USNRC, NUREG-0456, FBDU-224-10 (1978).
8. INTERA Environmental Consultants, Inc., Development of Radioactive
Waste Migration Model, report prepared for Sandia Laboratories
(August 1977).
9. J. 0. Duguid and M. Reeves, Material Transport Through Porous Media:
A Finite-Element Galerkin Model, Oak Ridge National Laboratory
report, ORNL-4929 (1976).
10. George F. Finder and William G. Gray, Finite Element Simulation in
Surface and Subsurface Hydrology, Academic Press (1977).
11. New York State Geological Survey, Computer Modeling of Radionuclide
Migration Pathway at a Low-Level Waste Burial Ground. West Valley,
New York, report prepared for USEPA (1980).
12. INTERA Environmental Consultants, Inc., Development of Radioactive
Waste Migration Model, report prepared for Sandia Laboratories
(August 1977).
13. J. M. Smith, T. W. Fowler, and A. S. Goldin, Environmental Pathway
Models for Evaluating Population Risks for Disposal of High-Level
Radioactive Waste in Geologic Repositories, USEPA report, 520/5-
80-002 (1980).
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76
o.i
0.5 1.0
5.0 10.0
50.0 100.0
Transport Number, RA L/V
d
Fig. 1. Results of the Health Effects Correction Factor Analysis
Using Eqs. (27) and (28).
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77
CJ
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78
10
-1
Cumulative Release, Case II-l
Rate of Release, Case II-l
Cumulative Release, Case II-2
Rate of Release at x=0
^— Result of Analysis, Optimum Model
——Result of Analysis, Exact Model
<-Rate Of Release, Case II-2
^Cumulative Release,
Case II-3
te of Release,
Case II-3
700
800
Fl9M ?', Con;Parison of the Results of Analyses Obtained from the
Optimum Model and the Exact Model Case II, *d =0.0693, td = 200 yrs
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79
L:
umulatrve Release, Case IIl-l
Cumulative Release,
Case III-2
ino
200 300
500 600
700
Time, yr
800
Fig. 4. Comparison of the Results of Analyses Obtained from the
Optimum Model and the Exact Model, Case II, xd = 0.000693, td = 400 yrs,
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MODELING OF WATER AND SOLUTE TRANSPORT UNDER VARIABLY
SATURATED CONDITIONS: STATE OF THE ART
E. G. Lappala
U.S. Department of the Interior
Geological Survey
Denver Federal Center
Denver, Colorado
ABSTRACT
This paper reviews the equations used in deterministic
models of mass and energy transport in variably saturated
porous media. Analytic, quasi-analytic, and numerical solution
methods to the nonlinear forms of transport equations are
discussed with respect to their advantages and limitations.
The factors that influence the selection of a modeling
method are discussed in this paper; they include the following:
1. The degree of coupling required among the equations
describing the transport of liquids, gases, solutes,
and energy;
2. The inclusion of an advection term in the equations;
3. The existence of sharp fronts;
4. The degree of nonlinearity and hysteresis in the
transport coefficients and boundary conditions;
5. The existence of complex boundaries; and
6. The availability and reliability of data required
by the models.
Analytic solutions are available for a variety of steady-
state nonlinear multidimensional problems such as infiltration
of liquid water from point or line sources. Analytic treat-
ment of heterogeneity and anisotropy is limited to spatial
variations in material properties (such as exponential depend-
ence of hydraulic conductivity with depth) that are not
generally realistic. Some quasi-analytic methods are numeri-
cally unstable, leading to oscillatory solutions, and in a few
cases, convergence to the wrong solution. The principal ad-
vantage of existing analytic and quasi-analytic methods is in
deriving general conclusions about the nature of solutions
that may be used to check the validity of numeric methods based
on finite differences or finite elements.
Numeric methods are reviewed as they have been applied
to variably saturated transport problems. The preferred
method with the most flexibility for extremely nonlinear water-
flow problems is a finite difference scheme applied to the
balance equation,- written in terms of pressure or total head,
with upstream weighting on the relative hydraulic conductivity.
81
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82
In addition, nonlinear coefficients should be evaulated
implicitly and improved by iteration. Galerkin finite element
schemes may result in oscillatory and non-convergent solutions
to these problems. Finite element methods based on orthogonal
collocation somewhat reduce these problems. However, the ad-
vantage of increased accuracy of finite element schemes over
finite difference schemes is reduced by the large computer
times required to formulate and solve the resulting algebraic
equations. Finite element schemes appear to be preferred
for linear and mildly nonlinear forms of the transport equa-
tions when sharp fronts are involved. Verified, documented
computer codes that use both finite element and finite differ-
ence schemes are available for a variety of realistic prob-
lems that arise in designing and managing waste disposal sites.
However, use of these models is not routine because of numeric
problems of solving the descriptive equations. Consequently,
most problems should be analyzed with more than one approach
or model to assure consistency.
INTRODUCTION
Quantitative description of movement of water and solutes through
variably saturated porous media is required to enable adequate design and
management of waste-disposal sites. Water originating as precipitation
or irrigation water percolates through overburden and the waste itself.
This water may dissolve pollutants which may reach underlying aquifers
or streams and lakes by downward and lateral movement. In addition,
gaseous waste forms may escape to the biosphere via upward diffusion.
Thus, the timing and location of pollutant loadings to potable water
supplies and the biosphere require quantitative description of the
movement of recharge waters, dissolved solutes, and gaseous pollutants.
Since this movement occurs through porous and fractured media that
may or may not be totally saturated with water, a three-phase system
comprising moist air, water, and solids must be analyzed. The equations
describing water and solute movement in these systems are complicated;
their solution usually requires some type of numerical approximation.
These approximations, the required transport coefficients, and initial
and boundary conditions comprise a model of a particular field problem.
Field problems generally include heterogeneity in the transport coef-
ficients and boundary conditions that vary in both time and space.
Until recently, solutions to these equations for realistic field
problems were impossible; however, with the development of modern computers,
many realistic problems can be analyzed. It is the purpose of this paper
to: (1) review the equations involved in variably saturated mass and
energy transport and the principal modeling methods that have been used for
the solution of these equations; (2) present problems encountered in the
principal solution methods; and (3) present needs for future work in the
application of models incorporating these methods. About 330 reports on
solutions to the equations for transport of mass and energy in variably
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83
saturated systems were reviewed. Primary emphasis of the review was on
problems involving flow of liquid water and dissolved solutes. However,
some discussion of problems involving combined movement of heat and
moisture in both liquid and vapor phases, and gaseous diffusion is in-
cluded. Some of the references through 1971 include comprehensive reviews
by other authors (Braester and others, 1971). Problems of transport in
systems that are saturated at all times will be covered in another paper
at this symposium (Grove and Kipp, 1980).
BASIC TRANSPORT EQUATIONS
Physics-based models of movement of water, solutes, and heat through
variably saturated porous or fractured media are derived by combining
mathematical statements of the conservation of mass and energy with
constitutive equations relating these statements to measurable variables,
such as pressure, temperature, and concentration. These equations are
generally written to apply over some finite space in which the transport
properties are constant or over which they can be assumed to vary in
some simple manner. The continuity of mass equation is:
/V
where
S => a general mass storage coefficient,
p = the mass density,
t = time,
V = volume of the region considered,
S = surface of the region considered,
q = the mass flux density orthogonal to the surface S.
A similar equation for the balance of energy is:
where
Su = a general enthalpy storage coefficient,
H
h « energy content or enthalpy,
q. => enthalpy flux.
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84
Constitutive equations that relate mass and energy flux to measurable
variables state that the flux is proportional to the first power of a
driving force which is expressed as a gradient in the measurable variable.
For the flux of liquid water (q ), a generalized Darcy's law is the
appropriate equation:
where
p. =* mass density of liquid water,
K =* liquid hydraulic conductivity coefficient,
K. « conductivity for thermally induced liquid flux,
K = conductivity for osmotically induced liquid flux,
$ > Q > $i.» aiu* $ are pressure, gravitational, thermal, and osmotic
p Tg h o
potentials,
7 = gradient operator,
i = subscript to indicate liquid water.
In most applications, the only potentials of significance are the
first three. Osmotic potential may be important where materials are
present that act as semipenneable membranes.
A similar equation for the flux of moist air (qfl), neglecting all
driving forces except gradients in pressure and thermal potentials, is:
a'
where all terms are as described above with the subscript a for air
replacing £. for water.
The flux of gases, including water vapor, and solutes dissolved in
liquid water CO in response to a concentration gradient is described by
Pick's law:
a = - L Vpc (5)
where
L is a general coefficient that accounts for the effects of diffusion
and hydrodynamic dispersion (Bear, 1972), and
p is the mass concentration of the gas or solute.
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85
This formulation of Pick' s law using mass concentration is valid for
diffusion of a constituent of a low enough concentration that the mass of
the solvent is not appreciably affected by the presence of the solute.
For most cases relative to waste disposal, this is true; one exception is
in the treatment of coupled heat and water-vapor flow under high tempera-
tures. In this case, when water vapor moves due to a concentration
gradient, the concentration should be expressed as a mass fraction (mass
of water vapor as a ratio to mass of moist air).
When a moving fluid is involved, an advection term is added to the
righthand side of equation 5:
^ = - L Vpc + qfpc (6)
where
q- = volumetric flux rate of the solvent fluid.
The flux of enthalpy (q^) due to heat conduction and advection of
enthalpy in a moving fluid is described by adding an advection term to
Fourier's law:
qh = - X VT + CpPf T qf (7)
where
X is a general thermal conductivity,
T is the temperature,
C is the mass specific heat, and
P
p is fluid density.
If phase changes are involved, the latent heat of vaporization or
the heat of fusion must be accounted for by incorporating them in the
general thermal conductivity, and in the advection term (in the case
of latent heat) .
Combining equations 3-7 with equations 1 and 2 and adding source/sink
terms results in the following balance equations for liquid water, moist
air, water vapor, or gaseous pollutants, solutes, and enthalpy, that are
applicable to most field problems in waste disposal.
Liquid water
'v S*dV + 'sP*Kp{V(f>P + VM*+ V*h dS + vQ,elV=0 (8)
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86
Moist air
dp
/v Sa -d dV + I P K V* + V* dS + / ^ dV =
v a B a a P ha v
Water vapor or gaseous pollutants
dp
/ S — j£ dV+f t L Vp -q p IdS+f Q dV = 0 (10)
'v g dt Js t g g a Kg f ;v xg
Solutes dissolved in liquid water
'v
s
s
dps
dt
dV
*/.
\
L
s
7ps ~ \
f.\
dS
+ 'v
Q,
dV = 0 (11)
Flux of energy
*/v SHl IV + /g {X VT - CpPf Tqf\dS + /vQH = 0 (12)
The source/sink term in equations 8-12 is generally a function of
space, time, and perhaps temperature, pressure, or concentration.
Equations 8-12 along with appropriate boundary and initial conditions
and measured or derived values of the transport coefficients comprise
the basis of models applicable to field problems. Boundary conditions
are required to specify the solution variable (s) or the flux of mass
or energy across the boundary of the modeled region as a function of
time and space. Examples are specified concentration at a contaminant
source, rainfall rate, and heat flux from a canister containing spent
nuclear fuel.
Equations 8-12 are partial differential equations that may be non-
linear owing to the dependence of the transport coefficients, boundary
conditions, and (or) source/sink terms on the solution variable. It is
principally the nonlinear nature of these equations when they are
applied to variably saturated flow problems that requires extension of
the solution methods and applicable models from those used for linear
problems. The nonlinearity in the transport coefficients is exemplified
by the general hydraulic conductivity in equation 8, which is a function
of properties of fluid, medium, and pressure potential:
V ' k Kr
where_ 2
K = the intrinsic permeability of the medium, L ,
y = dynamic viscosity of water, "ML T ,
Xr
K = a relative conductivity which depends upon the capillary
r pressure potential 0|») dimensionless,
4) = the difference between air pressure (P ) and liquid pressure
(P ) energy /unit mass.
Figure 1 shows typical functions of K for a sand and clay.
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87
The general storage coefficient in the water flux equation is also
highly nonlinear and depends upon the capillary pressure potential Cfig. 2)
The transport coefficients in the energy flux equation are also nonlinear
because of their temperature dependence. Nonlinearity in the transport
coefficients is further exacerbated when coupled equations are used.
For example, in a coupled water-flux solute-transport problem, hydraulic
conductivity of the medium may be affected by the solute concentration;
as, for example, when waters high in sodium infiltrate soils containing
clay minerals CFrankel and others, 1978; Pupiski and Shainberg, 1979).
Another example is the strong temperature dependence of dynamic viscosity
in the water flux equation for coupled heat and moisture flow problems.
In addition to nonlinearity in transport coefficients, certain
boundary conditions may cause the equations being solved to be nonlinear.
An example would be isothermal evaporation, where the evaporative flux
depends upon the unknown pressure head in the soil immediately below the
land surface. Source/sink terms may be nonlinear also; examples would
be the rate of water uptake by a plant root system, which depends on the
unknown pressure potential in the soil surrounding the root, and sorption
of a contaminant species which depends upon the solute concentration.
Table 1 shows the degree of nonlinearity generally associated with
equations 8-10 as they might be applied to variably saturated conditions
at a waste-disposal site.
MODELING APPROACHES
Depending upon the problem being analyzed, any subset of equations
8-12 may be required to describe the physical system.
The solution of the equations that are part of a model of a given
field problem may be accomplished in one of three ways: (1) By use of
analytic or quasi-analytic methods; (2) by use of an analog model such as
resistance-capacitance networks or deformable membranes; (3) by the use of
approximate numerical techniques. Because of the nonlinearities in
equations 8-14, analog models have found very limited application. Since
only one reference to the use of electric analog model to analyze variably
saturated flow problems was found (Bouwer and Little, 1959), further
discussion of analog models will be omitted.
Although analytic and numeric solution methods will be discussed
separately, this subdivision is arbitrary. In general, analytic methods
utilize direct integration or series solutions to equations 8-12, and
treat time and space as continuous; numeric methods involve discretization
of the solution domain and the time span of interest into finite intervals.
A section is also included on parametric or input-output models that may
incorporate some features of both analytic and numeric methods.
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o
g
u
i
o
>-
1
h-
i
o
8
-2
01234
LOG OF PRESSURE HEAD. IN CENTIMETERS OF WATER
Figure 1. — Relative hydraulic conductivity to
liquid water for a sand and clay.
M
i
o
0123
LOG OF PRESSURE HEAD, IN CENTIMETERS OF WATER
Figure 2. — Nonlinear storage coefficient for
a sand and clay.
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Table 1.—Degree of nonlinearity found in moat formulation® of equations 8-12
[V » very nonlinear, S » slightly nonlinear, L » can be assumed linear]
(8)
(9)
(10)
(11)
(12)
Equation
Liquid water
Moist air
Water vapor and
Solutes in liquid water —
Enthalpy
Transport coefficients
Storage Dispersion, diffusion,
or conductance
V V
s v
V V
L L
S S
Source and Boundary
sink terms conditions
s-v s-v
s s
s-v s-v
s s
v s-v
00
vo
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90
FACTORS THAT DETERMINE A MODELING APPROACH
There are several factors that determine the modeling method used to
analyze a given field problem:
1. The degree of nonlinearity and hysteresis in transport co-
efficients, boundary conditions, and source/sink terms (see
table 11;
2. Inclusion of an advection or gravity term;
3. Existence of sharp fronts;
4. Heterogeneity and anisotropy;
5. Existence of complex boundaries;
6. Data availability;
7. Time and funding available;
8. Necessity to couple two or more of equations 8-12.
Factor 8, coupling of equations, will be discussed first, with
factors 1 through 7 discussed in following sections.
Equations 8-12 are all coupled to some degree. The following two
examples illustrate coupling. The liquid and solute flux equations are
coupled, because the liquid flux is required to evaluate the advection
term in the solute flux equation. Hydraulic conductivity in the liquid
flux equation may also depend on the solute concentration (as discussed
previously). The thermal potential appearing in the liquid flux equation
(equation 8) depends upon the temperature distribution derived from the
energy flux equation (equation 12).
The degree of coupling required in solutions to the pertinent
equations depends principally upon the difference in time scales in the
phenomena being modeled. For example, air pressure response is generally
very rapid compared to liquid response, so that implicit consideration
of simultaneous air and water flow due to pressure gradients is not
required unless air pockets are entrapped. Large differences in time
scales may be used to advantage in solving field problems by specifying
the values of a rapid response variable as boundary conditions for a
variable that responds less rapidly. Sometimes the sensitivity to time
variations in a secondary variable in an equation is low with respect to
the response of the principle variable. For example, it has been shown
by Wierenga (1977) and Beese and Wierenga (1980) that, under certain
conditions, the pollutant concentration at depth in a soil profile
can be simulated as well using a time-averaged water-flux rate (q_) (in
equation 11), as with a fully transient treatment yielding the time
variations in q- that are caused by cyclic variations in rainfall and
evaporation.
When it is required to couple two or more of equations 8-12 for a
given application, care must be taken in proper boundary-condition
specifications with respect to all coupled equations.
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91
Coupled equations are complex enough to generally preclude any ana-
lytical or quasi-analytical approach. The numeric methods couple the
equations by either solving them sequentially or simultaneously. Coupling
of equations using a sequential solution is usually accomplished through
the use of source and sink terms. For example, in a problem of combined
heat and liquid flow (equations 8 and 12) , the liquid movement due to
thermal gradients is incorporated as a source or sink in equation 8, and
the movement of heat by moving fluid may be incorporated as a sink in the
enthalpy flux equation (equation 12). Sequential solutions are exem-
plified by water and solute flow problems where the water flow equation
(equation 8) is solved first, and the resultant fluid velocities are used
in the solute flow equation (equation 11). If the degree of coupling is
slight, this two-step sequential procedure is generally adequate. For
strongly coupled problems such as coupled heat and moisture flow problems,
however, the solutions from the two-step process must be improved by
iteration. This iteration can be accomplished either separately or
combined with the iteration required to linearize nonlinear forms of the
equations, and the iteration that may be used to solve the linearized
algebraic equations. Table 2 summarizes the degree of coupling of equa-
tions 8-12 for most problems relative to waste-disposal studies.
Table 2.—Degree of coupling of equations 8-12 exhibited in problems
relating to radioactive waste disposal
[S = strongly coupled, M = moderately coupled, W = weakly coupled]
Equation number
10 11
12
Equation
Moist
air
Water vapor Solute
and gaseous dissolved
pollutants in
liquid
Enthalpy
(8) Liquid water M
(9) Moist air — ~
(10) Water vapor and
gaseous pollutants —
(11) Solute dissolved in
Tln,,4A
W-M
W-M
*+
—
— —
S
W
M
••— —
M-S
M
S
W-M
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92
Analytic and Quasi-analytic Methods
When the problem to be solved can be cast in simple form, a closed
form analytic or quasi-analytic method may be available. These can be
classified as direct, similarity, quasilinear, and integral methods.
These categories are attributed to a review by Philip (1974). The
discussion that follows is written with emphasis on solutions to the non-
linear form of equation 8, the water-flow problem. However, the methods
apply equally well to nonlinear forms of equations 9-12. Analytic solu-
tion methods for linear forms of equations 8-12 will not be explicitly
addressed in this paper; solutions to linear problems are, in general,
special cases of the nonlinear methods presented. A review of analytic
solutions to the solute flux equation (equation 11) will be given by
another paper at this symposium (Grove and Kipp, 1980).
Direct Methods
Direct methods involve direct integration of some form of the
descriptive equation, solving for the constants of integration using
boundary and initial conditions. A simple example would be the problem
of isothermal isohaline steady upward flux from a shallow water table at
a depth I. The governing equation is Darcy's law (equation 3), because
the time derivative appearing in equation 8 is zero. If the desired flux
rate is q, then the problem can be written as:
/
q + K 0(0
*.
d*. (14)
which for certain analytic forms of K (iji) can be integrated directly.
Gardner (1958, 1959) and Ripple and others (1970) detail this solution
method for both uniform and layered media. The integral in equation 14
may not be integrable in closed form because of the nature of the K (i|0
function. In this case, numerical integration methods such as Gaussian
quadrature must be used, but this solution type can still be termed a
direct method. An application of direct solution methods that is
important in transient infiltration problems is the assumption of a step
function shape for the wetting front. This has been shown to be equiva-
lent to the assumption to a step function for hydraulic diffusivity,
D - K/S (Philip, 1973). The resultant formulation, which is attributed
to Green and Ampt (1911), has been shown to accurately represent the
infiltration rate under conditions of ponded water on the surface. Mein
and Larson (1973) have extended the method to a flux boundary condition
imposed by constant rainfall rates.
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93
Direct integration methods require the descriptive equation to be a
function of only one independent variable; thus, transient problems are
not immediately amenable to this solution method. However, the use of a
similarity variable that combines two independent variables may render
the problem tractable by direct methods. The most common of these trans-
formations is the so-called Boltzman transformation (Ames, 1965):
* - xt*2 (15)
where
$ is the transformed variable,
x is a spatial coordinate, and
t is time.
Even though a similarity transformation reduces the number of inde-
pendent variables by one, direct solution still requires the transport
coefficients to be constants or to have simple linear or step function
dependence on the transformed independent variable.
Linearization
When the transport coefficients cannot be expressed as simple
functions the descriptive equations must be linearized. The most common
linearization method used in the reviewed reports consists of using an
exponential function to describe hydraulic conductivity for steady-flow
problems, and hydraulic diffusivity* (D) for transient problems.
The exponential dependence of hydraulic conductivity on pressure
potential for example can be described by
K = K exp (ct ^) (16)
s
where
K = the hydraulic conductivity at saturation, and
S
a • a constant.
The second step in the linearization process is to introduce a new variable
termed the matric flux potential, by using the Kirchoff transformation
(Ames, 1965):
J
.
K (a) da (17)
These steps result in a linear form of equation 8 written with v as the
dependent variable, and render essentially all steady-flow problems
solvable using linear methods (Philip, 19741 . It should be noted that
for a wide range of soils, the £(.*) function is very well represented
by equation 16 (Laliberte and others, 1966; Mualem, 1976).
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Approximate Analytic Methods
When a linearized form of the descriptive equations cannot be solved
by direct integration, approximate methods must be used. These can be
categorized as series expansion methods and integral methods. Series
expansion methods involve expressing the solution as a series with unknown
coefficients that usually involve integrals of the transport coefficients.
The form of the series is often based on empirical observation. Philip
(1955) was the first to apply this method of solution to nonlinear
equations described by equation 8. Philip's solution is of the form:
i< "Ml
X = <|>t^ + xt + cat ' + . . . (18)
%
where , x> w» are t^ie unknown coefficients. The expansion in terms of t
is baaed upon the observed square root of time dependence of the advance
of a wetting front in the absence of gravity. The coefficients in equation
18 can be determined by an iterative trial and error process proposed by
Philip for arbitrary forms of the transport coefficients. When this
process Is followed, the method is essentially a numerical rather than
an analytic method. Brutsaert (1968a, b, 1977) has derived exact
expressions for the first two coefficients in the above equation for
one-dimensional absorption (gravitational potential ignored), and infil-
tration (gravitational potential included), by using a flux potential
formulation and assuming a simple form for the nonlinear storage
coefficient. The principal drawback to Philip's method is its divergence
from the true solution at long times. The maximum applicable time may be
a few days or less (Brutsaert, 1977).
Series solutions may also be derived by expressing the series in
terms of a perturbation variable, or some variable that is assumed to vary
a small amount over the range of the solution (Ames, 1965). Babu
(1976a, b, c, and 1977), has used this solution method, with the terms
in the series involving integrals of the hydraulic diffusivity function.
The other general category of approximate analytic methods that can
be applied to transient as well as steady-state problems involves an
iterative evaluation of an integral form of equation 8. The method is
useful in reducing the number of independent variables in problems that
cannot meet the similarity requirements (Philip, 1974). The method in-
volves making a first approximation to the form of the solution to
equation 8. This may be accomplished by ignoring the time-dependent
first term and solving the resultant steady-state problem by use of the
previously discussed methods. This first approximation is inserted in
the integral of the transient equation with respect to the variable to be
eliminated. The resultant equation is solved by a direct or series
solution method.
These steps may be repeated to iteratively improve the solution.
Ames (1965) gives a further explanation of these iterative methods.
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95
Parlange, in a series of papers (1971a, t>, c; 1972a, b, c; 1973a)
has applied the iterative method to a variety of problems ranging from
one-dimensional transient absorption to axisymetric transient infiltration
problems. Cisler (1974) derived a correction to Parlange's method which
improved the accuracy significantly. These iterative methods are very
sensitive to the initial estimate of the solution. Knight and Philip
(1973) have shown that these methods can fail to converge if the initial
estimate is very far from the solution. Parlange and Braddock (1978)
have shown that iterative methods may converge to the wrong solution
after a finite number of iterations.
Availability of Analytic Solutions
The length restrictions for a symposium paper preclude listing all
the analytic solutions found in this review. An excellent compilation of
the available solutions through 1971 is given by Braester and others (1971).
Solutions since 1971 are summarized in Babu (1976a, b, c, and 1977); Philip
(1974); Parlange and Babu (1976a, b); Raats (1971, 1972, and 1977);
Lomen and Warrick (1976a, b, and 1978a, b); Warrick (1974); and Warrick
and Lomen (1977). A summary of the types of problems that have been
solved for the nonlinear water flow equation is given in table 3. One-
dimensional problems have been solved for a variety of realistic boundary
conditions. The principal restrictions are: (1) The requirement to use
simple forms for the functional dependence of the transport coefficients
upon the solution variable; (2) the limiting of heterogeneous media in
transient cases to simple functional dependence upon depth; and (3) the
failure to include source and sink terms of a general nature.
For multidimensional cases, the geometry is restricted to two-
dimensional planar and three-dimensional cylindrical or spherical symmetry.
Few transient multidimensional problems have been analyzed, and no
solutions to multidimensional transient problems that involve sources and
sinks were found.
While it may appear that the application of analytic solutions to
field problems that relate to waste disposal is somewhat limited, there
are three main advantages to these methods. First, they give closed-form
solutions that allow general conclusions to be drawn regarding the move-
ment of mass and (or) energy in variably saturated systems. Second, these
conclusions can be used in a qualitative sense to check the results of
approximate numeric methods. Third, analytic solutions that exist for
simple problems can be used quantitatively to check approximate numeric
methods applied to the same simple problems. Examples of these general-
ities are the square root of time dependence of horizontal absorption
and the initial stages of vertical infiltration, and acceptable forms
of moisture and potential profiles for simple systems (Childs, 1969).
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96
Table 3.—Types of variably saturated flow problems for which.
analytic and quasi-analytic solutions are available
Number of reported solutions
Factor
One-dimensional
Steady Transient
Two and three-
dimensional
Steady Transient
No gravity term:
Homogeneous :
Source/ sink
No source/ sink
Heterogeneous :
Source/sink
No source/ sink
1 0
1 24
0 0
0 0
0 0
0 5
0 0
0 0
Gravity term:
Homogeneous:
Source/sink— 2
No source/sink—— 2
Heterogeneous:
Source/sink 0
No source/sink 4
1
21
0
2
3
11
0
2
1
3
0
0
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97
Numeric Methods
Numeric methods require subdividing the spatial and temporal domains
into discrete intervals over which the variables such as pressure,
temperature and concentration are assumed to be either constant or to
vary in a simple manner. This discretization results in a set of non-
linear algebraic equations that must first be linearized and then solved
simultaneously, usually using high-speed digital computers.
The emphasis of the following discussion is on solutions to the
liquid flow equation Cequation 8); however, the methods are applicable
to nonlinear forms of any of equations 8-12.
The principal advantage of numeric methods is their flexibility. In
principle, methods that are suitable for one dimension can be simply
extended to two and three dimensions. Conversely, computer codes incor-
porating these methods for multi-dimensional problems can collapse to
handle one-dimensional cases. Complex boundaries, nonlinear transport
coefficients and boundary conditions, heterogeneity and anisotropy, and
spatial and temporal source/sink variations can all be handled. The
principal disadvantages of numeric methods for nonlinear problems are the
large computer core and time requirements and the complexity of computer
codes.
The following discussion is pertinent to the solution of the water
flow equation. By suitable rearrangement, this equation can be written
in terms of one of three variables. If the system under consideration
remains unsaturated at all times, the flow equation can be written in
terms of the volumetric moisture content ($)• This is referred to as
the '0-based1 approach, in which the driving forces are gradients in
moisture-content concentration. The 0-based equation is identical in
appearance with the solute-flux equation Cequation 11), with the
gravity-driven flux term in equation 8 being equivalent to the advective
term in equation 11. When part of the system is saturated, this approach
fails, because the volumetric moisture content remains constant at satura-
tion, but flow still occurs due to gradients in pressure, gravitational,
and thermal potentials. For partially saturated systems, equation 8 can
be written in terms of the matric flux potential Cequation 17), the
pressure potential , or the sum of the pressure and gravitational
potentials (<|> + )? The pressure-potential approach can also be refer-
red to as the pressure-head Ch) approach by using units of length for the
pressure potential: h = —^-, where g » gravitational acceleration.
Similarly, the combined prlssure and gravitational potential approach is
often termed the total head CH) approach, where H =» — C4> +
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98
Spatial Discretization Methods
There are two basic methods of subdividing the spatial domain of a
problem described by equations 8-12: the use of finite differences or
finite elements.
Finite-difference methods require subdivision of the region considered
into subregions that have a regular geometry. These regions are usually
rectangular, but arbitrary polygonal subregions have also been used
(Narasimhan and Witherspoon, 1977). In finite-difference methods, spatial
derivatives are approximated by assuming that the solution variable varies
linearly between the geometric centers of each of the subregions, and the
storage coefficients are assumed to be constant over each subregion. Any
sources or sinks present are assumed to contribute uniformly over each
subregion in which they are present.
Because the gradients in the solution variable are assumed to be
linear between adjacent subregions, the use of finite-difference methods
may require many subdivisions in problems that involve rapidly changing
gradients; many problems in variably saturated flow described by
equations 8-12 are this type. Examples are wetting fronts in infiltration
problems; sharp solute concentration fronts resulting from a step or pulse
input of contaminant; and evaporation-condensation fronts in combined
heat and moisture flow from a buried heat source. The existence of sharp
fronts can result in oscillatory solutions even if a fine grid spacing is
used CFinlayson and others, 1978). For these types of problems, a
numerical dispersion term is often added. For the water-flow problem,
the use of upstream weighting on relative hydraulic conductivity is
commonly used (Brutsaert, 1971; Finlayson and others, 1978). Similar
terms must often be used in finite difference solutions to equations
with a strong advection term (equations 11 and 12) to prevent oscillations.
The requirement for fine subdivision to accurately represent sharp
fronts may result in a large number of simultaneous equations that must
be solved. This restriction, when combined with restrictions on time-step
length to prevent oscillation and nonconvergence caused by nonlinearities
and (or) presence of an advection term, can result in formulations of
field problems that are very expensive to solve.
Another disadvantage of finite-difference methods is that, due to
the regular shape of the subregions, many subregions may be required to
accurately represent the geometry of a given field problem.
Accurate representation of complex geometries is an advantage of the
finite-element method of spatial discretization. In this method, the
region of interest is subdivided into subregions or elements that may
have irregular shapes. These may be simple linear quadralaterals,
triangles, or complex quadralaterals with curved sides.
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99
In solving the balance equations 8-12 for these irregular subregions
with a finite-element method, the surface integrals are converted to
volume Integrals using Gauss' divergence theorem. The resultant equations
are evaluated at a finite number of points or nodes that coincide with the
vertices of and points interior to the subregions or elements. For
example, the solute transport equation Cequation 10) would become:
30r
e " Sc IT + 7Cqf(5c) ~ V'LV^c- (19)
where
e is a residual between the approximate and true solutions, and
p" is the approximate value of p .
c c*
There are two types of finite element methods: those derived using vari-
ational calculus, and those based on the method of weighted residuals.
Only the latter will be discussed in thia paper as most of the published
solutions to equations 8-12 use them, and because the resultant equations
from both methods are the same in cases where a variational formulation
exists.
The weighted-residual methods multiply the residual, e, by a weight-
ing function, which is a polynomial in the spatial coordinates only.
To minimize the error of approximation, it is then required that:
/ew. dV = 0, 1-1, m (20)
V
where there are m weighting functions over the solution domain. This
step in the finite-element method results in a powerful tool to control
the accuracy of the approximate solution by choosing suitable weighting
functions (w.). This has been an important advance in simulating some
sharp front problems CBender and others, 1975).
The next step in weighted-residual finite-element methods is to
approximate the solution variable with a series of the form:
M .
h = I h (t) a) (x) (21)
j=>l J 3
where^
h (t) is an unknown coefficient which becomes the desired solution
•* variable, and
a) (x) are basis functions that are a function of space only.
The choice of basis functions defines further the type of finite-element
method used. If the basis functions are chosen to be the same as the
weighting functions (w. in equation 20) , then the method Is a Galerkin
method (Finlayson and others, 1978). If the basis functions are chosen
to be proportional to dirac delta functions, then the method is an
orthogonal collocation method CFinlayson and others, 1978).
Galerkin methods applied to forms of equations 8-12 that include a
nonlinear storage term can result in oscillations when sharp fronts are
involved (Mercer and Faust, 1977). Since the solution of the algebraic
-------
100
equations resulting from either finite-difference or finite-element
methods often requires iteration due to nonlinearity of the problem,
these oscillations very often result in a failure of the iterative
process to converge. The oscillations are caused by the evaluation of
the nonlinear storage term at points adjacent to the point of interest.
These oscillations can be minimized by a Galerkin formulation that
evaluates the storage terms as an Integrated average over each element
(Neuman, 1973). This is sometimes called capacitance lumping. Ortho-
gonal collocation on finite elements results in a formulation that does
not include the extra storage terms, and hence, is free of this type of
oscillations (Finlayson and others, 1978).
The following conclusions regarding the preferability of finite-
difference or finite-element methods are adapted from Finlayson and
others (1978, p. 72) in a study of numerical methods applied to very
dry soils.
1. "***finite elements*** (Galerkin and orthogonal collocation)***
are capable of tracking steeper fronts than finite difference
methods***."
2. Most predictive simulations in hydrology using finite element
methods have not involved fronts as steep as those encountered
in infiltration into extremely dry soils.
3. "***for linear problems with steep fronts, finite element
methods are much faster for equivalent accuracy (than finite
difference methods). For nonlinear problems, the Galerkin
finite element methods are slower than finite difference
methods***."
A. "***for both finite difference and finite element methods,***
very steep fronts can only be handled by including numerical
dispersion in the form of averaged (or upstream)(conductivities),
special weighting functions or some other techniques***"
"***As the front becomes steeper, the number of spatial nodes
or grid points increases and the time step (required to mini-
mize oscillations) correspondingly decreases***."
5. "***The preferred method for dry soils is a finite difference
method with averaged (conductivities) because it introduces
numerical dispersion and allows calculations without a
prohibitively large number of nodes***."
Time Integration Schemes
Once the spatial derivatives in equations 8-12 have been approximated
by either a finite-difference or finite-element scheme, time derivatives
must be approximated In some fashion. For both finite-difference and
finite-element spatial discretizations, this is usually done using finite
differences in time. It is possible to use the Galerkin approximation
(similar to equation 21) for the time derivative, as described by Gray
and Finder (1974) and Zyvoloski (1973). However, it has been shown that
no significant accuracy Improvement is obtained by this method and that
additional computation is required to formulate and solve the resulting
algebraic equations (Yoon and Yeh, 1975).
-------
101
Commonly used time-differencing schemes evaluate the spatial deriva-
tive approximations at the previous time-step (explicit method), the
current time-step (implicit method) or as a weighted average of these
(Crank-Nicolson method). The explicit method results in the simplest set
of equations to solve, but is subject to stability restrictions on the
step length that often make its use infeasible (Haverkamp and others,
1977). The Crank-Nicholson method is the most accurate, but Jeppson
(1974a) has shown that, if the nonlinear transport coefficients are used
with a Crank-Nicholson method that is valid for constant coefficients,
the solution is grossly in error. If a Crank-Nicholson scheme that is
valid for varying coefficients is used, Jeppson (1972) also showed that,
if the storage coefficient is divided into the conductivity terms
before the equations are solved, the resulting system of equations is
difficult if not impossible to solve, due to rapid oscillations in the
neighborhood of the true solution. In addition to these problems,
Neuman (1975b} showed that a Crank-Nicholson approach cannot be used in
solution of equation 8 for problems in which part of the system is
saturated and boundary conditions change in the saturated zone.
The fully implicit method, which evaluates the spatial derivative
approximations at the current time-step, appears to be the preferred
method (Jeppson, 1974a; Neuman, 1975b; Haverkamp and others, 1977).
While it is slightly less accurate than the Crank-Nicholson approximation,
it permits solution of the widest range of problems likely to be encoun-
tered in waste-disposal problems.
Recently, Narasimhan and Witherspoon (1977, 1978a and b), and
Narasimhan and others (1978) developed models that utilize a so-called
mixed explicit-implicit time-integration scheme. This scheme is based on
the following: stability criteria for explicit methods depend on the
rate of change of the solution variable, being more stringent for small
grid spacings and for rapid changes in the solution variable. Many
problems may involve subregions over which changes are slow due to a large
ratio between the storage coefficient and the conductance or to large
elements or finite-difference cells. The equations for these subregions
can o.ften be evaluated explicitly, requiring implicit evaluation only
where rapid changes are occurring. However, in many cases, the computer
time required to evaluate the status of all points in the system may be
greater than if the entire region had been evaluated implicitly.
Another time-integration scheme takes advantage of the fact that
after the spatial derivatives have been approximated by finite differences
or finite elements, the result is a set of ordinary differential
equations of the form: ^ _
S dt ~ Ax h>
where A2 h is the approximation to the spatial derivatives. This set of
equations can be integrated in time using methods developed for ordinary
differential equations. The most common of these methods are fourth-order
Runge-Kutta and fifth-order Milne (James and others, 1968; International
Business Machines, 1975). These'are essentially explicit methods and are
-------
102
subject to time step restrictions. Haverkamp and others (1977) found
that 5 to 10 times more computer time was required to solve the nonlinear
one-dimensional water-flow problem with these methods than with a fully
implicit scheme.
The principal advantage of these methods of time integration is that
they have been incorporated into an applications-oriented language by
International Business Machines (1975) entitled: "Continuous System
Modeling Program" (CSMP). This language makes it very simple to code a
model for a particular application. Hillel (1977) gives sample programs
using CSMP for solving one-dimensional finite differenced forms of the
water flow equation (8); water and solute flow equations (8 and 11); and
moisture and heat flow equations (8, 10, and 12). CSMP solutions to
equation 8 have also been reported by Van der Ploeg and others (1974) and
Bhuiyan and others (1971). Combined one-dimensional solute and water-
flow problems have been solved by CSMP and reported by Wierenga (1977),
Beese and Wierenga Q.980), and Siddle and others (1977).
Linearization Methods
Any or all of equations 8-12 may exhibit some degree of nonlinearity
in the transport coefficients or boundary conditions. Since the equations
resulting from any spatial and temporal discretization schemes must be
linear to solve, some form of linearization is required. Three main
linearization methods commonly have been used: explicit methods, pre-
dictor-corrector methods, and fully implicit methods.
Explicit linearization involves evaluation of the nonlinear co-
efficients and boundary conditions using values of the solution variable
from the previous time step. As would be expected, this method may
require very small time steps to achieve acceptable accuracy (Jeppson,
1974a; Rubin, 1968; Haverkamp and others, 1977). For infiltration prob-
lems, the general effect of explicit linearization with large time steps
is for the solution to lag behind the true solution (Haverkamp and others,
1977).
Predictor-corrector methods involve two Iterations for each time
step. The first step solves the equations for one-half the step length
using the coefficients evaluated at the previous time step or by extrapo-
lating to one-half the time step. Extrapolation may use the previous
two or three time steps (Rubin, 1967). The solution Is then repeated
for the full time step, using the values of the solution variable at one-
half the time step to evaluate the coefficients. Jeppson (1974a) has
shown that oscillations in sharp fronts can result from this method.
The accuracy of the method is dependent on the accuracy of the extrapo-
lation; since Iteration is not continued until convergence, instability
can result (Jeppson, 1974a; Braester and others, 1971), particularly
with highly nonlinear problems.
-------
103
Fully implicit schemes involve evaluating the nonlinear co-
efficients and boundary conditions at the current time step by
iteratively solving the equations until convergence occurs. The value
of the solution variable to evaluate the coefficient for the first
iteration may be obtained by extrapolation from previous time steps
(Freeze, 1971b; Neuman, 1975b) . While implicit linearization may be the
most time-consuming method, it results in the most stable, accurate
scheme with the least computational effort (Haverkamp and others, 1977).
In addition, the iterative process gives a means of controlling the
accuracy of the solution by specifying the convergence tolerance.
Although all numeric schemes require a large number of computations,
iterative linearization requires an even larger number. This would be
only of passing interest, except the earliest reference this author
found to the solution of a form of equation 8 describing vertical one-
dimensional water flow was accomplished by Klute in 1952, using hand
calculations for a fully implicit iterative scheme.
Convergence of fully implicit iterative linearization schemes is not
guaranteed, and may be slow. The most effective means of accelerating
convergence is to apply a Newton-Rap hs on scheme to the matrix equations
(Jeppson, 1974b; Brutsaert, 1971; Mercer and Faust, 1977; Finnemore,
1970) . This scheme may be applied to both the conductivity and storage
terms, or to the more nonlinear of the two. This author has found it
sufficient for many highly nonlinear problems to apply this linearization
technique to the storage term only.
The iteration required to solve nonlinear equations often fails to
converge when the solution is approached. The convergence failure takes
the form of oscillations around the true solution, and often occurs when
static equilibrium is approached. There is no panacea for this problem;
however, the use of an adjustable damping factor often will result in
convergent solutions. This is an ad hoc procedure; a more rigorous
approach to optimize the iterative process for nonlinear problems is
currently being investigated by R. L. Cooley, U. S. Geological Survey
(oral commun. , 1980) .
Matrix Solution Methods
All numerical schemes except those involving explicit time and co-
efficient evaluation result in a set of algebraic equations that must
be solved simultaneously. These equations can be written in matrix
form as: HVl
Direct methods involve some form of Gaussian elimination and take
advantage of the sparse nature of the coefficient matrices IA] and [B].
These methods generally require large computer storage and time require-
ments. To obtain accurate solutions, roundoff and truncation error must
be minimized by the proper use of single and double-precision arithmetic,
-------
104
and such features as pivoting to assure diagonal dominance of the
matrices {A] and [B] (Nobel, 1969). Recent advances in equation
ordering (Price and Coats, 1973) have significantly optimized the storage
and time requirements for using direct solution methods, thus making
their use more feasible for large problems.
The most commonly used iterative matrix solution methods are the
iterative alternating direction implicit (IADI), line successive over
relaxation (LSOR), and the strongly implicit (SIP) methods. These methods
solve sets of simultaneous equations by iteratively improving an initial
guess at the solution. The initial guess uses values of the solution
variable from the previous time step. As Braester and others (1971,
P- 113) point out, "***unfortunately however, none of the methods
(LSOR, IADI) will guarantee convergence of the unsaturated flow solution
in all cases***."
It should be noted that if the matrix equations are solved by an
iterative procedure, this procedure must be embedded in the iteration
required to handle the nonlinearity of the equations. Some authors
(Freeze, 1971; Brutsaert, 1971) have accomplished this requirement with
an "inner iteration loop" to solve the matrix equations and an "outer
iteration loop" to handle the nonlinearity. This author and others have
found it efficient to combine the two iterative loops.
The principal advantages of iterative-solution methods over direct
methods are: (1) smaller computer core requirements; and (2) ability to
control the accuracy of the solution (and hence its cost) by specifying
the convergence criteria. For example, this author has found it computa-
tionally more efficient to solve strongly nonlinear problems to the same
degree of accuracy with LSOR with a very restrictive convergence criteria
rather than a direct solution with a less stringent criteria.
Availability of Verified Numeric Models
Over 280 references to numeric solutions to nonlinear forms of
equations 8-12 were found in the literature. The earliest solutions were
for simple one-dimensional liquid-water absorption problems. Present
capability includes solutions of complex three-dimensional water and
solute-flow problems and coupled-heat and moisture-flow problems. Figures
3 through 6 summarize chronological development of models incorporating
these solutions.
Although many real-world problems can be solved by making the one-
dimensional approximation to the required equations, general codes should
have multidimensional capabilities. Many of these codes have been applied
to a variety of field problems. Table 4 lists the capabilities, applica-
tions, and limitations of using some of these codes. While the codes in
table 4 are documented and verified, their use is not necessarily
straightforward because of the many numerical tricks that may be required
to obtain a solution. Extreme nonlinearities can preclude solutions in
many cases. These problems are inherent to the nature of the nonlinear
-------
105
2
5
65
60
55
50
45
40
35
30
25
20
15
10
5
ONE-DIMENSIONAL
1960
1965
1970
YEAR
1975
1980
Figure 3. — Development of variably saturated
finite difference models since 1960.
1970
1980
Figure 4. — Development of variably saturated
finite element models since 1970.
-------
106
3 -
1970
1980
Figure 5. — Development of variably saturated
water and solute transport models since 1970.
10
(ONE-DIMENSIONAL MODELS ONLY)
I
1970
1975
YEAR
1980
Figure 6. — Development of variably saturated
water and heat transport models since 1970.
-------
Table t.—Ftaturti ef a"kilobit gmtrol eenputtr codtt far nntrla tolution of variably tatimttd
vattr, tolutt, and hat trantpert probltnt rttating to ndloaetiv* HWt« dttpotal
Hodtl
BUM Author!
Baca, King*
and Norton
U97J)
Brandt and
othtri (1971)
Brealer
(1»7S)
CSU trutaacrr
(1971)
Brutaaert,
BrcUenbach and
Sunada (1970,
1971, WJ»
Cauaeade.
and other*
(197»)
Cool*?
(1981)
Soluta
Haxlxn trane-
dtmnalom port
1 No
2
(3-»»liT»- Mo
•atrle)
2
(3-axlajm- Taa
metric)
I No
2 Ho
(J-axlajna-
•etrie)
2 Mo
2
(3-utayB- No
Mtrle)
Haat Tlaw
trana- Spatial Inta-
port dliccetiittlon (ntton
CaUrkin-
T» ilntt* alcMnt Illicit
Tinlea
Mo dlffarmci
rinlt* Crank-
No dlffaraau KUhoUon
Flnlt*
No diffartnco Implicit
No Tlniti bpllclt
dlffarane*
Mo rinlti Explicit
dltfaranc*
Orthogonal
Ho collocation Implicit
on Plnlta
•laawato
Llnttdutton Solution
Implicit Cauaalaa
•itti Mavtoa- oliaiination
Hapliaon
lapllclt
with Mavtoo- ADI
lUphaon
tBn1i«<*
inpl ic 1C
vlth Ncvton- ADI
kaphion
lapllclt Una SOI
with Navton- with
Raphaon cornctloo
Inpllclt
with NMton- Llna sot
Raphaon
Alternating
linear
oolutlon with Mono
nonlinear
correction
Implicit SI?
Sourco/ Boundary
alnk condition
typ«» tjrpa*
bpliclt All
Nona All
Explicit All
All,
Explicit Including
aaepago
facia
Explicit All
Nona All
Explicit All.
Including
aaapaga
facaa
Verification
field data, lao-
therMl coluam
data, aulytie
eolution
Unknown
Coluin data
Ub. field data.
other aodela
Ub. field data.
analytic
aolutlona
Analytic
aolut ion
Ub. field data.
other aodela
Ritnitrka
Log tranaforrution of
prenaure head •
tinaAturat<*d
conditions c
-------
TakU t.—ftatuiva of mafoblt amerat otfuUf oodtt for nmnto tolutitm of variably tuturatad *jt«T,
•oluti, and Awl tfmtfart protbmi Mtattnff to radtoaatl* *ut*
tolttte Halt Tim*
Modal Kaxlmum trana- trana- Spatial Intt-
•a«M Authora dimenolen* port pert diecrotlutlon (ration
Finlayaon,
XeUon. and 1 Ma Toe Finite Implicit
aaca (l«7«) difference
Fre«o
(19/1) 1 Mo Mo Finite Implicit
difference
Ftlod, Cnletkin-
Clllham, ana J Xo po finite a lament Implicit
Flclunu (H77) vith lumped
capacitance
Frlnd and Calerkln-
Verja (U7») 1 -Mo Mo finite element Implicit
vith lumped
capacitance
Hamnel Finite Creak-
Cm*) 1 Mo Tea difference Micholno*
Khaleol end
Redden 1 Ten Mo Finite Implicit
(1»76) difference
Uppala 2
(1M1) (J-axiar»>- Wo Mo Finite Implicit
metric) difference
! Llixnrliattom (olutian
Implicit
vlth Itettto*)* ttakmovn
Kaphaon
Implicit
vlth Ume tot
extraaolatla*
Implicit
vlth Oauatia*
functional «lt»im»tiom
axpanaln
in apace
and time
extrapolation
Implicit
•1th Cauailaa
functional olimlnatlom
expanalon
in apace
and time
extrepolatlo*
Implicit Cauaalaa
elimination
Implicit Cauaalan
elimlnatiom
Explicit or
Implicit UOK. tit.
vtth full Ceueelan
or partial ellmlBatiom
Kevton -
Source/ ioundarjr
aink condition
txpea typea Veriflcetlon
(xpliclt All Field data
All, Uh, field data,
Explicit Includlni analytic
aatpaia aolutloM
(one All On«-dlmen«lon
column dralnaea
data
Column data.
bcpliclt All Finite
difference
•olut lone
Explicit All Uh ami
field data
Analytic and
Explicit All numeric
•olut lone,
experimental
data
Explicit plua All Other modala.
extraction analytic
»y note eolutlona
Kcaarkii
Temperature ecuation
aolvfd analytjcall))-
heat conduction cnljr
Autcmatlc tine ftej*
control.
hyiteretil
Gravity term
trenri*d explicitly*
waate dtapoeal
application
alnulata*
Application to
radioactive vavce
manacement are.i
Tranigwrt >olv
-------
Tabu ^,~r*aturl• of availabU gmtral eonputtr oodff for mtmria tolutlon of variably taturattd aittr,
aolute, and h*at transport problmt rtlatinff to radiaaativ* aattt dfajxwal--Continued
Solute Neat
Model Maximum trana- Irani- Epatlal
•aw Authore dimenelona port port dlacretliatlon
Lappala and 2
rolled O-axla/v Mo Ye* Finite
{1981) Mtrlc) difference
Marlao CalerkU-
(197S) 2 Tea Ko finite element
with lumped
capacitance
T»BST Rerailmhan 2 Integrated
(197)) O-axleym- Ko »» finite
•«trlc) difference
and Jeppeon 2 No No Finite
(w,j) dlffarenc.
Xaruimhan 2 finite element
(1977) O-axleym)- «0 «o with lumped
Htrlc) capacitance
IMSAT Heuman 2 Calerkln-
2 (1972, 1»7J (J-»xiay»- do »o finite element
197*. 197$) metric) with lumped
ferrena Unit*
Time
Inte-
gration
Implicit
Implicit
Mixed
explicit-
Implicit
Tine
centered
Mixed
explicit-
Implicit,
implicit-
explicit, at
time centered
Implicit
or time
TlM
Llnearltetlom,
Implicit
with Newton-
taphaon
Implicit
with
functional
expanelon
in apace and
extrapolation
In time
Explicit or
Implicit
with
extrapolation
Implicit
with Hevton-
Upheon
Implicit
or
explicit
Implicit
with
extrapolation
Implicit
Solution
LSOd, SIP,
Gauaalaa
elimination
Cauiela*
elimination
rolnt
sot
rolnt
SO*
rolnt
>0x
Cauaalan
elimination
IADI
Source/
elnk
typea
Explicit
plua note
and
evaporation
Explicit
Explicit
•one
Explicit
evapo-
ration,
implicit
explicit
Boundary
condition
typea
All
All,
Including
aeepage
faeee
All
All
All
All,
Including
aeepaga
face*
All
Verification
Analytic
aolutiona
lab data
Unknown
other modela.
analytic
aolutiona
Held
experimental
data
data, analytic
aolutieni, other
modela
Ub, field data,
othrr numeric
modela, analytic
aolutlona
Remarka
Unpubllahed research
model
matlc time ate?
contrnl, defornuble
media, may be alow
Sloping eoordtnJtea
Automatic tli* met
control, mar or
a lew
reported aprlic'.ition
to very dry aeiln.
well bore calculation*
-------
Tabla t.—Ttatunt of availablt gmvnt eatputtr aodti for ntvrto tvtution ff variably taturatid water,
•olutt, and tuat traniport pmhtfre rtlating to nxHoaetit* itutt iUtpo«tl~Cmtl
Linoarliatlo* folutloa
Inpllcit
vttti CaviatM
functional •llnlaatlo*
•xponilM
In »p«c«
and tlM
axtrapoUtlon
lofllclt Oautlan
altei nation
Inpllclt
with C«u««l««
•xtrapolatlo* allnlnatlon
txpllelt ADI
Inpllclt Ron
Iipllclt
with *0«
Nevtoo.-
tathaon
Ixpllelt Hone
tourc*/ Boundary
• ink condition
typea typaa Verification Xeaarka
Mnt All, Coluan data,
including finite
aeepaga difference
facaa oodcli.
annlylic
aolutiona
txpllclt All, finite (me r«r»Tta r< n»
including dKferrnca obtainable aolvllni
aeepaga nnlrla, for very *on-
facaa (laid data linear problem*
one-JIrwneion
txpllcit All coluum, lab
data. Brealera
finite
dlfforenc*
•ndel
Explicit All Unknown
Nona All, lab tank
including data
aoepaga
facea
Coiw«Tgcnce
Explicit Ail Unknown problem for vtry
nonlinear problcna
Explicit All Other -odola, e-b..ed -
analytic unaaturated
eolutloni condition! only.
Ko gravity tent.
very long CrT tlm
-------
T«bl« ^.—•f^atUf»^ of avallablt gmaral oofitttr oodit far nmHe lolutlon of variably taturatcJ uater,
toluti, and htat trantpart proMewe nlatlng to radtoaaHvt Halt* oK*poea{--Contlnued
Model
aaae Auchora
Vauclln
and othera
(1975)
W.1 and
Jeppaon
(1971)
Zyvoloikl
anil Iruch
(M73)
.Solute Heat Tine
Maxima; trana- trane- Spatial Inte-
elMMlona port port dlicretliatlon gratlon
2 No Ho Finite Implicit
difference
2
O-axU]W Mo Ro Finite (taady
•atrlc) difference etata
2 Mo Ho Calerkln- TlM
finite element centered
Llnearlaatlon Solution
Implicit IADI
Mwton-
lapheon *O*
tBpllclt
«lth *hepa Cauaalan
function ellmioatloo.
expanalon
In aptfca
Source/ Boundary
• ink condition
typea typea
Ron* All,
Including
aiepaga
facea
Ron* Dirlchlet
ROM M.1
Verification
tab tank
data
None
reported
Finite
difference
awdela
Reaarka
Im
-------
112
forms of equations 8-12 and the methods of numeric approximation used
for their solution. A thorough understanding of the physical phenomena
described by the equations themselves, the numeric methods involved, and
the definitions of valid solutions (in a qualitative or semi-quantitative
sense) are requisite for the responsible use of these models.
One check of the validity of numerical solutions, when the results
cannot be anticipated a priori, was suggested by the National Academy of
Sciences in a report on risk assessment in radioactive waste disposal
studies (National Academy of Sciences, 1979). Their suggestion was the
application of several models to the same problem to give a consistency
check. This check is necessary because of uncertainties in a numerical
model owing to roundoff and convergence errors as well as uncertainties
in material properties and transport coefficients for a given field
problem. Such duplicate simulation might be an affront to the authors
of particular computer codes, and might appear a duplication of effort
to some involved in the management of scientific studies. However, in
the light of the stakes involved in proper selection and management of
disposal sites for extremely hazardous wastes for very long times, such
checking seems mandatory.
Parametric Models
The use of solutions to equations 8-12 discussed above depends upon
several factors: (1) the availability of an analytic solution or compu-
ter code using a particular numerical solution; (2) the availability of
measured values of the nonlinear transport coefficients as a function of
the solution variable as well as a function of space; (3) the availability
of data to calibrate the models over the range of expected conditions dur-
ing the life of the disposal site; and (4) funding to adequately simulate
nonlinear transport processes over periods that may exceed 1,000 years.
Restrictions imposed by the last three factors may preclude the use of
detailed physics-based models. In these cases, adequate predictive
capability may be met by the use of parametric or input-output models.
Parametric models simulate the movement of mass or energy by consid-
ering the modeled system to consist of a series of empirical functions
that transform an input into an output:
Input - Transfer ^
Aupuu functions v
Cfor example, rainfall) (for example, infiltration rate)
Transfer functions may be purely empirical or based on some physical
principle. Often, transfer functions are derived from analytic solutions
to simple formulations of one part of the simulated system. As an example,
the translation of rainfall into infiltration can be accomplished by using
the Green-Ampt infiltration equation (see p. 19) as the transfer function.
-------
113
Parametric models are most applicable to systems in which the
physical and chemical processes simulated are so weakly coupled that they
can be considered independent. However, some feedback mechanisms can be
incorporated in these models to allow some degree of coupling.
Eagleson, in a series of papers (1978a, b, c, d, e, f, and g) gives
a comprehensive overview of parametric models applied to rainfall-
infiltration-runoff modeling. Comprehensive parametric models that in-
clude the movement of water in variably saturated soils are described by
Skaggs (1978), Afonda (1978), Crow and others (1977), Fitzpatric and Nix
(1969), Neibling (1976), and Neibling and others (1977),
Parametric models of solute movement in variably saturated systems
have been described by King and Hanks (1973), and Davidson and others
(1978). Dutt and others (1972) combine a numeric solution to the liquid-
water flow problem with a parametric representation of the chemical
transformations for an unsaturated system.
The principal advantages of parametric models are: (1) the reduced
number of dependent variables and parameters; and (2) that they are
usually structured to give directly the desired output in terms of mass
and energy flux rates. Most analytic or numeric methods solve for the
spatial and temporal variation in pressure, concentration, or temperature.
Flux rates must then be computed as an additional step. Hence, para-
metric models directly provide the desired fluxes with less effort than
numeric models. Another advantage of parametric models is that because
their mathematical structure is relatively simple, the effects of uncer-
tainties in the input can be quantitatively related to uncertainties
in the output.
Parametric models usually are given some physical basis by fitting
transfer functions to observed data from field experiments; this practice
often limits the generality of these models. The principal disadvantage
of this approach is that data collection and analysis must be pursued
over long time periods to assure an adequate model.
An approach that minimizes this expensive process consists of using
detailed numeric models as the "experimental facility" rather than a field
site. Once a numeric model is verified, if it is sufficiently general, it
may be used cheaply to conduct numerical experiments to derive the appro-
priate transfer functions. An example of this use of numeric models is
the work of Smith (1972) in deriving rainfall-infiltration transfer
functions. This seems a particularly promising area for future work.
SUMMARY
Physics-based or deterministic models that describe mass and energy
transport through variably saturated porous media can be expressed by one
or more nonlinear partial differential equations. These equations are
derived by combining equations for continuity of mass and energy with
constitutive relationships. The latter express mass or energy flux as
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proportional to a driving gradient in measurable variables, such as tem-
perature, pressure, or concentration. The proportionality functions are
defined as general transport coefficients. The equations can be nonlinear
because of dependence of the transport coefficients, boundary conditions
and sink strengths upon temperature, pressure, or concentration.
The most flexible formulation of the nonlinear liquid-water flow
problem is obtained by writing the equation in terms of pressure or total
potential rather than as a concentration-dependent problem with volumetric
moisture content as the solution variable. The pressure or total poten-
tial formulation allows simple unified treatment of both saturated and
unsaturated conditions.
Analytic, quasi-analytic, and numeric methods are available for
solving the descriptive equations for a given waste-disposal problem.
Analytic and quasi-analytic solutions are generally available for most
multidimensional steady-state water-flow problems, if the hydraulic con-
ductivity can be described by an exponential function of head or moisture
content. Analytic solutions that consider heterogeneity in the transport
coefficients are available if the heterogeneity is restricted to layering
for one-dimensional, steady-state problems, and to linear or exponential
dependence on the spatial coordinates in multidimensional transient
problems. Analytic solutions that include source and sink terms that are
simple functions of spatial coordinates and the solution variable are
available for some simple geometries.
If the assumptions required for a particular analytic solution are
reasonable for analyzing waste-disposal problems, it should be used.
However, most waste-disposal problems are sufficiently complex that
analytic solutions are not flexible enough to provide reasonable answers.
Few, if any, analytic approaches are available when two or more coupled
equations are needed.
The complexities of most waste-disposal problems can usually be
handled by approximate numeric methods. These methods consist of spatial
discretization, using finite differences or finite elements combined with
suitable time integration, linearization, and matrix solution schemes.
For problems with very mild nonlinearities in the storage term,
Galerkin finite-element methods are the most flexible. These methods are
particularly accurate for tracking sharp fronts. When the storage term
is more than mildly nonlinear, the Galerkin finite-element schemes must
be modified by lumping the storage term over each element. Orthogonal
collocation on finite elements and finite differences are the most
flexible solution methods for the nonlinear transport equations. The
application of upstream weighting on the conductivity that may be
required to prevent numerical oscillation in some problems is more
straightforward in finite-difference methods than in finite-element
methods.
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Time-integration schemes that are fully implicit in the solution
variable are preferred for nonlinear problems, and required for the vari-
ably saturated water-flow equation, when boundary conditions change in
the saturated zone. Crank-Nicholson methods are the most accurate, but
a scheme that is valid for nonconstant transport coefficients must be
used. Explicit time-integration schemes for nonlinear problems have such
stringent time-step restrictions that their use is generally not feasible
for simulating large systems over long time periods.
Linearization methods that evaluate nonlinear terms using the current
values of the solution variable (implicit linearization) with iterative
improvement are the most accurate and flexible. Evaluating the nonlinear
terms using values of the solution variables from the previous time step,
and extrapolation using previous time steps without iterative improvement,
can result in erroneous solutions. Implicit Newton-Raphson linearization
of the storage and Cor) conductance terms often speeds convergence, and
is required for a convergent solution in many extremely nonlinear prob-
lems. Ad hoc damping coefficients may have to be applied to obtain con-
vergence in many problems as they approach steady state. The matrix
equations that result from the foregoing steps can be solved by some form
of direct Gaussian elimination or by an iterative scheme. Direct solution
methods are the most accurate if they include matrix conditioning and the
proper use of single- and double-precision arithmetic. The major dis-
advantage of direct methods are large computer core and time requirements.
Iterative matrix solution methods require less core and often less time
to achieve the same accuracy. Two additional advantages of iterative
methods are: (1) accuracy of the solution can be controlled by specifying
convergence criteria; and (2) the methods can be efficiently embedded in
the iteration required for linearization and coupling of equations.
Documented computer codes are available to handle the following
types of variably saturated transport problems relative to waste
disposal:
1. Three-dimensional movement of liquid water and solutes in
unfractured, heterogeneous, anisotropic media;
2. Two-dimensional planar and three-dimensional axisymmetric fully
coupled movement of liquid water, moist air, water vapor, and
heat in unfractured heterogeneous anisotropic media;
3. Three-dimensional uncoupled movement of liquid water, moist air,
and solutes in unfractured heterogeneous, anisotropic media.
Some of the multidimensional, multiphase, multicomponent codes are
so general that their application to a simple problem may be inefficient.
However, several codes are available that are efficient for simpler
problems. The capabilities of the most flexible documented codes are
summarized in table 4.
Although there are many existing computer codes with capability to
simulate many realistic problems in radioactive waste disposal, there are
several areas in which considerable effort should be expended to assure
adequate ability to predict the fate of radionuclides and other high-level
wastes emplaced in variably saturated media. These include the following:
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1. The physical mechanisms of flow of water and solutes through
variably saturated, fractured, and structured media need to be incorpo-
rated in simulation models. Movement through large openings in these
media dominate infiltration and solute movement.
2. The direct use of detailed physics-based models for long-term
(on the order of 100 to 1,000+ years) prediction of the fate of radio-
active pollutants may not be defendable on the following grounds:
a. Uncertainties in data due to spatial variation of transport
properties and boundary conditions may be so large that
collection of the required data for adequate simulation with a
detailed model would be prohibitively expensive. This data
uncertainty is further compounded by the necessity to determine
many transport properties in the laboratory at a generally
smaller scale from the field problem; the transferability of
laboratory determined data to field conditions must be
established.
b. Insufficient data are available to adequately calibrate detailed
models for particular sites.
c. Computer costs for running realistic problems may be prohibitive.
The need for usable, reliable prediction tools for waste-disposal
problems can perhaps be better met by the use of simplified parametric or
input-output models, in which the functions that translate input into
output have some physical basis. These functions can often be determined
by numerical experimentation with physics-based models. The physics-based
models used to train the parametric models need to be verified by
simulating physical and chemical transport from controlled laboratory and
small-scale field experiments.
3. For both physics-based and parametric models, the uncertainty in
any prediction of the fate of pollutants should be quantified, based on
all the assumptions in the model and uncertainties in the data. This
area of modeling of ground-water systems is in its infancy. However, it
will be required so that confidence limits can be placed on any predictive
results and incorporated in risk analyses used in design and management
decisions.
Whether physics-based models are used directly to simulate waste-
disposal problems or used to train parametric models, their adequacy
should be checked by duplicate simulation of the same problems with
another model or models. This step seems mandatory in the light of the
complexity of most waste-disposal problems, uncertainty in model
parameters, and the necessity to use numerical tricks to obtain a
solution.
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SELECTED REFERENCES
General
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James, M. L., Smith, G. M., and Wolford, J. C., 1968, Applied numerical
methods for digital computation with FORTRAN: International Text
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Laliberte, G. E., Corey, A. T., and Brooks, R. H., 1966, Properties of
unsaturated porous media: Colorado State University Hydrology
Paper 17, 40 p.
Mualem, Y., 1976, A new model for predicting the hydraulic conductivity
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National Academy of Sciences, 1979, Implementation of long term environ-
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prepared by the panel on The Implementation Requirements of
Environmental Radiation Standards, Committee on Radioactive Waste
Management, 65 p.
Nobel, B., 1969, Applied linear algebra: Prentice Hall, Inc., 523 p.
Narasimhan, T. N., and Witherspoon, P. A., 1976, An integrated finite
difference method for analyzing fluid flow in porous media: Water
Resources Research, v. 12, p. 57-64.
Price, H. S., and Coats, K. H., 1973, Direct methods in reservoir
simulation: Society of Petroleum Engineers of the American Institute
of Mining Engineers, Paper No. 4278, 20 p.
United States Environmental Protection Agency, 1978, Utilization of
numerical groundwater models for water resource management: U.S.
Environmental Protection Agency, R. S. Kerr Environmental Research
Laboratory.
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118
REFERENCES
Analytic Solutions
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1979, Steady state moisture flows in unsaturated soil containing a
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1974, More on an approximate solution for non-linear diffusion:
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119
Brutsaert, Wilfried, and Weisman, R. N., 1970, Comparison of solutions
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Heaslet, M. A., and Alksne, A., 1961, Diffusion from a fixed surface
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Kobayashi, H., 1966, A theoretical analysis and numerical solutions of
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1976b, Time-dependent linearized infiltration II; Line sources: Soil
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1978a, Linearized moisture flow with loss at the soil surface: Soil
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_1978b, Time-dependent solutions to the one-dimensional linearized
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Merrill, S. D., Raats, P. A. C., and Dirksen, C., 1978, Laterally confined
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Morel-Seytoux, H. J., 1978, Derivation of equations for variable rainfall
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Morel-Seytoux, H. J., and Khanji, J. J., 1974, Derivation of an equation of
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1975, Prediction of inhibition in a horizontal column: Soil Science
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121
Parlange, J. Y., 1971c, Theory of water movement in soils, 3, two and
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1972a, Theory of water movement in soils, 4, two and three
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1972b, Theory of water movement in soils, 5, unsteady infiltration
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1972c, Theory of water movement in soils, 6, effect of water depth
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1972e, Theory of water movement in soils, 8, one dimensional
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_1973a, Theory of water movement in soils, 10, cavities with constant
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__1973b, A note on three parameter soil water diffusion function.
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1980, An application of Brutsaerts and optimization techniques to
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REFERENCES
Numeric Solution (Water Flow Only)
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135
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136
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137
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VERIFICATION OF PARTIALLY SATURATED ZONE MOISTURE
MIGRATION: MODEL AND FIELD FACILITIES
S. J. Phillips
T. L. Jones
Rockwell International
Rockwell Hanford Operations
Energy Systems Group
Rich!and, Washington 99352
Battelle Memorial Institute
Pacific Northwest Laboratory
Richland, Washington 99352
ABSTRACT
Two large field-experimental facilities, designed to evaluate
moisture migration under partially saturated geohydrologic conditions,
have been constructed on the Hanford Site. The function of these
facilities is to monitor the rate and magnitude of moisture migration.
Models have been used to simulate and predict one dimensional liquid and
vapor phase water migration. Parametric data from these facilities are
being used for verification of the UNSAT and MPHASE codes. To date,
these models have shown only limited success in the simulation of:
volumetric water content, water storage, water drainage, liquid flux and
vapor flux. Continued studies will enhance the capability to simulate,
predict and verify moisture migration and radionuclide transport under
field conditions at the Hanford Site.
139
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INTRODUCTION
The Hanford Site located in south central Washington state was
established in December, 1942. The Site was constructed for production
of nuclear defense materials. For nearly four decades, nuclear fuels
fabrication plants, reactors, chemical separations and processing plants,
laboratories, and waste storage and waste disposal facilities have been
operated. These facilities have produced substantial quantities of low-
level radioactive waste materials that have been stored or permanently
inturned, i.e., disposed within the partially saturated geohydrologic
system at the site. Nuclear waste materials from other defense and
commercial facilities have also been sent to the Hanford Site for storage
or disposal.
Low-level wastes have been released within the partially saturated
geologic media of the site by several disposal methods, e.g., shallow
land burial. These methods rely on the geochemical sorption of long-
lived radionuclides and the attenuation of liquid- and vapor-phase flux
of radionuclides in arid geohydrological environments. Theoretical
investigations of radionuclide and moisture transport in partially
saturated systems have suggested a potential for loss of containment
and migration of radionuclides away from disposal facilities. This
may result in wastes entering the biosphere. As a result, numerous
laboratory and field studies have been conducted to evaluate the rate,
magnitude, and consequences of radionuclide and moisture migration at
the Hanford Site.
Following is a generalized review of specific numerical modeling
efforts and field experimental facilities that have been used in part,
for the validation of these models. The purpose of this report is to
show the initial results and complexities of modeling and experimental
studies to understand and verify migration of liquid- and vapor-phase
materials in the geohydrologic system, relative to low-level waste
disposal. The results of this report cannot be considered conclusive
relative to validation of the models. Results are given only to
enumerate progress toward this goal.
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Much of the information reported here has been taken directly from
published reports originating from the Hanford Site.
METHODS AND MATERIALS
SHALLOW CAISSON, LYSIMETER, MICROMETEOROLOGICAL FACILITY
A Shallow Caisson, Lysimeter, Micrometeorological experimental facil-
ity was constructed on the Hanford Site in 1978. U-»2»3J This facility
was expressly designed for: (1) development and calibration of monitoring
instruments and (2) development of methods to evaluate mass and energy
balance factors that control radionuclide transport in and around
shallow, partially saturated, geohydrologic systems. In addition, the
Shallow Facility was designed to function analogous to a one-dimensional,
shallow land, waste burial structure. Experimentation and monitoring
activities were also conducted that were amenable to model validation,
i.e., one-dimensional, nonisothermal, diphasic, partially saturated water
and radionuclide transport.
The Shallow Facility, Figure 1, shows micrometeorological and hydro-
geological monitoring instrumentation as it exists at present.
The Shallow Facility consists of three buried caissons 8.1 m by
2.7 m, and four buried caissons 7.5 m by 0.6 m. These caissons are
attached to form a seven-caisson array. The caissons are interconnected
with steel tubing to provide access to the outer caissons from the cen-
tral caisson. These tubes are installed at 15-cm vertical intervals
between the central caisson and large caissons and between the central
caisson and small caissons at 30 cm intervals. The tubes provide access
for instruments and destructive sampling of radiocontaminants and other
materials.
Weighing lysimeters to determine mass balance are installed below
grade and attached to the central caisson. Uniform sand was backfilled
into the six peripheral caissons of the array between 8.1 m and 6.3 m
deep. A layer of coarse quartz sand was then introduced into the cais-
sons between 6.0 m and 6.3 m. This layer provided for evaluation of
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ro
FIGURE 1. Oblique View Schematic of Shallow Facility Showing
Caissons, Lysimeters, Micrometeorologic Instrument Sensors, etc.
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layering effects. The remaining 6.0 m of the caisson were then back-
filled with uniform sand. The four small caissons were excavated to a
depth of 60.5 cm and a 1.0 cm layer of either cobalt-60 (60Co), -
ethylenediaminetetracetic acid (EDTA) or tritium (3H) tagged backfill
material was emplaced in alternating small caissons. The concentrations
of respective radionuclides that were designed to function as liquid and
vapor phase migration tracers were 6.4 yCi Kg and 6.37 m Ci Kg"1
at volumetric moisture contents equal to that of the larger caissons at
the 60.0 cm depth. Excavated backfill material was then emplaced over
the tagged layer.
During and subsequent to backfilling operations, active and passive
monitoring access tubes, sensors and transducers were installed for:
(1) neutron thermalization and sodium-iodide well logging, (2) geohydro-
logical thermal and matric potential measurement, and (3) micrometero-
logical thermal relative humidity, wind, precipitation, solar radiation,
evaporation, etc. Passive monitoring procedures were developed and
active data collected for computer processing.
Moisture migration and radionuclide transport occurs as a function
of moisture content induced by changes in precipitation. In order to
evaluate this function one-half of the facility field, in bilateral
symmetrical configuration, was irrigated at twice the ambient precipita-
tion rate at nearly ambient frequency.
MODIFIED UNSAT MODEL APPLICATION
Soon after construction of the Shallow Facility, modeling activities
were completed to produce an initial prediction of water migration. A
modified one-dimensional, isothermal, partially saturated, finite dif-
ference model was used'-4'5-'. Equations for total water flux were
developed to account for thermal and matric potential gradients. Using a
-1.0 cm cm"1 water potential gradient and equations for thermally
induced water flux, a nonisothermal water flux in response to thermal
gradients was calculated for various water contents in geologic media.
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This suggested that simulation of isothermal conditions could adequately
describe the flux of liquid water within the geologic media.
The following model simulating conditions in the caisson facility
were used: (1) an 8.0 m profile of uniform sand with a coarse quartz
sand layer, (2) a static water table at 8.0 m depth, (3) measured
retentivity and saturated hydraulic conductivity values for the uniform
and coarse quartz sand, (4) partially saturated hydraulic conductivity
values as determined by the Millington and Quirk method"- , (5) the
vapor flux from the surface was taken as 0.001 cm day , (6) measured
atmospheric temperature, wind, solar radiation, cloud cover, and precip-
itation, calculated potential and actual evaporation and (7) standard
initial conditions of depth, suction head, volumetric water content,
initial storage, final storage, infiltration, evaporation, and drainage.
A model sensitivity analysis was also conducted to evaluate such factors
as hydraulic conductivity, total precipitation, precipitation distribu-
tion, and layering.
DEEP CAISSON/DEEP WELL FACILITY
In 1971 a Deep Caisson/Deep Well Facility was designed to evaluate
the migration of naturally occurring moisture in the partially saturated
zone'- . This Deep Caisson Facility has served as a one-dimensional
structure amenable to partially saturated zone model simulation and
verification. The Caisson Facility, Figure 2, shows geohydrological
monitoring instrumentation. Near the Deep Caisson Facility is a deep
instrumented well designed to monitor matric potential and temperature in
geologic media between the surface and the water table. This
instrumented well is also illustrated in Figure 2.
The Deep Caisson Facility consists of two thermally insulated cais-
sons 3.0 m in diameter by 17.0 m in length buried below grade. Centrally
located between these caissons is an instrument access caisson. Neutron
thermalization access tubes and matric potential sensors were installed
vertically through the two outer caissons. One caisson was sealed at its
lowest extremity to form a liquid barrier,while the other remained open
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RCP8012 120
FIGURE 2. Oblique View Schematic of Deep Caisson/Deep Well Facility
Showing Caissons and Geohydrological Instrument Sensors, etc.
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146
so that direct communication between the ambient geohydrologic system
would result. Uniform sand excavated during caisson construction and
emplacement was backfilled into the two caissons.
A deep instrumented well was constructed to a depth of 97.0 m. Ma-
tric potential and thermal sensors were affixed to a cable and lowered
into the well. The well was then backfilled with geologic media removed
during construction.
MPHASE MODEL APPLICATION
In 1975 a computer code, MPHASE, was developed to simulate water
migration through the Deep Caisson Facility. A one-dimensional, steady
state, nonisothermal, liquid and vapor phase, partially saturated, finite
element model formulation was used.'- ^ Equations were developed and
modified for determination of saturation, total water flux and liquid and
vapor phase fluxes. These equations accounted for thermal and diphasic
water potential gradients. Dimensionless equations were formulated for:
(1) water balance, (2) dry air balance, (3) energy balance, (4) liquid
mass flux, (5) wet air mass flux and (6) dry air mass flux. Depth and
time were also made dimensionless. A variable pressure, nonispthermal
gradient for liquid and vapor phase flux was determined for all depths
from the ground surface to the water table.
The MPHASE model used the following conditions: (1) 97° m deep pro-
file of homogeneous geologic media (2) static water table, (3) measured
liquid and vapor hydraulic conductivities, (4) measured temperature,
(5) measured capillary pressure and (6) fixed and/or time varying boundary
conditions.
RESULTS AND DISCUSSION
MODIFIED UNSAT MODEL SIMULATION RESULTS
Sensitivity analyses of geologic media sand dependence on hydraulic
conductivity to moisture migration at the shallow facility were performed
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147
using ambient, 0.1 and 0.01 factors. Simulations demonstrated that:
(1) water penetration into the geologic media was reduced by reducing the
hydraulic conductivity by 0.1 or 0.01, (2) water storage and seepage
below 2.0 m was not significantly affected, (3) a reduction in hydraulic
conductivity reduced the water flow into the Shallow Facility so that it
tended to remain closer to the ground surface and (4) a reduction in
hydraulic conductivity resulted in reduced evaporation. Table 1 shows
the results of varying hydraulic conductivity.
TABLE 1. Simulation Results of Varying
Hydraulic Conductivity Values.a>&
Variable
Ambient
Ambient/ 10
Ambient/ 100
Infiltration
(cm)
24.1
24.5
24.5
Evaporation
(cm)
26.3
23.3
22.7
Drainage
(cm)
-0.4
-0.2
-0.1
Storage
(cm)
42.6
45.7
46.2
Modified after Gee and Simmons, 1979
Initial storage - 44.6 cm
Model simulations indicate that by varying the hydraulic conductiv-
ity there can be a pronounced effect on water content and flux near the
ground surface, indicating these effects are reduced with depth. In
addition, modification of hydraulic conductivity can, over time, influ-
ence the migration of water and nonsorbed radionuclides within geohydro-
logic systems.
The effects of annual precipitation in the form of rainfall were also
evaluated by sensitivity simulations. The model, as anticipated, predic-
ted that an increase in precipitation would increase drainage to the water
table. The model results indicated: (1) changes of precipitation from
one year to the next are reflected in the migration of moisture through
and from the geologic media, (2) a significant decrease in annual precipi-
tation below mean annual precipitation will propably result in negative
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148
drainage of moisture to the water table whereas an increase will result
in positive drainage, (3) moisture stored in the profile will not be dras-
tically affected. The effect of total precipitation over six years using
a one standard precipitation year value are shown in Table 2.
TABLE 2. Simulation of Total Precipitation Effects on
Six Consecutive Years Based on Multiples of
One Standard Precipitation Year.a»b,c
Year
1
2
3
4
5
6
Infiltration
(cm)
49.1 (2.0 x)
36.8 (1.5 x)
8.1 (0.33 x)
24.5 (1.0 x)
36.8 (1.5 x)
18.5 (0.75 x)
Evaporation
(cm)
34.9
31.5
13.1
23.2
30.9
21.1
Drainage
(cm)
7.5
6.8
0.6
-0.3
1.9
1.0
Storage
(cm)
51.0
49.3
43.9
45.5
49.3
45.6
Modified after Gee and Simmons, 1979
"Initial storage - 44.6 cm
Potential evaporation - 131.0 cm
Results of this simulation demonstrate that moisture migration is
significantly affected by changes in total annual precipitation. Fur-
thermore, annual precipitation history additionally affects moisture
migration.
The sensitivity of the temporal distribution of precipitation on an
annual basis was simulated in addition to the total annual precipitation.
Two years with different temporal precipitation were selected, i.e., one
year with precipitation occurring predominantly in the fall, year a, and
the other with predominantly spring and winter precipitation, year b, as
in Table 3. Model simulation results have indicated that by alternating
years a and b variable increases or decreases of moisture evaporation,
drainage and storage result depending on the alternation sequence. After
using developed standard initial conditions and sequentially repeating
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149
year a, moisture, evaporation, drainage and storage all increase. Model
simulation results for alternative and repetitive years are shown in
Table 3 and 4 respectively.
TABLE 3. Simulation of a Three-Year
Precipitation Sequence.a,b,c
Year
a
b
c
Infiltration
(cm)
23.5
24.6
24.0
Evaporation
(cm)
20.1
25.0
24.6
Drainage
(cm)
-0.1
0.7
0.2
Storage
(cm)
48.0
46.7
46.4
Modified after Gee and Simmons
"Initial storage - 44.6 cm
Potential evaporation - 147.7 cm, year a; 131.2 cm, year b
TABLE 4. Simulation of a Three-Year
Repetitive Precipitation.a,b,c
Year
a
a
a
Infiltration
(cm)
23.5
23.5
23.5
Evaporation
(cm)
20.1
22.0
22.1
Drainage
(cm)
-0.1
1.2
1.3
Storage
(cm)
48.0
48.2
48.2
Modified after Gee and Simmons
blnitial storage - 44.6 cm
Potential evaporation - 147.7 cm, year a; 131.2 cm, year b
The simulation indicates that the temporal distribution of precipi-
tation significantly affects the hydrogeologic system. Furthermore, it
illustrates the complexity of predictive modeling of systems with
variable upper boundary layer conditions.
Sensitivity simulations were also completed to evaluate the effect
of layering, relative to partially saturated moisture migration. Hetero-
geneity of geologic media in texturally alternating layers was found to
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150
have a pronounced effect on moisture flux and drainage. Layering reduces
the flux of liquid phase moisture across lateral layer boundaries during
drainage, until the water saturation in the upper layer reaches some
critical valve wherein the water will freely drain into the next lower
vertical layer.
Model results indicate: (1) the interface between course and fine
textured geologic media at a depth of 6.0 m significantly perturbates the
moisture content profile, (2) drainage is reduced where a coarse layer
exists in the profile while drainage is increased when this layer is
omitted from the profile (3) storage of moisture in the profile is
increased where the coarse layer is present; conversely, storage is
reduced with omission of the coarse layer.
Simulation results of layering cases are shown in Table 5 for a non-
homogeneous coarse geologic media layer between 6.0 m and 6.5 m in depth,
(a) and a homogeneous profile (b).
TABLE 5. Simulations of Layering Effects.a»b»c,d
Water
Table
(a)
(b)
Infiltration
(cm)
23.5
23.5
Evaporation
(cm)
20.1
20.4
Drainage
(cm)
-0.01
0.3
Storage
Initial
44.6
50.0
(cm)
Final
48.0
52.7
Modified after Gee and Simmons, 1979
Potential evaporation - 147.7 cm
cCoarse sand layer 600-650 cm
dSand layer
Results of this sensitivity simulation demonstrate some of the
effects of layering on the flux of moisture through, and the retention
of moisture within, geologic media. These results may also provide
insight into the utility of design of engineered barriers relative to
the disposal of nuclear materials.
-------
151
The UNSAT model simulations, using the configuration and data from
the Shallow Facility, were used to provide an initial understanding and
prediction of moisture migration and nonsorbed radionuclide transport in
partially saturated geologic media.
Field data collected at the Shallow Facility over the past two years
have shown: (1) the gravity potential dominates the flux of moisture
through geologic media under both ambient and accelerated precipitation
conditions, (2) low distribution coefficient 60Co-EDTA has not been
transported under ambient or accelerated precipitation conditions as
o
calculated, (3) H has been transported both in the vapor and liquid
3
phase within the partially saturated profile of the facility, (4) H
has been transported in the liquid phase (data that suggests H has
also been transported in the vapor phase) within the partially saturated
profile of the facility, (5) moisture in the liquid phase has accumulated
in the bottom of the caissons receiving ambient plus two times ambient
precipitation, where no accumulation of moisture has been found in the
ambient precipitation caissons.
These simulation results are not intended to be conclusive. The
results were derived using actual and assumed data from the Shallow
Facility and adjacent experimental sites soon after construction. The
purpose of modeling this system was to provide an approximation of mois-
ture migration and nonsorbed radionuclide transport at the Shallow
Facility. These data could then be used to develop monitoring systems.
The model results also increased our understanding of the mechanism of
flow in geologic media under various conditions.
MPHASE MODEL SIMULATION RESULTS
Steady state and transient simulations of liquid and vapor phase
moisture flux were performed for both the open and closed bottom caissons
of the Deep Caisson Facility. Model solutions for the open bottom cais-
son during steady state conditions yielded applicable results while solu-
tions for the closed bottom caisson were found to require further model
-------
152
development. Steady state simulation results indicated: (1) near the
ground surface moisture migration is predominantly in the vapor phase,
(2) liquid phase moisture flux in the caisson is upward but is extremely
small, (3) vapor phase migration throughout the profile is upward at a
nearly constant rate, liquid phase flux below the caisson is downward at
a rate lower than that of the vapor phase, and (5) the total flux is
upward.
Table 6 shows dimensionless values for liquid and vapor fluxes
versus saturation at various depths. The caissons extend to a dimension-
less depth of 0.16.
TABLE 6. Vapor and Liquid Phase Flux vs. Saturation at
Various Depths in Deep Open Bottom Caisson
(Dimensionless Values).a»b
Depth
0.0000
0.1154
0.3846
0.5000
0.6154
0.8846
1.0000
Saturation
0.0600
0.0600
0.0600
0.0600
0.0600
0.0611
1.0000
Liquid Flux
-1.843 (-12)
-8.670 (-12)
2.019 (-10)
1.186 (-9)
2.997 (-9)
1.091 (-8)
-1.307 (-7)
Vapor Flux
-1.307 (-7)
-1.307 (-7)
-1.309 (-7)
-1.319 (-7)
-1.314 (-7)
-1.393 (-7)
0.000
Total Flux
-1.307 (-7)
-1.307 (-7)
-1.307 (-7)
-1.307 (-7)
-1.284 (-7)
-i;282 (-7)
-1.307 (-7)
Modified after Finlayson, 1975
bDimensionless flux x 5.5 x 10"4 = g sec -1
The total flux was found to be quite dependent on the pressure gra-
dient that determines the direction of liquid mass flux. This relates
whether the pressure gradient is greater or less than the gravitational
force acting on the liquid.
Continued monitoring of variables controlling moisture migration at
the Deep Caisson Facility could yield results amenable to model calibra-
tion and verification.
-------
153
CONCLUSIONS
Field monitoring and model simulations of moisture migration and
radionuclide transport and controlling factors are exceedingly complex.
This is indicated by the variability of model simulations and model
sensitivity simulations reported herein. Presently only limited simu-
lation and prediction estimates of migration and transport can be made.
Further field, laboratory and model development efforts will enhance
the capability to simulate, predict, error bound and verify moisture
migration and radionuclide transport under field conditions at the
Hanford Site.
-------
154
REFERENCES
1. S. J. Phillips, L. L. Armes, R. E. Fitzner, G. W. Gee, G. A.
Sandness, C. S. Simmons, Characterization of the Hanford 300 Area
Burial Grounds: Final Report Decontamination and Decomissloning.
PNL-2557 (January, 1980).
2. S. 0. Phillips, A. E. Reisenauer, W. H. Richard, and G. A. Sandness,
Initial Site Characterization and Evaluation of Radionuclide
Contaminated Solid Waste Burial Grounds, BNWL-2184 (February. 1977).
3. S. J. Phillips, A. C. Campbell, M. 0. Campbell, G. W. Gee, H. H.
Hooper, and K. 0. Schwarzmiller, A Test Facility for Monitoring
Water/Radionuclide Transport Through Partially Saturated Geologic
Media: Design, Construction, and Preliminary Description, PNL-3226
and PNL-3227 (November 1979).
4. G. W. Gee and C. S. Simmons, Characterization of the Hanford 300
Area Burial Grounds: Fluid Transport and Modeling, PNL-2921
(August, 1979).
5. S. K. Gupta, K. K. Tanji, D. R. Nielsen, J. W. Biggar, C. S. Simmons
and J. L. Maclntyre, Field Simulation of Soil-Water Movement with
Crop Water Extraction, Water Science and Engineering Papers No. 4013.
University of California at Davis (January, 1978).
6 A. E. Reisner, Calculations of Soil Hydraulic Conductivity From Soil
Water Retention Relationships, BNWL-1710. 1973.
7. T. L. Jones, Sediment Moisture Relations: Lysimeter Project
1976-1977 Water Year, RHO-ST-15 (June, 1978).
8. J. J. C. Hsieh, A. E. Reisenauer, and L. E. Brownell, A Study of
Soil Water Potential and Temperature in Hanford Soils. BNWL-1712
(1973).
9. R. E. Isaacson, L. E. Brownell, R. W. Nelson and E. L. Roetman, Soil
Moisture Transport in Arid Site Vadose Zones. ARH-SA-169, 1974.
10. J. J. C. Hsieh, L. E. Brownell and D. E. Reisenauer, Lysimeter
Experiments. Description and Progress Report on Neutron
Measurements. BNWL-1711, 1973.
11. B. A. Finlayson, MPHASE - A Computer Model Simulating Air-Water
Movement in Desiccated Soils. RHO-SD-31. 1978.
-------
RADIONUCLIDE TRANSPORT UNDER CHANGING SOIL SALINITY CONDITIONS*
T. L. Jones
G. W.. Gee
Pacific Northwest Laboratory
Richland, Washington 99352
and
P. J. Wierenga
New Mexico State University
Las Cruces, New Mexico 88001
SUMMARY
This paper presents experimental measurements and computer simula-
tions of strontium-85 movement through laboratory soil columns. There
are two primary objectives of the work. The first is to examine the
enhanced mobility of strontium-85 when the soil salinity is raised. The
second is to compare the ability of two solute transport models to
describe this phenomena. The experiment consisted of injecting a pulse
of what will be termed a "low salt" solution spiked with strontium-85
into a soil column. The column was then leached with over eighteen pore
volumes of the low salt solution resulting in no leaching of the
strontium-85 from the column. Subsequently, the column was leached with
a "high salt" solution, causing the rapid release of the strontium-85
within five to six pore volumes.
These data were analyzed using models based on two sets of equations.
The first used the traditional convective dispersive equation. This
equation contains the three parameters; water velocity V, dispersion
coefficient D, and the distribution coefficient Kd. The first two may
be combined to form the dimensionless Peclet number (P)' and the Kd is
often expressed as a dimensionless retardation factor (R). This equation
in dimensionless form is a two parameter solute transport equation. The
second model uses the mobile-immobile water equation described by Van
Genuchten and Wierenga (1976a). This model uses a set of two equations
which contain six parameters which may be combined into four dimensionless
parameters.
These models involve a large number of parameters that must be
evaluated. The techniques used in this work used the curve fitting
routine described by van Genuchten (1980), as well as some batch Kd
measurements described by Gee and Campbell (1980). The curve fitting
was done on breakthrough curves obtained from separate column Teachings
with strontium and tritium using the high salt solution. The parameter
*Prepared for the U.S; Department of Energy under contract DE-A-
C06-76RLO-1830.
155
-------
156
estimates were then used in the low salt-high salt experiment described
above. This follows the reasoning outlined by van Genuchten and Wierenga
(1977a, 1977b).
The results demonstrate that strontium-85 is subject to greater
mobility when the soil salinity is increased. This confirms the neces-
sity of evaluating the soil salinity very carefully in modeling efforts.
The results also indicate that batch Kd values, combined with column Kd
values and parameters obtained by curve fitting procedures do a fairly
good job in describing the "snowplow effect." The comparison of the two
models reveal insignificant differences when describing the movement of
tritium and strontium-85 under high salinity conditions. There was a
significant improvement, however, when the mobile-immobile water equations
were used to describe the combined low salt-high salt experiment.
INTRODUCTION
The characterization of sorption reactions that remove radionuclides
from solution in soils and sediments is a very active area of research.
Among the diverse processes being studied are oxide and carbonate pre-
cipitation, precipitation of hydroxide compounds, coprecipitation with
amorphous hydrous oxides, lattice replacement, and ion exchange. In
recent years the relative emphasis placed on ion exchange has decreased,
however, it still is an important process for certain radionuclides. A
fundamental property of ion exchange reactions is that they are dependent
on the solution concentration of the exchanging ion as well as the ionic
strength and composition of the bulk solution. The general trend of
strontium-85 observed by Gee and Campbell (1980) was for the sorption
(i.e., Kd value) to increase with increasing ionic strength of the bulk
solution. This would indicate that ion exchange plays some role in the
sorption of strontium-85.
The implication for shallow land burial sites is that errors can be
made in predicting the movement of some radionuclides through sandy soils
when only average salt concentrations are used in predictive equations
rather than actual time dependent salt concentration. Burial ground
soils with relatively high concentrations of salt may, under increased
rainfall, flush sorbed radionuclides from the soil exchange sites creat-
ing a much higher effluent concentration than predicted. This flushing
action by a high salt concentration influent solution has been called the
"snowplow effect" (Starr and Parlange 1979). Included in this report is
verification of the "snowplow effect" for strontium-85. Conventional
solute transport equations and those that include the mobile and immobile
water concept (Van Genuchten and Wierenga 1976b), are used to analyze
the data.
Materials and Soil
The soil used in this experiment was a coarse sand found at the
Hanford site. This soil is classified as a typic torripsament. The
clay content is <4%, the cation exchange capacity is approximately
5 meq/100 g and the soil pH is 8.2.
-------
157
The solutions used in the tests were two mixed salt solutions
designated as low (L) and high (H) salt. The two solutions had the
following compositions:
Low salt (L) -0.15M NaN03, 0.01M KN03> 0.002M Ca (N03)2
High salt (H) -0.15M NaN03, 0.15M KN03> 0.01M Ca (N03)2
The experimental equipment used to demonstrate the snow plow effect
was designed after that used by Van Genuchten and Wierenga (1977a,b).
This is the same design described in Gee and Campbell (1980). Influent
solutions were applied at the upper end of soil columns packed in plexi-
glass cylinders of 5.4 cm inside diameter. Columns 27.5 cm long of
Rupert sand were packed to a dry bulk density of 1.73 g/cm3. At the
lower end of the vertically positioned column, a porous plate for con-
trolling vacuum and supporting the soil was attached. The column was
connected to a vacuum chamber enclosing an automatic fraction collector.
Figure 1 shows the apparatus used for measuring radionuclide transport
in unsaturated soil.
The low salt solution containing a 50 nci/m£ spike of strontium-85
was pumped through the column. After 4.3 pore volumes were injected,
the strontium-85 was removed from the influent solution. The column was
maintained at a moisture content of 0.17 cm3/cm3, and the flow rate was
6.8 cm/day. After 18.25 pore volumes of effluent were collected, the
relative concentration of strontium-85 (C/C ) was <0.01. The high salt
solution containing no strontium-85 was then added to the column at a
flow rate of 7.0 cm/day. The effluent was collected and the break-
through curve determined.
The results of the low-salt, high-salt experiment are shown in
Figure 4. The data show quite clearly the rapid release of strontium-85
after the addition of the high salt solution at 18.25 pore volumes. The
rapid rise in effluent concentration and the C/C0 value >1 are manifes-
tations of this showplow effect. The two curves shown represent the
curves predicted by the two and four parameter models. The parameters
used in the simulations are discussed in the next section. Figure 4
shows a distinct improvement in describing the snowplow effect by using
the four parameter model. Neither model predicts the fast rise of the
curve or the C/C0 > 1. The four parameter model does however do a good
job of describing the tailing.
COMPUTER SIMULATION
The computer models used for this simulation are described in
detail by Van Genuchten and Wierenga (1974, 1976b) and Van Genuchten
(1980). They were written using the IBM system 360 Continuous Systems
Modeling Program (S/360 CSMP), simulation language. The first model
used the convective dispersive equation:
-------
NEGATIVE PRESSURE
SOURCE
FLUID
MANOMETER'
DIFFERENTIAL
PRESSURE
REGULATOR PRESSURE
C-, REGULATOR
REGULATED
NEGATIVE '
PRESSURE
POSITIVE
PRESSURE
SOURCE
PARTIALLY
SATURATED
SEDIMENT
COLUMNS
FRACTION
COLLECTOR
on
c»
TAGGED SOLUTION
STEPF
SYRING
C
-X.
ULSE
EPUMP
O
T
VACUUM
CHAMBER
Fig. 1. Apparatus for measuring radionuclide movement in unsaturated soil columns.
-------
159
o
fc>
O
0.70
0.60
0.50
0.40
0.30
0.20
0.10 -
STRONTIUM -85
HIGH SALT
O DATA
2 PARAMETER
, 4 PARAMETER
I
3.00 6.00 9.00 12.00 15.00 18.00
PORE VOLUMES
Fig. 2. High salt strontium breakthrough curve.
-------
160
o
O
1.00
0.80 -
0.60 -
0.40 -
0.20 -
0 1.00 2.00 3.00 4.00 5.00
PORE VOLUMES
Fig. 3. High salt tritium breakthrough curve.
o DATA
4 PARAMETER
2 PARAMETER
17 18 19 20 21 22 23 24 25 26 27 28 29
PORE VOLUME
Fig. 4. Comparison of two and four parameter fit for snowplow effect.
-------
161
ax
where
C = concentration of solute
t = time
x = vertical distance from surface
D = dispersion coefficient
v = pore water velocity
P = bulk density
0 = volumetric water content
Kd = radionuclide distribution coefficient
in dimensionless form this becomes
D aC 1 a C aC
(2)
m+ d-f )p« ^ - «m D 1^ - vra «m ^
and
TT ' « (Cm - Ci»,' (8)
-------
162
where
e = mobile water fraction
f = fraction of exchange sites in contact with mobile water
C , C. = solute concentration in mobile and immobile water phase
V = mobile water velocity
m
a = mass transfer coefficient
in dimension!ess form these equations become:
(9)
(1-B) R — = a (Cm - C.J (1Q)
(ID
(12)
* +
-------
163
of the bulk solution and also the radionuclide solution. As the soil
salinity increased from low to high as the high salt solution moved
through the column, the Kd value was assumed to vary linearly from
6 to 0.6.
The parameter values for the mobile-immobile water equations were
obtained in a similar way. An analytical solution (Van Genuchten and
Wierenga 1976b), of Equations 9 and 10 was curve fit to the data in
Figures 2 and 3. The CSMP model used Equations 7 and 8 so a conversion
using equations 3, 4, 11, and 13 was done. The only conversion which is
not straightforward is the use of Equation 13. The value of B and R from
the tritium experiment was used first. Assuming 0 * f * 1, we calculate
a range of 0 of 0.84 to 0.95. Assuming 0 is the same for tritium and
strontium, we then use B and R from the high salt strontium data. Assum-
ing 0.84 < 0 * 0.95, we obtain a range for f of 0.69 to 0.7. The simula-
tion of the snowplow experiment used 0 = 0.9 and f = 0.7.
CONCLUSIONS
Radionuclide transport under changing soil salinity conditions was
investigated by using two solute transport equations to describe the
transport of strontium-85 through laboratory columns. The two parameter
convective dispersive equation was compared to the four parameter mobile-
immobile water equations of Van Genuchten and Wierenga (1976a).
Both models were relatively successful in describing the rapid flush
of strontium-85 from a column with a high salt solution, a phenomena
called the snow plow effect. This effect was simulated by using a Con-
tinuous System Modeling Program (CSMP) simulation model to solve the two
equations. The four parameter mobile-immobile water model predicted the
release of the strontium more accurately, but neither model predicted
the peak effluent concentration well. Both models confirm enhanced
mobility of strontium-85 with increased salt concentration of leaching
waters.
Table 1. Parameter Estimates for Computer Simulations
R 3 a D a Kd 0m
Tritium 106 1.1 0.9 0.4 - - - 0.9 0.5
Strontium-85
2 parameter 19 7.2 - 58 - 0.6
4 parameter 29 8.4 0.7 0.5 41 0.12 0.74 0.9 0.7
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164
REFERENCES
Gee, G. W., and A. C. Campbell. 1980. Monitoring and Physical Charac-
terization of Unsaturated Zone Transport-Laboratory Analysis. PNL-
3304. Pacific Northwest Laboratory Richland, Washington.
Starr, J. L., and J. Y. Parlange. 1979. "Dispersion in Soil Columns:
The Snowplow Effect." Soil Soi. Soo. Am. J. 43: 448-450.
Van Genuchten, M. Th., and P. J. Wierenga. 1974. "Simulation of One-
Dimensional Solute Transfer in Porous Media." Agricultural Experi-
ment Station Bulletin 628. New Mexico State University, Las
Cruces, New Mexico.
Van Genuchten, M. Th., and P. J. Wierenga. 1976a. "Mass Transfer Studies
in Sorbing Porous Media. I. Analytical Solutions." Soil Soi. Soo.
Am. J. 40: 473-480.
Van Genuchten, M. Th., and P. J. Wierenga. 1976b. "Numerical Solution
for Convective Dispersion with Intra-Aggregate Diffusion and Non-
Linear Adsorption." In System Simulation in Water Resources
(G. C. Vansteenkiste, ed.). North Holland Publishing Company,
Oxford, New York.
Van Genuchten, M. Th., and P. J. Wierenga. 1977a. "Mass Transfer Studies
in Sorbing Porous Media. II. Experimental Evaluation with Tritium
(3H20)." Soil Soi. Soo. Am. J. 41: 272-278.
Van Genuchten, M. Th., and P. J. Wierenga. 1977b. "Mass Transfer Studies
in Sorbing Porous Media. III. Experimental Evaluation with 2, 4,
5-T." Soil Sci. Soo. Am. J. 41: 278-285.
Van Genuchten, M. Th. 1980. "Determining Transport Parameters from
Solute Displacement Experiments. U.S. Salinity Laboratory Research
Report No. 118. USDA/SEA, Riverside, California.
-------
ON THE COMPUTATION OF THE VELOCITY FIELD AND MASS BALANCE
IN THE FINITE-ELEMENT MODELING OF GROUNDWATER FLOW
Gour-Tsyh Yeh
Environmental Sciences Division
Oak Ridge National Laboratory
Oak Ridge, Tennessee 37830
ABSTRACT
Darcian velocity has been conventionally calculated in
the finite-element modeling of groundwater flow by taking the
derivatives of the computed pressure field. This results in
discontinuities in the velocity field at nodal points and
element boundaries. Discontinuities become enormous when the
computed pressure field is far from a linear distribution.
It is proposed in this paper that the finite element proce-
dure that is used to simulate the pressure field or the
moisture content field also be applied to Darcy's law with
the derivatives of the computed pressure field as the load
function. The problem of discontinuity is then eliminated,
and the error of mass balance over the region of interest is
much reduced. The reduction is from 23.8 to 2.?% by one
numerical scheme and from 29.7 to -3.6% by another for a
transient problem.
INTRODUCTION
To study the transport of dissolved constituents in a subsurface
flow system, the velocity field therein must be determined first.
Several finite-element Galerkin models have been developed to obtain
the flow field [1-8]. The continuity equation of water-mass governing
the distribution of pressure head was solved by the Galerkin
finite-element method, subject to appropriate boundary and initial con-
ditions. The flow field was computed with Darcy's law by taking the
derivatives of the calculated pressure field [3-6]. Inherent in that
approach, however, was the resulting discontinuity in the velocity at
nodal points and element boundaries, which unfortunately lead to a vio-
lation of the conservation of mass in a local sense. Results from two
hypothetical sample problems showed that the discontinuity in the ve-
locity field obtained by the conventional approach ranges from very
small to several hundred percent depending on the location in the
region. Furthermore, the overall mass balance is not preserved. When
spatial distribution of the velocity is significant, inputting this
discontinuous flow field to the contaminant transport computation could
conceivably produce large error.
165
-------
166
It is proposed to solve Darcy's law for the velocity field at
nodal points by the same finite-element method used for the pressure
field rather than simply to take the derivatives of the approximate
pressure field. This approach is consistent with the spirit of
finite-element modeling of groundwater flow. It also yields a con-
tinuous velocity field over the region of interest including nodal
points and element boundaries. Accordingly,
use of the finite-element method, as applied
the discontinuity described above. The two
reexami ned
presented.
this paper describes the
to Darcy's law, to remove
hypothetical problems are
in the light of the proposed approach, and the results are
MATHEMATICAL STATEMENT
The governing equations to describe the water movement in variably
saturated-unsaturated porous media have been derived in detail [4]in the
following form:
F -
- Q = 0
(1)
and
(2)
where
(3)
and
H - h -4-
(4)
in which h is the pressure head, 0 is the moisture content, n is the
effective porosity, « ' and 3' are the modified coefficients of com-
pressibility of the medium frame and water, respectively, Kfj is the
hydraulic conductivity tensor, Vj is the Darcian velocity vector, t
is the time, Q is the artificial recharge or withdrawal, z is vertical
coordinate, and H is the total head. In Eqs. (1) and (2),
-------
167
the partial differentiation with respect to the spatial
-j; and 2^x3 is the vertical coordinate. In general,
8-j denotes
coordinate,
Eq. (1) is nonlinear because both the soil property,
Eq. (3)), and hydraulic conductivity tensor, K,-
functions of the pressure head, h.
(represented by
are nonlinear
The initial condition of Eq. (1) may take the following form:
in R
(5)
where h0 is a
prescribed function of the spatial coordinate, Xj,
region of interest enclosed by the boundary, B(xi, x?, x
h, '
R
is the
= 0 as shown in Fig. 1. The function, h0, may also be obtained by
simulating the steady-state version of Eq. (1) with time-invariant
boundary conditions. In the groundwater flow model, three types of
boundary conditions are generally encountered. On the first type
(Dirichlet) boundary, the pressure head is prescribed:
on B,
(6)
where B] is a portion of the boundary B and h-| is a given input
function of time and spatial coordinate on B-). On the second type
(Neumann) boundary, the flux is prescribed as:
on B,
(7)
where n- is the directional cosine of the outward unit vector normal
-j
B
to the B£ portion of the boundary B. The third type is the variable
boundary in the sense that either the Dirichlet or the Neumann
conditions may prevail :
h = h3(xi,t)
on B,
(8a)
or
= q
on
(8b)
-------
168
ORNL-DWG 78-20110R
t
B
B
B= B, U B2 U
B-
Fig. 1. Spatial boundaries, BI , 83, 83, and B-|, of flow
region R. (B.C. = Boundary Conditions).
-------
169
where 113 and Q3 are two known input functions of time and spatial
coordinate x-j on the 83 portion of B. The boundaries, B-j, Bj>,
83, and the impervious boundary, Bj, constitute the entire
boundary, B(XT, X2, ^3) = 0. Initially Eq. (8a) is applied to
the boundary 83 when the exact boundary conditions cannot in general
be predicted a priori. Such a case would arise at the ground surface
where either ponding (Dirichlet) or infiltration (Neumann) conditions
could prevail [6]. This can only be determined in the cyclic process
of solving Eqs. (1) and (?).
Applying the Galerkin finite-element procedure to Eq. (1), one
obtains the following matrix equation:
(9)
where the temporal derivative of the head, hj is given by
dh.
The matrix equation coefficients are defined as:
= /„ N.FN. dR , (ID
» 1 J
D4 = /_ [K -O N ) - QN ] dR, and (13)
i R mJ mi i
Q± = / N±q2 dB + / N±q3 dB , (1A)
B2 B3
where Nj is the basis function at the nodal point i.
Eq. (9) can be solved by any of the six alternative numerical
schemes listed in Table 1 [8]. They are dependent on the method of
time marching and the treatment of the mass matrix, L?1jJ- For
example; schlne 1 uses the Central difference time marching with no
mass lumping for the matrix,
-------
170
After the pressure field, h, is obtained, the Darcian velocity can
be obtained by Darcy's law, Eq. (2). Conventionally, it is evaluated
numerically by taking the derivative of the approximate pressure field,
h = hN, as follows:
V1 = -K1j [3j(hkNk' + V (15)
This approach naturally yields the discontinuity in the velocity, V-j,
at nodal points and element boundaries. If continuity is to be
achieved, the nodal gradient must also be treated as an unknown [6].
This in turn will increase, by a factor of 3 in the two-dimensional
problems or 4 in the three-dimensional problems, the number of equa-
tions and the bandwidth of the system and thereby increase drastically
the computing time and computer storage requirements. To circumvent
this problem, it is proposed that the finite-element method be applied
to Eq. (2) to yield:
[Sijl{Vnj} = {Dni} n - 1, 2, or 3 , (16)
where
S' - / N N. dR (17a)
ij R 1 J
and
D^,- = - / N-[K mOm(N.h.) + 6mo> ] dR (17b)
Eqs. (9) and (16) can be solved simultaneously by iteration. This will
not complicate the problem, since Eq. (9) has to be solved iteratively
because of its nonlinearity for each time step. It will not require
any additional storage either. However, it will increase the CPU time
by a factor of less than 3 or 4 for two- and three-dimensional problems
respectively. If the storage for the matrix [sfj ] is available, the
CPU time wijl remain practically the same as the conventional approach,
because [Sjj] needs to be evaluated and triangular!zed only once.
-------
171
The mass balance over the whole region of interest is obtained by
integrating Eq. (1):
i
/ F ^ dR = / F dB , (18)
R 8t B n
where Fn is the normal flux through the global boundary B(x,z) ~ 0.
In fact, Fn denotes:
F - n. K.. 3.H . (19)
n i ij j
Having obtained the pressure head field, h, one could integrate the
rightand left-hand sides of Eq. (18) independently. If the solution
for h is free of error, one would expect the equality of the two inte-
grals. In this paper, the integral of the right-hand side is broken
into five components:
dB . (20
F = / F dB , (2D
N J n
f F dB , (22)
B3S n
F - / F dB , and
K. _ n
B3R
Fn dB ,
B
(23)
-------
172
where Fp, FN, FS, FR, and Fj represent the fluxes through the
Dirichlet boundary, B]; the Neumann boundary, 82; the seepage
boundary, 635; the rainfall-infiltration boundary, 83^; and the
impervious Neumann boundary, Bj; respectively. On the other hand,
the integral on the left-hand side of Eq. 18,
dR ,
(25)
represents the volumetric increasing rate of the moisture content in
the region. For exact solution, the net flux across the whole
boundary, defined by
Fnet = FD + FN + FS + FR + FI
(26)
should satisfy the following equation:
F = F
net V *
(27)
In addition, FI should theoretically be equal to zero. However, in
any practical numerical simulation, Eq. (27) will not be satisfied and
FI will be nonzero. Nevertheless, the mass balance computation
should provide a means to check the numerical scheme and consistency in
the computer code.
RESULTS
Two sample problems are used to compare the results from the
models represented by Eqs. (15) and (16), respectively. The first
example is a hypothetical seepage pond problem described earlier [9].
The second one is the Freeze's transient problem reported elsewhere
[4,10]. In addition, results by all six alternative numerical schemes
(Table 1) are compared in both examples.
Seepage Pond Problems
A seepage pond is assumed to be situated entirely in the unsatu-
rated zone above a water table (Fig. 2a). This pond provides a source
-------
173
Table 1. Listing of Alternative Numerical Schemes
Numerical
schemes
1
2
3
4
5
6
Time -marching
Central Backward Mid-
difference difference difference
X
X
X
X
X
X
Mass
Without
lumping
X
X
X
matrix
With
lumping
X
X
X
-------
ORNL-DWG 79-15951R2
NEUMANN
NODE NO. 179
ELEMENT NO. 153
ELEMENT NO.
30m
ELEMENT NO. 160
ELEMENT NO. 159
ELEMENT NO. 522
NODE NO. 587
ELEMENT NO. 528
ELEMENT NO. 521
j
P»
180m
ELEMENT NO. 1
NODE NO. 2
ELEMENT NO. 2
Fig. 2. Spatial discretization of the seepage pond problem,
-------
175
of water which infiltrates into subsurface aquifers. After water
reaches the water table, it flows toward a stream. Is is further
assumed that the system is composed of highly permeable sand with soil
properties shown in Fig. 3. For the finite-element computation, the
entire region is discretized by 595 nodal points and 528 elements
(Fig. 2b). Seven nodal points on the stream-soil interface are desig-
nated as Dirichlet nodes. Seven nodal points on the bottom of the
seepage pond are considered as constant Neumann flux points and are
assigned a constant infiltration rate of 4.0 x 10~4 cm/s. The top
sides of all elements on the sloping surface, except the two elements
immediately to the right of the seepage pond, are considered variable
boundary surface, i.e., seepage-rainfall boundary. In other words, the
nodal points on this surface are either Dirichlet or Neumann points
with the infiltration rate equal to the excess rainfall rate.
The Darcian velocity field computed by model Eq. (15) shows the
discontinuity at every nodal point as indicated by multivectors at any
point in Fig. 4. The severity of this discontinuity depends on the
location. This discontinuity is completely eliminated with model
Eq. (16) as can be seen from Fig. 5, which shows the unique velocity
vector at all nodal points. Table 2 shows the comparison of the
computed Darcian velocity components simulated by model Eq. (15) and
(16), respectively, for the three selective nodal points 2, 179 and 587
(Fig. 2). These three sample points are taken from computer output to
illustrate the difference between the two models. It is seen that at
nodal point no. 2, the vertical velocity component as computed from
element no. 2 is about 2.58 times that computed from element no. 1.
The values of the horizontal component, Vx, at nodal point no. 179 as
computed from element nos. 159 and 160 are about 1.41 times those
computed from element nos. 152 and 153; while the values of the verti-
cal component, Yz, at the same point as computed from element nos.
153 and 160 are about 4.69 times those computed from element nos. 152
and 159. At nodal point no. 587, the vertical velocity component,
Vz, as computed from element no. 522 is about 1.27 times that
computed from element nos. 521 and 528. On the other hand, results
from model Eq. (16) show that the values of velocity components are
identical at the same point as is expected. However, the CPU times on
IBM 360/91 by model Eq. (15) and (Ifi) are 0.92 and 2.51 minutes, re-
spectively. A saving of computer storage of about 150 K is achieved by
model Eq. (16) though. Since the steady-state solution is sought, nu-
merical scheme nos. 1 through 6 (Table 1) yield identical results as
expected.
Freeze Transient Problem
A very small (6 x 3 m) laboratory-sized watershed was monitored by
Freeze [10] to test his finite difference computer code. The same data
were also used by Reeves and Duguid [4] to debug and test their
finite-element model. These watershed data are again used in the
present paper to compare model Eqs. (15) with (16).
-------
176
ORNL-DWG 74-3908
-400 -300 -200 -100 0
A, PRESSURE HEAD (cm of water)
100
Fig. 3. Hydraulic conductivity and soil moisture characteristics
of a hypothetical sandy soil.
-------
ORNL-DWG 79-13061R
(a] VELOCITY VECTOR PLOT
Fig. 4. Flow velocity plot as simulated by model Eq,
seepage pond problem.
(15) for the
-------
ORNL-DWG 79-13060R
(a) VELOCITY VECTOR PLOT
oo
Fig. 5. Flow velocity plot as simulated by Model Eq. (16) for the
seepage pond problem.
-------
179
Table 2. Comparison of Velocity (cm/sec) Components Simulated by
Model Eqs. (15) and (16), Respectively, at Three Selected Points
Node Element
number number
1
2
2
152
153
179
159
160
521
587 522
528
Model Eq.
Vx
2.38E-8
2.33E-8
2.26E-5
2.26E-5
3.31E-5
3.31E-5
6.28E-5
6.28E-5
7.89E-10
(15)
vz
-2.253E-8
-6.54E-8
-9.15E-5
-4.31E-4
-9.15E-5
-4.31E-4
1.85E-4
2.36E-4
1 .85E-4
Model Eq.
Vx
1.24E-8
1.24E-8
3.03E-5
3.03E-5
3.03E-5
3.03E-5
1.23E-5
1.23E-5
1.23E-5
(16)
vz
-4.44E-8
-4.44E-8
-2.94E-4
-2.94E-4
-2.94E-4
-2.94E-4
1 .84E-4
1.84E-4
1 .84E-4
-------
180
The flow system is shown in Fig. 6. It is composed of highly per-
meable sand, the unsaturated properties of which are shown in Fig. 3.
To obtain initial conditions, pressure-head values were pre- scribed
along the stream channel, part of the slope, and the upper plateau.
Taking all other boundaries to be impermeable, a steady-state solution
was determined which was the initial condition for the transient
calculation.
Using Freeze's transient boundary condition (Figure 6b) and Reeves
and Duguid's finite-element discretization #2 (Figure 6c), results ob-
tained by models Eqs. (15) and (16), respectively, are shown in Figs. 7
and 8. Model Eq. (15) again displays the discontinuity of velocity
vectors at all nodal points, while model Eq. (16) has completely elimi-
nated this inconsistency. Furthermore, Table 3 shows that the mass
balance has not been satisfied by model Eq. (15). At the end of about
3-h simulation time, the total net mass through all boundaries is only
about 76.2% of the mass accumulated in the media as computed by numeri-
cal scheme no. 1 of model Eq. (15). In other words, 23.8% of the mass
has not been accounted for, i.e., has been lost through boundaries.
Reeves and Duguid [4] speculated that this large loss of mass might be
eliminated by adding triangular elements. However, without using tri-
angular elements, model Eq. (16) only yields a 2.?% mass loss by elimi-
nating the discontinuity of the velocity and using Eq. (25) to evaluate
the moisture rate change. An even larger mass loss of 29.7% is
obtained by numerical scheme no. 2 of model Eq. (15). Model Eq. (16),
on the other hand, renders a 3.6% mass gain. Thus, error of mass
balance (positive for loss, negative for gain) by model Eq. (16) is
much smaller than that by model Eq. (15). The CUP time on IBM 360/91
for models Eq. (15) and Eq. (16) are 2.3 and 4.F minutes, respectively,
but a saving of computer storage of about 140 K is obtained by the
latter.
Table 3 also shows the percentage of mass change by all alterna-
tive numerical schemes. It is noted that the central difference
standard Gale rid n scheme in model Eq. (16) yields the best results.
This is not surprising since the water transport equation does not
contain advection terms.
CONCLUSION
A continuous velocity field is required for the simulation of
waste transport in the subsurface aquifer system. The conventional
approach used in finite-element modeling, that is, taking the deri-
vative of the approximately computed pressure head field, results in
discontinuity of velocity at element boundaries and nodal points. It
may also yield large error in overall mass balance computation. Thus,
it is imperative that the same finite-element method that is employed
to simulate the pressure field be applied to Darcy's law to obtain the
velocity field. This will ensure a Continuous velocity field and
greatly reduce the overall mass balance error. The central difference
Galerkin scheme yields the lowest error, since the governing equation
is basically the Laplacian type in spatial coordinates.
-------
181
ORNL-DWG 79-13083R
• Comparison with FD Solution
(a) STEADY STATE
BOUNDARY CONDITIONS
IMPERMEABLE
IMPERMEABLE
(b) TRANSIENT BOUNDARY
CONDITIONS
RAINFALL
IMPERMEABLE
(c) SPATIAL MESH DISCRETIZATION
T
70cm
10
13C
)cm
) cm
UlOO cm-
X
*•
f
/ /
*£''''
'
' / /[ /
s / /
*.
s "
----
. - • - -
—100 cm—
Fig. 6. Configuration of Freeze's experimental watershed:
(a) steady state boundary condition, (b) transient boundary condition,
(c) spatial finite element discretization after Reeves and Duguid (197F),
-------
ORNL-DWG 79-13082R
(a) VELOCITY VECTOR PLOT
Fig. 7. Flow velocity plot at time equal to 2.96 h of Freeze's
transient problem as simulated by Model Eq. (15).
00
N>
-------
ORNL-DWG 79-43067R
(a) VELOCITY VECTOR PLOT
Fig. 8. Flow velocity plot at time equal to 2.96 \\ of Freeze's
transient problem as simulated by Model Eq. (16).
00
CO
-------
184
Table 3. Comparison of Percentage of Mass Change of Freeze's
Transient Problem as Simulated by Model Eq. (15) and (16)
Scheme
Model
Eq.
Eq.
(15)
(16)
1 2 3
23.8 29.7 N/A
2.2 -3.6 8.9
4
N/A
3.0
5
N/A
-3.2
6
N/A
-3.3
-------
185
REFERENCES
1. S. P. Neumann and P. A. Witherspoon, "Analysis of nonsteady flow
with a free surface using the finite element method," Water
Resour. Res., 7(3). 611-623 (1971). -
2. G. F. Pinder and E. 0. Frind, "Application of Galerkin procedure
to aquifer analysis, Water Resour. Res.. £(1), 108-120 (1972).
3. G. F. Pinder, "A galerkin finite element simulation of groundwater
contamination on Long Island, New York," Water Resour. Res., 9(6),
1657-1669 (1973). - — ~
4. M. Reeves and J. 0. Duguid, Water Movement Through
Saturated -Unsatu rated Porous Media: A Finite Element Galerkin
Model, ORNL-4927. Oak Ridge National Laboratory, Oak Ridge,
Tennessee (1975).
5. S. K. Gupta, K. K. Tanji, and J. N. Luthin, A Three-dimensional
Finite Element Groundwater Model, University of California, Davis,
Water Resources Center Contribution Series #1F2 (1975).
6. G. Segol , A Three-dimensional Galerkin Finite-Element Model for
the Analysis of Contaminant Transport in Variably
Saturated -Unsatu rated Porous Media, Dept. of Earth Sciences,
University of Waterloo, Canada (1976).
7. Y. H. Huang and S. J. Wu, "Comparison of Three-dimensional Finite
Elements for Aquifer Simulation, Finite Element in Water
Resources," Proceedings of the First International Conference on
Finite Elements in Water Resources, edited by W. IT. Gray,
G. F. Pinder and (H A~T Brebbia, Pentech Press, London,
pp. 2.87-2.105 (1977).
8. G. T. Yeh and D. S. Ward, FEMWATER: A Finite Element Model of
Water Flow through Saturated-Unsaturated Porous Media, ORNL-5567,
Oak Ridge National Laboratory, Oak Ridge, Tennessee (1980).
9. J. Duguid and M. Reeves, Material Transport in Porous Media: A
Finite Element Galerkin Model, ORNL-4928, Oak Ridge National
Laboratory, Oak Ridge, Tennessee (1976).
10. R. A. Freeze, A Physics-based Approach to Hydro!ogic Response
Modeling; PI
Contract NoT
D.C. (197217
Modeling; Phase"I - Model Development, Completion Report for OWRR
Contract No. 14-31-001-3694, Pept. of the Interior, Washington,
-------
FRACTURE FLOW MODELING FOR LOW-LEVEL NUCLEAR WASTE DISPOSAL
Brian Y. Kanehiro Varuthamadhina Ruvanasen
Charles R. Wilson Paul A. Witherspoon
Lawrence Berkeley Laboratory
University of California
Berkeley, California 94720
ABSTRACT
Fracture flow modeling as applied to low level radio-
active waste disposal sites is discussed. Low level waste
disposal sites presently exist in fractured basalts, crys-
talline igneous rocks, and limestones. The primary objec-
tive of forecasting the migration of radionuclides from
these sites requires the consideration of fracture flow
modeling. Such modeling is also an integral part of the
planning and analysis of field work at the site.
Fundamental mathematical models describing discrete
fracture flow and equivalent anisotropic porous medium
flow are reviewed. Consideration is given to the theoret-
ical and field aspects of the problem of when and how to
treat a fractured medium as an equivalent anisotropic
porous medium. The input parameters required for each
type of model are discussed.
Finally a brief description of some of the models
developed at the Lawrence Berkeley Laboratory is given.
The models include those based on the integrated finite
difference, polygonal finite difference, and finite-
element schemes.
INTRODUCTION
Fracture flow modeling is an attempt to consider the effects of
discrete fractures in approximating the nature of fluid movement in a
fractured medium. Fracture flow predominates whenever the fluid flux
through the fractures is appreciably greater than that through the
unfractured, porous matrix. This condition implies that the hydraulic
conductivity of the fractures, whether completely open or partially
filled, is substantially greater than that of the matrix. Fracture
flow strongly influences groundwater particle flow velocities, flowpath
directions and the mechanisms of sorption-desorption of migrating radio-
nuclides, and thus is of considerable importance to the safe disposal
of low-level nuclear waste within media where an interconnected network
of open fractures exists. This paper discusses the fundamental prin-
ciples of fracture flow and the circumstances under which fracture flow
187
-------
188
must be considered in modeling low-level waste disposal sites. Brief
descriptions of the various types of fracture flow codes available at
LBL are given.
The most important fractured media in terms of existing low-level
waste disposal sites are basalt, crystalline igneous rocks, and lime-
stone. This is not to suggest that these rock types are the best for
disposal or the most commonly occurring types of fractured media, but
rather that many low-level waste sites have been constructed in them.
Indeed, basalt and limestone are generally poor choices for containment
and problems requiring remedial efforts are quite likely to occur at
such sites unless they are properly designed for the existing fracture
system. Matrix flow in each of these rock types is likely to be rela-
tively small in comparison to the flow in the fractures.
The most obvious reason for the utilization of a model in studying
a low-level waste site is to forecast the future migration of radionu-
clides from the site. There are, however, other reasons for employing
a model that are just as important to the full assessment of the site.
Because of the geologic and hydrologic complexities of most sites, the
input parameters for forecasting the future of a site are generally
difficult and costly to obtain. The availability of a model during the
field phase of site assessment, when data for input parameters are being
collected, can insure that the most useful information is collected in
the most efficient manner. In particular, parameter sensitivity studies
can be used to plan field work and provide assurance that characteriza-
tion of some aspects is not being overemphasized to the neglect of others.
Further, the availability of a model permits analysis of field test data,
such as complex tracer tests, to an extent that would not be practical if
only more conventional analytical techniques were available.
FUNDAMENTAL MATHEMATICAL MODELS
Flow through a fractured medium is generally mathematically described
in one of three ways: (1) by considering each fracture as a discrete
hydraulic conduit; (2) by assuming a hydraulically equivalent anisotropic
porous medium and using the appropriate porous medium model; or (3) by a
combination of (1) and (2).
Flow through a single fracture is normally represented by a parallel
plate analogy [1]. If the flow is assumed to be laminar, the fluid single-
phase and Newtonian, and the parallel plates smooth, the steady flux from
a single fracture may be described by:
where q = flux per unit length of fracture (L/T)
b = fracture half-aperture (L)
p = fluid density (M/L3)
g = acceleration due to gravity (L/T2)
-------
189
y = dynamic viscosity of the fluid (M/LT)
h = piezometric head (L]
I = length along the fracture plane (L]
Utilizing the parallel plate analogy, it can be seen that the factor
(2b)2/12 in eq. (1) directly corresponds to the intrinsic permeability of
a porous medium (k^). The extent of this correspondence is demonstrated
in the following parallel development of the fundamental equations govern-
ing fluid flow and nonreactive solute transport in both porous media and
fractures.
The equations governing flow in an anisotropic porous medium can be
derived from continuity and Darcy's law and may be tensorially written as:
3 i/ oh , n _ c- on
^r i\ • • ^r I \j — j ^ .
^v T t jv " C *JT
8x. ij 3x.j s 3t
where K.. = hydraulic conductivity tensor
k-Hpg
= -^— (L/T)
Q = flow rate of source or sink per unit volume of rock (1/T)
S$ = specific storage (1/L)
t = time (T)
k.. = intrinsic permeability tensor (L2)
Along a fracture plane, consideration of continuity and eq. (1)
requires that:
u
3 0 3h
X/
- Ss %
where ?. = rectangular coordinates along the fracture plane
K = fracture hydraulic conductance
, o/n
g = fracture specific storage coefficient (1/L)
q. = flow rate per unit volume of fracture across the
1 fracture walls (1/T)
m. = component of the unit vector outward normal to the
1 fracture plane
The second term in the above equation refers to the difference
between influx and efflux at the upper and lower fracture walls.
The equation governing the transport of a nonreactive solute in
porous media is tensorially written [2]:
(4)
-------
190
where C = solute concentration (M/L3)
u. = seepage velocity in the ith direction (L/T)
Dii = hydrodynamic dispersion tensor (L2/T)
S'n = so^ute concentration at sources or sinks (M/L3)
The velocities and coefficients of hydrodynamic dispersion are
calculated from
e ax.
J
where a^i = convective dispersivity tensor (L)
]u| = seepage velocity magnitude (L/T)
D. = molecular diffusiyity coefficient (L2/T)
T.. = tensor of tortuosity
6^. = Kronecker delta
9 = effective porosity
Along a fracture plane the nonreactive solute transport equation can
be written ,
3u.C
" = 2b -^ D^. ^- (7)
I i .1
in which the third term refers to solute influx and efflux across the
fracture walls. At present there exists no experimental data for DV..
the hydrodynamic dispersion coefficient tensor. For approximation J
purposes, however, D^j can be considered to vary from D^ii for a clean
fracture to D^j for a fully-filled fracture. Transport of reactive
constituents can be treated by the source/sink terms in eqs. (4) or (7).
THE EQUIVALENT POROUS MEDIUM MODEL
A model based on only discrete fractures provides a theoretically
correct flow pattern if all fractures and connections between fractures
are considered. But, while such a model may be relatively easy to con-
struct from a numerical point of view, the mass of input data required,
the computational effort, and the difficulty of obtaining data for many
real-world problems makes it generally difficult, if not impossible to
apply to regional analysis.
One way to circumvent this problem is to represent the fracture
system by an equivalent porous medium. An equivalent porous medium is a
fictitious porous medium with material properties so selected that, under
specific hydrologic and geometric conditions, the hydraulic behavior of
the porous medium is in some respects the same as the hydraulic behavior
-------
191
of the fractured rock. The validity of the equivalent porous medium concept
is thus very site-specific and flowpath-specific. If it is assumed that
for the purposes of the problem at hand, the domain being considered can
indeed be represented by an anisotropic porous medium, then the major
problem becomes that of determining the permeability tensor. The mathe-
matics of calculating directional permeabilities from fracture orientation
and aperture data based on the assumption that permeability is a function
of aperture squared were first discussed by Romm and Pozinenko [3]. Exten-
sive work in this area has been done by several others [1, 4, 5].
Under a general field gradient, flow in a single conduit may be des-
d as [1]:
cribed as
- V
where q- = flow rate per unit width of the fracture plane
in the jth direction (L2/T)
6.. = Kronecker delta
M.J. = 3x3 matrix formed by direction cosines of the
1J unit normal to the fracture plane
x. = Cartesian coordinates (L)
j
It is further assumed that for a large number of fractures intersect
ing a sample line, each fracture has its image at a distance L equal to
the length of the sample line and in the direction of the sample line.
Thus, the spacing between the fracture and its repetition is given by [6]:
w=L|niD.| (9)
where w = spacing between fractures (L)
L = length of sample line (L)
n- = component of unit normal to fracture plane
D! = component of sampling line direction
The effective intrinsic porous medium permeability of a fracture is
then given by:
The equivalent permeability of the medium at a given sampling station
is obtained by summing the contributions from the individual fractures.
Where more than one station is used, the average of all stations is calcu-
lated. Knowing the permeability tensor, its principal components and dir-
ections may then be calculated.
This approach assumes that each fracture completely penetrates the
volume being characterized, without changes in aperture or orientation.
-------
192
Since in nature many fractures are observed to terminate over relatively
short distances, this model would generally be expected to overestimate
rock mass permeability.
If a similar procedure is followed to determine Dji based on field
data, then the anisotropic porous medium model describes by eqs. (2) and
(4) may be used directly.
In some cases where a relatively small number of large fractures
exist together with a large number of small fractures with different
orientation, it may be useful to employ both the discrete fracture and
porous medium models. The equivalent tensorial properties would be
applied to that portion of the medium dominated by the smaller fractures
and the problem would be treated as a fractured anisotropic porous medium.
Eqs. (2), (3), (4), and (7), and consideration of continuity would govern
the system.
MODELING OF A FIELD SITE
Selection of a model for a particular field application requires two
primary considerations. First, the type of output required of the model
must be clearly defined, and second, the type of input available for the
model from the field site must be identified. Put in another way, the
data needs for decision-making that are to be supplied to the model must
be compatible with the capabilities of the model, which in turn must be
compatible with the physical and economic realities of obtaining the
necessary data from the field and operating the model. Clearly, the out-
put needs, the choice of model, and the input needs are all interdependent.
The types of output parameters required of models for safe disposal
of nuclear waste will involve dose rates and contaminant concentrations
under environmental conditions to be prescribed by the Environmental
Protection Agency. Our purview as earth scientists will be to recommend
the most appropriate approaches to obtaining the required information.
This means balancing the applicability of models against the availability
of input parameters to best achieve the output requirements.
Groundwater movement and nuclide transport in fractured media can be
modeled with either a discrete fracture model or an equivalent porous
medium model. The types of input parameters that are required for these
two models are not the same. The fracture model requires fracture geometry
data such as fracture orientation, spacing, and aperture while the porous
medium model requires the large-scale averaged parameters of permeability
and dispersion.
The question of which model to use for a given field site is, in
general, not a simple one. The limitations of each model must be evaluated,
and its applicability with regard to the model output requirements must be
determined. From a theoretical point of view, the choice of models is
generally a question of scale. If the representative elementary volume [2]
concept is valid for the site, and if both the size of the site and the
-------
193
scale of required output parameters are larger than the representative
elementary volume, then the anisotropic porous medium equivalent can be
employed. The other extreme of a case where there are a rather small num-
ber of identifiable distinct fractures within the entire site, calls for a
discrete-fracture model. Both types of models suffer from difficulties in
acquisition of input data. Parameters suitable for equivalent porous
medium models must be obtained from field tests designed to average the
individual effects of many fractures over volumes larger than the repre-
sentative elementary volume. On the other hand, parameters suitable for
discrete-fracture models must accurately identify the specific geometry
of the principal flowpaths. Unfortunately, many near-surface low-level
waste field sites will probably not clearly fit into either category and
a hybrid model may be required.
In principle, the most straightforward method for determining the
size of the representative elementary volume is to perform hydro!ogic
tests in different portions of the site starting with small-scale tests
(i.e., closely-spaced wells) and expanding the scale of each of the tests
in increments. As the tensors of permeability and dispersion, computed
from tests of progressively increasing scale, approach some type of con-
sensus in magnitude and orientation at a particular test site, it could
be assumed that the scale of the individual test at that particular loca-
tion is approaching that of the representative elementary volume at that
location. Further, if the results of the individual tests at different
locations are seen to converge, then it could be concluded that the site
as a whole can be described as a single anisotropic porous medium with
similar properties throughout.
s
While this procedure appears to be straightforward, it may be imprac-
tical for many real-world problems. First, the procedure might be costly
and time-consuming. Second, most field sites are likely to be heterogen-
eous, implying that this procedure would have to be performed for each
fractured geologic formation at the site. Finally, determination of ten-
sorial properties in the field, particularly for a three-dimensional case,
would be quite difficult. Despite these difficulties, the alternative of
measuring the geometry of all significant discrete fractures is likely to
be impossible over any reasonably-sized field site, leaving us with little
choice but to employ an equivalent porous medium model or a hybrid model
at many sites.
An alternative to the direct large-scale field measurement approach
described above is to compute equivalent porous medium properties based
on many individual measurements of single fractures, usinn a procedure
similar to that proposed by Snow [1] and discussed above. An extensive
field evaluation of this approach is now being conducted by LBL at.the
Stripa test station in Sweden, and the results will be compared with
those of a large-scale test involving 105 to 106 m3 of rock [7].
For the modeling of a real field site, it may be best to not decide
on the type of mathematical model to be used until substantial field data
are available. Information on discrete fractures will be required in the
design of field tests and in evaluating the applicability of the equivalent
-------
194
porous medium concept, thus the field work will of recessity provide input
parameters for both equivalent porous medium and discrete fracture models.
Such an approach will provide more flexibility and will ultimately result
in a better understanding of the site. After, and to a certain degree
during the field-work phase of the study, both types of models, together
with a hybrid having both discrete fractures and equivalent porous media,
can be tried and the best model identified.
^From a practical point of view the need for fracture data will
require the addition of a limited number of activities to those required
for the standard porous medium type characterization and assessment of a
low-level site. In particular, fracture mapping and fracture core logging
will have to be added to the geologic activities. Hydraulic and tracer
testing will have to be performed in a somewhat more complicated fashion
so as to test individual fractures. Finally, preparation of two or three
types of mathematical models will be required.
MODELS AT LAWRENCE BERKELEY LABORATORY
%
The Lawrence Berkeley Laboratory is presently using three types of
numerical models that have application to the modeling of low-level waste
deposit sites. The first group of models is based on the integrated
finite difference scheme. Examples of these programs are TERZAGHI [8],
CCC [9], and SHAFT79 [10]. Of these, TERZAGHI may be the most useful for
low-level site modeling. It solves a nontensorial form of eq. (2) in
three dimensions. It has been used to model discrete fractures and some
types of anisotropic porous media. It also has the capability to deal
with one-dimensional consolidation.
The second group of models is based on the polygonal finite differ-
ence scheme. Examples are BIFDA and GENIE [11]. These models solve a
three-dimensional form of eqs. (2) and (4). They can be used to deal with
both discrete fractures and anisotropic porous media. In addition, they
can handle heat transport.
The third group of models is based on the finite-element scheme.
Examples of these models are ROCMAS [12]. FRACFLOW [13], and LINE ELEMENT
[14]. These programs solve various forms of eqs. (2), (3), (4), and/or
(7), for two-dimensional cases. The first two can be used for either
discrete fractures or porous media. In addition, the first and second
can deal with the effects of stress and the second can also deal with
heat and anisotropic problems. The third program is a steady state line
element code presently being used in research on the definition of the
permeability tensor in random systems of discrete fractures.
The first group of models evolved from a coupled fluid-heat flow
model. The second group was developed to provide good chemical transport
and anisotropic media capabilities while maintaining three-dimensional
capabilities. The third group was developed to deal specifically with
discrete fractures with stress-dependent properties.
-------
195
ACKNOWLEDGEMENTS
Work performed under the auspices of the U. S. Department of Energy
through Lawrence Berkeley Laboratory, University of California, under
contract W-7405-ENG-48.
REFERENCES
1. D. T. Snow, "A Parallel Plate Model of Fractured Permeable Media,"
Ph.D. Thesis, University of California, Berkeley, California (1965).
2. J. Bear, Dynamics of Fluids in Porous Media. Elsevier, New York (1972),
3. E. S. Romm and B. V. Pozinenko, "Investigation of Seepage in Frac-
tured Rocks," Trudy VNIGRI, 214 pp (in Russian), (1963).
4. C. Louis and M. Parnot, "Three-Dimensional Investigation of Flow
Conditions and Grand Mai son Damsite," Proc. Sym. Percolation Through
Fissured Rocks, Stuttgart, Paper T4-F, (1972).
5. M. A. Caldwell, "The Theoretical Determination of the Permeability
Tensor for Jointed Rocks," Proc. Sym. Percolation Through Fissured
Rocks. Stuttgart, Paper Tl-C, (1972).
6. L. Bianchi and D. T. Snow, "Permeability of Crystalline Rock Inter-
preted from Measured Orientations and Apertures of Fractures,"
Annals of Arid Zones. 8^(2), (1969).
7. J. Gale and P. A. Witherspoon, "An Approach to the Fracture Hydro-
logy at Stripa—Preliminary Results," Sem. on In-Situ Heating
Experiments in Geologic Formations. Org. Econ. Coop. Dev. Ludvika,
Sweden. Also Lawrence Berkeley Laboratory report LBL-7079, SAC-15,
Berkeley, California (1978).
8. T. N. Narasimhan and P. A. Witherspoon, "Numerical Model for Saturated-
Unsaturated Flow in Deformable Porous Media. 1. Theory," Water
Resources Research, 13_(3), 657-664, (June 1977).
9. M. J. Lippmann, C. F. Tsang, and P. A. Witherspoon, "Analysis of the
Response of Geothermal Reservoirs under Injection and Production
Procedures," SPE 6537, presented at the 47th Annual California
Regional Meeting SPE-AIME, Bakersfield, California, (April 1977).
10. K. Pruess, R. C. Schroeder, P. A. Witherspoon, and J. M. Zerzan,
"SHAFT78, a Two-Phase Multidimensional Computer Program for Geo-
thermal Reservoir Simulation," Lawrence Berkeley Laboratory report
LBL-8264, Berkeley, California 94720 (1979).
-------
196
11. B. Y. Kanehiro, "A Three-Dimensional Polygonal Solution of Fluid Flow
and Chemical Transport in Anisotropic Porous Media," (in preparation).
12. J. Noorishad, M. S. Ayatollahi, and P. A. Witherspoon, "A Finite-
Element Method for Stress and Stress and Fluid-Flow Analysis in
Fractured Rock Masses," submitted to ASCE Journal of Engineering
Mechanics Division, (1980).
13. V. Guvanasen, "Finite-Element Solution of Fluid Flow, Heat and Solute
Transport, and Thennomechanical Deformation in Fractured Porous Media,"
(in preparation).
14. C. R. Wilson and P. A. Witherspoon, "An Investigation of Laminar Flow
in Fractured Porous Rock," Dept. of Civil Engineering, Pub. #70-6,
University of California, Berkeley (1970).
-------
A CONCEPTUAL ANALYSIS OF THE GROUNDWATER FLOW SYSTEM
AT THE MAXEY FLATS RADIOACTIVE WASTE
BURIAL SITE, FLEMING COUNTY, KENTUCKY
D. W. Pollock
H. H. Zehner
U.S. Geological Survey
Reston, Virginia
Louisville, Kentucky
ABSTRACT
The complex groundwater system at the Maxey Flats
radioactive waste burial site in Fleming County, Kentucky
was investigated with the aid of a highly simplified, steady
state, groundwater flow model. An analysis of the local
hydrogeology suggests that at least two saturated groundwater
flow systems are present at the site: a lower system in the
bottom part of the Ohio Shale, and an upper, perched system
extending from the base of the Bedford Shale up to a water
table in the weathered part of the Nancy Member of the Bordon
Formation. Unsaturated conditions may also exist in the
Sunbury Shale. Simulations indicate that most of the water
which recharges the upland surface of Maxey Flats is even-
tually discharged as seepage along the hillside through the
upper part of the Farmers Member of the Bordon Formation and
the Sunbury Shale.
INTRODUCTION
This report discusses the groundwater flow system at the Maxey Flats
radioactive waste burial site in Fleming County, Kentucky. The ground-
water system at Maxey Flats is complex; flow occurs mainly through
fractures in hydraulically "tight" shales and sandstones. In addition,
water levels from wells at the site indicate the possibility of unsaturated
conditions at depth. At the present time the available data do not
warrant a detailed model of groundwater flow; however, careful examination
of the hydrogeology and the hydro!ogic setting of the site can help
provide a conceptual understanding of the groundwater system.
Maxey Flats is topographically an isolated knob located in the
Appalachian plateau region of eastern Kentucky. The Maxey site is
located on one part of this knob and is bounded on three sides by streams
(fig. 1). Relief between the upland surface and bordering valley bottoms
is about 300 feet.
Maxey Flats is underlain by fractured shales and sandstones which
dip gently (25 to 40 feet per mile) toward the southeast. The rock units
197
-------
198
38° 16'
Topographic Divide
83° 34'
Trench Area
38° 15'
\ Jock Lick Creek
Fig. 1
Kentucky.
SCALE
CONTOUR INTERVAL = 100 FEET
Generalized topographic map of Maxey Flats, Fleming County,
-------
199
directly underlying the Maxey Flats site are the Nancy and Farmers Members
(predominantly a shale and sandstone respectively) of the Mississippian
Bordon Formation, the Henley Bed (shale) of the Farmers Member, the
Mississippian Sunbury Shale, the Mississippian and Devonian Bedford Shale,
and the Devonian Ohio Shale, plus the upper part of the Silurian Crab
Orchard Formation (shale). In this report, the Nancy and Farmers Members
will be referred to as the Nancy shale and Farmers sandstone, and the
Henley Bed will be referred to as the Henley shale.
The weathered and unweathered Nancy shale are separated by a thin
(1 to 2 foot thick) sandstone bed, which is present over most of the site.
This bed will be referred to as the sandstone marker bed. The upper part
of the Farmers sandstone is a sequence of alternating, thin (1 to 3 foot
thick) sandstone and shale beds, whereas the lower part of the unit is
predominantly sandstone. The two parts probably have different hydraulic
characteristics so, in this report, they are separated into the upper and
lower Farmers sandstone. Figure 2 is a generalized geologic section taken
along line A-B, shown in figure 1. Radioactive wastes were buried in
trenches excavated in the Nancy shale on the upland surface.
The distribution of wells at Maxey Flats is shown in figure 3. E-wells
were drilled by Emcon Associates. UA^-wells, northeast of the burial area,
and. UB-wells, in the trench area, were drilled by the U.S. Geological Survey
in 1976 and 1977. Well construction and water-level data are summarized
in figures 4 and 5.
ANALYSIS
Conceptual Model
Eleven hydrogeologic units are exposed at Maxey Flats. The sequence
of units, from top to bottom, and their thicknesses in feet, are:
1. Colluvium, which partially covers the hillsides (0 to 5)
2. Weathered Nancy shale, which covers the hilltops (1 to 25, with
an average of about 20)
3. Sandstone marker bed, which is usually at the base of the
weathered Nancy shale (1 to 2)
4. Unweathered Nancy shale (20 to 40, with an average of about 25)
5. Upper Farmers sandstone (15 to 20)
6. Lower Fanners sandstone (15 to 20)
7. Henley shale (7)
8. Sunbury shale (20)
-------
HI
u.
UJ
1100
1000
•ft
CO
UJ 900
O
ffl
<
g soo
1
Alluvium
Ohio Shale
Henley Shale
Sunbury Shale
Colluvium
UJ
700
Crab Orchard Shale
0
I
I
I
1000
I FEET
SCALE
Fig. 2. Geologic section along line A-B in figure 1
B
-------
201
SCALE
Fig. 3. Location of wells.
-------
1050
1000
950
u_
z
of
Q
6 8
3'/'
< 800
750
700 -
-Weathered
--Nancy Shale
Unweathered
Nancy Shale
Upper Farmers
Sandstone
"Lower Farmers
Sandstone
Henley Shale
Sunbury Shale
Bedford Shale
Ohio Shale
• Cemented Interval
Open Interval
Water Level
Crab Orchard Formation
Fig. 4. Well construction information and water level data for
wells drilled by Emcon Associates.
ro
o
t\j
-------
1050
1000
s
"- 950
Z
LJ
o
D
< 900
UA-1 UA-2 UA-3 UA-4
s
800
750
UB-1 UB-1AUB-2 UB-3 UB-4
Weathered Nancy Shale
Unweathered Nancy Shale
Upper Farmers Sandstone
Lower Farmers Sandstone
^Henley Shale
Sunbury Shale
Bedford Shale
1
^
1
•\
1
(_
1
V
I
I
L
h
i ^
E
s
'Ohio Shale
Cemented Interval
Open Interval
Water Level
Fig. 5. Well construction information and water level data for wells
drilled by U.S.G.S.
ro
o
UJ
-------
204
9. Bedford shale (20)
10. Ohio Shale (185)
11. Crab Orchard Formation, with only the upper few feet exposed
in the valley bottoms (approximately 120)
No quantitative data are available regarding hydraulic properties
of these units. Qualitative, visual observations of some properties
(including porosity, fracture density, and groundwater discharge) were
made at outcrops. The hydrogeologic units above the valley bottoms are
grouped into four categories, based on visual observations and water-level
recovery rates in wells finished in some of the units (Table 1). The
colluvium is not considered in this study. It undoubtedly plays a role
in controlling seepage along the hillsides; however, because there are no
wells on the hillsides, the groundwater regime in the colluvium is unknown.
The hydrogeology of Maxey Flats can be thought of as a series of extremely
poorly permeable rock units alternating with a few layers of somewhat higher
permeability. The upper part of the Crab Orchard Formation, which crops
out in the valley bottoms, is a clayey shale with very few fractures; in
this report, it is considered to be the "impermeable" base of the flow
system.
The only significant groundwater flow at Maxey Flats occurs through
fractures spaced from 1 to 40 feet apart. One way of modeling flow through
fractured rock is to treat the fractured rock mass as a porous medium.
That is, instead of distinguishing individual fracture-flow pathways, each
point in the rock mass is assigned an hydraulic conductivity which repre-
sents a measure of the "average" water-transmitting ability of the fractures
in the vicinity of that point. Fractured rocks appear to behave as porous
media on a scale where the distance between fractures is small compared
with the dimensions of the flow system being considered. Fractured rocks
are usually treated as porous media for regional flow studies, but for
some small-scale studies (such as dam site investigations) it is sometimes
necessary to consider flow through a network of discrete fractures (e.g.,
Wilson and Witherspoon, 1970). The major drawback with fracture-network
flow models is that they require detailed information about the spatial
distribution and hydraulic properties of individual fractures which is
rarely available. In the case of Maxey Flats, it is probably possible to
obtain an adequate representation of the head distribution in the fracture
network by assuming porous media flow; however, it may be very difficult
to measure the head distribution in the fracture network. In fractured
rocks that have extremely low intergranular permeability, the water level
in a well may represent the head in a single fracture at the point where
the fracture intersects the open interval. If more than one fracture
intersects the well, the water level will represent a composite of the
heads in the individual fractures at the points of intersection. Water
level data from fractured rocks is therefore often difficult to interpret
because the location of fracture intersections is usually not known.
Water levels from wells at Maxey Flats indicate a large decrease in
head with depth. This fact has led previous workers to suggest that
-------
205
Table 1. Qualitative classification of the hydrogeologic units
Medium
Hydraulic Conductivity
Low Very Low
Extremely Low
Colluvium
Weathered Nancy
shale, including
the sandstone
marker bed
Sunbury Shale
Ohio Shale
Unweathered Nancy
shale
Lower Farmers
sandstone
Upper part of
Crab Orchard
Formation
Table 2. Distribution of discharge for the simulations
illustrated in figures 7 and 8.
Percent of Total Discharge
Upper Farmers sandstone Sunbury Shale Bedford Shale
Figure 7
Figure 8
60
66
37
23
3
11
-------
206
unsaturated conditions may exist within as many as four hydrogeologic
units: the Nancy shale, Farmers sandstone, Sunbury Shale, and the Ohio
Shale (Papadopulos and Winograd, 1974). Isolated lenses of saturation
bounded above and below by unsaturated conditions can occur in sequences
of rock which have layers of contrasting hydraulic conductivity. These
lenses of saturation, often referred to as "perched" groundwater, form
whenever a layer of relatively low conductivity (a confining bed) is
unable to transmit enough water to maintain complete saturation throughout
an underlying layer of higher conductivity which drains laterally. Seepage
outflow along the hillsides tends to drain the more permeable layers and
create a favorable situation for the occurrence of perched ground water
zones, separated by drained unsaturated zones.
In theory, saturated-unsaturated flow models (e.g., Freeze, 1971)
can accurately simulate the groundwater flow field for a hydrogeologic
system containing perched groundwater. At Maxey Flats, however, the lack
of data, especially with regard to the hydraulic properties of the
unsaturated zones, precludes the use of a complicated saturated-
unsaturated flow model. An obvious alternative is to treat each perched
groundwater lense as a distinct, saturated flow system. The entire
groundwater system can then be modeled by determining the flow field for
each of the perched systems sequentially, from top to bottom, using the
discharge from the basal confining bed in one system as recharge to the
underlying system.
It is assumed here that the transition from saturated to unsaturated
conditions occurs at the base of the confining bed, where the pressure is
assumed to be atmospheric. It can be shown (Zaslavsky, 1964) that the
pressure in the lower part of the confining bed will actually be slightly
less than atmospheric, and that the lowermost part of the confining bed
can even be unsaturated in certain situations. For the purposes of this
study, however, the error due to neglecting these additional complications
is small compared to the other uncertainties in the analysis.
It should also be noted that what is interpreted as an unsaturated
zone in a confined layer may actually be a zone saturated at less-than-
atmospheric pressure. In such a case, a well completed in a rock unit
saturated partly with water at less-than-atmospheric pressure and partly
with water at greater-than-atmospheric pressure would accumulate water only
from that part of the unit at greater-than-atmospheric pressure. The
water level in the well could therefore be far below the top of the
saturated unit. For purposes of brevity, the upper part of such a unit
will be referred to in this report as unsaturated, but it is understood
that the "unsaturated" zone may actually be saturated, but partly at less-
than-atmospheric pressure.
For an isolated hill such as Maxey Flats, the potential for a perched
water table to exist within a confined layer increases as the contract
in hydraulic conductivity between layers increases. Thick layers are also
more likely to contain unsaturated zones simply because, they require a
greater inflow from the overlying confining bed to stay completely
-------
207
saturated. Furthermore, because discharge may occur through seepage
faces in each layer, inflow from above to successively deeper layers
decreases; consequently, deep layers are more likely to contain unsaturated
zones than are shallow layers of equivalent conductivity and thickness.
The Ohio Shale possesses many of the characteristics important for
the development of unsaturated conditions; it is thick, has adequate
permeability, and is located at a low stratigraphic position in the hill.
There is, in fact, evidence that water table conditions probably do exist
within the Ohio Shale. Three wells open to nearly the entire 185-foot
thickness contain less than 50 feet of water. If the shale were completely
saturated at greater than atmospheric pressure, these low water levels
would indicate a large vertical head gradient, and therefore a large com-
ponent of vertical flow. However, the Ohio Shale is underlain by the
poorly permeable upper part of the Crab Orchard Formation; consequently,
flow in the Ohio Shale (especially the lower part) should be nearly hori-
zontal due to the high contrast in hydraulic conductivity. The most
plausible explanation for the low water levels appears to be that the upper
part of the Ohio Shale is unsaturated. At least two wells drilled in the
Ohio Shale encountered isolated zones of water some 30 to 50 feet above
the ultimate static water levels, which could indicate that there is
actually more than one saturated zone within the Ohio Shale. There is no
way to check this hypothesis, however, because all of the wells in the Ohio
Shale are open to the entire 185-foot thickness.
The existence of perched water tables within the upper Farmers
sandstone and the Sunbury Shale is problematical. Dry wells finished in
these units probably fail to intersect water-bearing fractures, and others
with extremely low water levels may intersect fractures only at deep levels.
A number of wells do not have relatively high water levels, which suggests
that the upper Farmers sandstone and the Sunbury Shale could be completely
saturated. Due to its lower stratigraphic position in the hill, the Sun-
bury Shale is the more likely of these two units to contain an unsaturated
zone.
Based on the preceding analysis, Maxey Flats could be considered to
consist, conceptually, of two saturated flow systems: an upper system
which extends from just above the sandstone marker bed down to the base of
the Bedford Shale, and a lower system which occupies the bottom part of the
Ohio Shale. Only the upper flow system is considered in detail because
water level data is insufficient to warrant a model analysis of groundwater
flow in the Ohio Shale.
Mathematical Model
Idealized, two-dimensional vertical cross-sectional flow models are
useful tools for studying general conceptual features of complex flow
systems. The governing equation for two-dimensional, steady-state ground-
water flow is:
-------
208
*\ J4 h Ji H h
aF * zz aT' ay * yy ay'
where K and KZZ are the hydraulic conductivities in the horizontal and
vertical directions, respectively, and h is the hydraulic head. For this
report, equation 1 was solved numerically using a finite difference
approximation.
A representative vertical cross section was taken through the central
part of the hill and oriented perpendicular to the topographic divide
(line B'-B in figure 1). Boundary conditions for the upper flow system
are summarized in figure 6. The left boundary is considered to be a
no-flow boundary corresponding to a groundwater divide near the center of
the hill. Heads along the lower boundary are set equal to the elevation
of the base of the Bedford Shale. Heads along the water table in the
weathered Nancy shale are assumed to vary linearly from a value cor-
responding to the water level in well UB-1A near the center of the hill to
a value equal to the elevation of the top of the sandstone marker bed at
the edge of the upland surface. The seepage boundary along the hillside
is slightly more complicated. As a first approximation, the head at each
point along the hillside was set equal to its elevation. Equation 1 was
then solved for the head distribution. Occasionally, some sections of
the hillside showed an anomolous horizontal component of flow back into
the hill, which indicates that the flow conditions within the system are
unable to suport the heads specified along those parts of the hillside.
These "back flow" regions were changed to be no-flow boundaries, and
equation 1 was solved again. This procedure was repeated until no more
anomolous "back flow" regions occurred. An iterative approach of this
type is frequently used to model steady-state flow in systems which contain
seepage faces (Freeze and Cherry, 1979).
DISCUSSION
Figures 7 and 8 present the results of simulations for two different
conductivity distributions which reproduce the important features of the
head distribution in the upper flow system at Maxey Flats. The flow system
is characterized by nearly vertical flow in the three major zones of low
hydraulic conductivity (the unweathered Nancy shale, the lower Farmers
sandstone and Henley shale, and the Bedford Shale) and essentially hori-
zontal flow in the upper Farmers sandstone and the Sunbury Shale. The
presence of the unsaturated zone below the Bedford shale assures a large
average vertical head loss in the upper flow system. In addition, the
simulations demonstrate how a well such as 14E, located near the center of
the hill and finished primarily in the upper Farmers sandstone, can have
a water level far above the top of the upper Farmers sandstone, whereas
similar wells located around the perimeter of the hill often have much lower
water levels.
-------
SPECIFIED HEAD, h = h (y)
A
-c -c
*O|iO
o
O
z
z
I,
SEEPAGE FACE
BOUNDARY
ro
o
SPECIFIED HEAD
h = 0
Fig. 6. Schematic diagram illustrating boundary conditions for the
groundwater flow model.
-------
500
1000
100
500
Weathered Nancy Shale
Sandstone Marker Bed
Q W
ZQ
Otr
oo
Ul
m
li
cc
LU
10
10
1000
Unweathered Nancy Shale
Upper Farmers Sandstone
Lower Farmers
Sandstone
Henley Shale
Sunbury Shale
Bedford Shale
ro
o
SCALE
VERTICAL EXAGGERATION - 10X
CONTOUR INTERVAL = 10 FEET
DATUM = BASE OF BEDFORD SHALE
Fig. 7. Simulated head distribution for the relative conductivity
distribution, 500: 1000: 100: 500: 10: 10: 1000: 1.
-------
Weathered Nancy Shale
Sandstone Marker Bed
Unweathered Nancy Shale
Upper Farmers Sandstone
Lower Farmers Sandstone
i
Henley Shale
Sunbury Shale
Bedford Shale
FEET
SCALE
VERTICAL EXAGGERATION = 10X
CONTOUR INTERVAL = 10 FEET
DATUM = BASE OF BEDFORD SHALE
Fig. 8. Simulated head distribution for the relative conductivity
distribution, 125: 250: 25: 250: 2.5: 2.5: 250: 1.
-------
212
The shaded areas in figures 7 and 8 indicate regions where the
predicted head is less than the elevation head; that is, regions where
the pressure is less than atmospheric. Because rocks tend to become
unsaturated when the pressure falls more than a few feet below atmospheric
pressure, regions of sub-atmospheric pressure predicted by these simula-
tions can be interpreted as an indication that unsaturated conditions
could exist even though the model used in this analysis has no way of
accounting for unsaturated conditions.
Figures 7 and 8 also illustrate two of many possible distributions
of unsaturated zones. In figure 7, the upper flow system is completely
saturated except, perhaps, for a small part of the Sunbury Shale and the
overlying confining layers near the hillside. It is probably reasonable
to assume that most of the more permeable units are unsaturated to some
extent near the edge of the hill. In figure 8, the head everywhere in
the Sunbury Shale is far below the top of the unit, which implies that
water table conditions could exist throughout the Sunbury Shale. All that
can be concluded, however, is that it is at least possible that water
table conditions exist in the Sunbury Shale beneath the entire area of the
hill.
Most of the water which enters as recharge to the upland surface of
Maxey Flats in the two simulations discussed above is discharged along
the hillside through either the upper Farmers sandstone or the Sunbury
Shale (Table 2). It should be noted that the discharge distribution is
a strong function of the anisotropy in hydraulic conductivity. For most
of the simulations made during this study it was assumed that the hori-
zontal conductivity was 100 times larger than the vertical conductivity
due to the fact that fractures appear to be much more continuous in the
horizontal direction than in the vertical. A few simulations were made
which assumed isotropic conditions (K = KZZ). In those cases, a much
larger fraction of the groundwater flow (over 75 percent) discharged across
the base of the Bedford Shale. However, the isotropic simulations were
not able to account for the large heads in the upper Farmers sandstone.
The sandstone marker bed, one of the most permeable units at the
site, occurs directly beneath the trenches; consequently, the possibility
of radionuclide movement in the sandstone marker bed is of major concern.
The models presented in this report cannot provide good, quantitative
estimates of flow in the sandstone marker bed, but they do emphasize that
the horizontal component of the hydraulic gradient in the sandstone marker
bed should be essentially equal to the slope of the upper water table.
Thus, in order to minimize horizontal flow in the sandstone marker bed
it is extremely important to maintain low water levels in the trenches over
the central part of the hill. In the simulations presented here most of
the flow in the sandstone marker bed is downward into the underlying
unweathered Nancy shale. Field data indicates that there is a significant
component of horizontal flow of contaminated trench water in the sandstone
marker bed.
-------
213
This investigation illustrates the difficulty and complexity of
defining the subsurface flow system in a layered, fractured-rock system
of extremely low permeability. The basic constraint in modeling the
system is the shortage of reliable field-measured parameters such as the
horizontal and vertical hydraulic conductivities in all the major layers
as well as the lack of adequate hydraulic head measurements. The model
itself is capable of incorporating many different combinations of input
parameters and boundary conditions. The results presented in this report
do not necessarily approximate "actual" conditions in the field; they
merely typify some possibilities based on qualitative estimates of
hydraulic conditions at the site.
REFERENCES
Freeze, R. A. 1971, Three-dimensional, transient, saturated-unsaturated
flow in a groundwater basin, Water Resources Res., v. 7, pp. 347-366.
Freeze, R. A., and Cherry, J. A., 1979, Groundwater, Prentice-Hall, Inc.,
Englewood Cliffs, New Jersey.
Papadopulos, S. S., and Winograd, I. S., 1974, Storage of low-level radio-
active wastes in the ground: Hydrogeologic and hydrochemical factors,
U.S. Geological Survey Open-File Report 74-344.
Wilson, C. R., and Witherspoon, P. A., 1970, An Investigation of laminar
flow in fractured rocks, Geotechnical Report No. 70-6, University
of California, Berkeley.-
Zaslavski, D. A., 1964, Saturation-unsaturation transition in infiltration
to a non-uniform soil profile, 8th Congr. Int. Soc. Soil Sci.
Bucharest.
-------
COMPUTER SIMULATION OF GROUNDWATER FLOW AT A
COMMERCIAL RADIOACTIVE-WASTE LANDFILL NEAR
WEST VALLEY, CATTARAUGUS COUNTY, NEW YORK
David E. Prudic*
U.S. Geological Survey
P.O. Box 1350
Albany, N.Y. 12201
ABSTRACT
Commercial low-level radioactive wastes were buried in
trenches excavated in a clay-rich till of low hydraulic conduc-
tivity near West Valley, N.Y., from 1963 to 1975. Groundwater
from the trenches flows out laterally and downward through
approximately 25 m of till.
A two-dimensional finite-element computer model was used to
evaluate factors controlling groundwater flow in two vertical
sections. Measured heads in 24 piezometers along these sections
were most accurately reproduced by simulating four geologic
units, each internally isotropic but differing in hydraulic con-
ductivity to reflect fracturing near land surface and increased
consolidation with depth. The model accurately simulated the
changes in head that followed removal of water from one of the
trenches. Specific-storage values required to calibrate the
model for transient conditions agreed well with values derived
from four consolidation tests of till samples.
Infiltration rates ranging from 3.8 centimeters per year near
swampy areas to 1.5 cm/yr along scraped or smooth, sloping sur-
faces were used to simulate observed area! variation of head in
the till. Simulations indicated that water flowing laterally
out of a trench would not intersect nearby intermittent streams
as long as the water level in that trench remained below the
trench cover.
* Present address U.S. Geological Survey
Room 227, Federal Bldg.
705 North Plaza St.
Carson City, NV 89701
215
-------
216
INTRODUCTION
Location and Physical Setting
A State-licensed landfill for commercial radioactive wastes is among
the facilities at the Western New York Nuclear Service Center, which
occupies 13.5 km2 (3,350 acres) in the northern part of Cattaraugus
County, about 50 km south of Buffalo, N.Y. (fig. 1). The center is
in a sparsely populated region of the glaciated Allegheny plateau (1)
that is characterized by rounded hills of Devonian shale capped with
thin deposits of till and separated by bedrock valleys containing as
much as 150 m of drift.
The climate at the center is humid content!al with an annual average
temperature of 7°C. Temperature extremes range from 38°C during summer
to -26°C during winter. Precipitation averages 100 cm/yr without
distinctly wet or dry seasons. During winter, precipitation is normally
in the form of snow.
Operation of the landfill began in November 1963 and continued until
May 1975. The 4-ha (11-acre) landfill contains a series of trenches
excavated in silty-clay till of low hydraulic conductivity and con-
taining scattered pods of silt and sand. Radioactive wastes were buried
in trenches that are generally 6 m deep, 10 m wide at the surface, and
180 m long (fig. 1). Trenches 6 and 7 were set aside for burial of spe-
cial material of high specific activity (2).
Purpose and Scope
This study was done as part of a national study by the U.S.
Geological Survey to determine the principal factors that control the
subsurface movement of radioisotopes at several radioactive-waste land-
fills. Results of this study have been used in an investigation begun
in 1975 by the New York State Geological Survey, as lead agency under
contract with the U.S. Environmental Protection Agency, and the U.S.
Nuclear Regulatory Agency, to evaluate all processes of radioisotope
migration at the site. This report describes results of computer
modeling of groundwater flow near the north trenches (fig. 1).
Test holes drilled or augered near the landfill during 1975-78 were
designed to provide information about the flow of groundwater as well
as the geology. Cores were collected and analyzed to determine the
distribution of radioisotopes and to detect possible movement of water
away from the burial trenches (3,4). Several laboratory tests were run
on selected core samples of the till to determine hydraulic properties.
Test holes were assigned a letter, and in the few instances when two or
more test holes were drilled in proximity (1 to 3 m of one another), a
sequential number followed the letter; for example, test holes I, 12,
and 13. (See fig. 1.)
-------
217
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EXPLANATION
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Test hole drilled 1975-78
Q2C
Test hole augered 1973
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Sump pipe in trench
10
Trench number
A A1
Cross-section profile
0 ^5 50 76 METERS
Base from U.S. Geological Survey
_J Ashford Hoi low, 1 964, 1 :24,000
NOTE: Map based on plane-table survey by R.J. Martin, U.S. Geological Survey, 1975-78
Figure 1. Layout of trenches and location of test holes
and wells used to monitor water-level trends.
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218
To determine head distribution in three dimensions, each test hole
was designed to contain from 1 to 4 piezometers 3 cm in diameter and
having a 30-cm-long screen. Each screen was finished in a sand envelope
about 45 cm thick. At least 3 m of grout separated piezometers when
more than one was finished in a test hole. Two small-diameter tubes (1
cm and 0.5 cm) extended from the screen to land surface. The smallest
diameter tubing ended near the bottom of the screen, whereas the larger
ended near the top of the screen. The small-diameter tubing was used to
minimize the amount of water needed to fill it and thus reduce the lag
time between head change in the till and response in the piezometer.
The double-tube system was used to obtain periodic water samples.
Piezometers were designated according to test-hole number followed
by a number preceded by a hyphen. The numbers designate individual
piezometers within a test hole, and piezometers were numbered sequen-
tially from shallow to deep; for example, G-l, G-2, and G-3.
During this investigation, hydrologic studies were limited to the
till for the following reasons: (a) the trenches were excavated only in
this till; (b) trench cover consisted of reworked till, and (c)
radioisotope migration was confined to the till (3,4).
A two-dimensional finite-element model developed by Reeves and
Duguld (5) was used to study factors controlling groundwater flow in
the till. The objective was to simulate groundwater flow in a cross
section perpendicular to the north trenches (nos. 2-5) from test hole G
eastward to Franks Creek (fig. 1) because these trenches had had high
Initial water levels in February 1976 that were lowered by pumping in
July 1976. This lowering produced a stress that normally would not have
occurred and allowed for better calibration of the model.
Acknowledgments
Mark Reeves (formerly with Oak Ridge National Laboratory) assisted
with the initial computer runs and helped adapt the program to the
conditions at West Valley. Nuclear Fuels Services, the site operator,
provided personnel to monitor radiation levels whenever Survey personnel
and their contractors were near or on the burial trenches; also their
staff answered many questions about burial procedures and practices.
Stephen Mollelo (New York State Geological Survey) and Robert Wozniak
(New York State Department of Environmental Conservation) assisted with
the data collection.
GEOLOGIC CONDITIONS NEAR LANDFILL
Surflcial geologic mapping (6) near the landfill defined the nature
and extent of the till in which the burial trenches are excavated. The
till, composed predominantly of silt and clay, is a valley fades that
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219
is widely distributed in the Cattaraugus Creek basin. It is found below
the level of saddles in the drainage divide to the south and must have
been deposited by an ice sheet partly suspended in ponded water.
The till is commonly covered by thin gravel deposits (6) that, at
the landfill, are largely absent. The till extends to about 30 m below
land surface near the trenches and overlies a bedded lacustrine unit 10
to 20 m thick consisting of fine sand and silt and locally capped by
gravel. The upper part of the lacustrine unit is unsaturated (fig. 2).
Bedrock is as much as 150 m below land surface near the trenches.
Although the geology and hydrology of the thick glacial deposits beneath
the lacustrine unit are largely unknown, the lacustrine unit probably
provides an avenue for slow lateral flow to points of discharge in the
bluff along Buttermilk Creek, 550 m northeast of the landfill (3).
Fourteen till samples from test holes near the landfill were ana-
lyzed for particle size. On the average, the samples contained 50 per-
cent clay, 27 percent silt, 10 percent sand, and 13 percent gravel;
LaFleur (6) reported that 10 to 20 percent pebbles and cobbles is
characteristic at most exposures. Typically the unsorted till inter-
fingers with a similar till having a pebble qontent of less than 5 per-
cent and a matrix containing tiny blebs and torn, deformed wisps of
quartz silt; this latter till constitutes between 20 and 25 percent of
the till thickness (6). The unsorted till also includes thin layers of
silt, sand, and rarely coarser sand, which together constitute about 7
percent of the core footage logged (3). Both subunits were encountered
in all test holes (fig. 2). However, stratified materials penetrated by
nearby test holes commonly differ appreciably in altitude and(or) litho-
logy, and even where altitude and lithology are similar, the steep dips
indicated by many of the cores and exposures seem to negate the possibil-
ity of a continuous subhorizontal layer that could indicate a widespread
inplace glaciofluvial deposit.
Distribution of Fractures in Till
A network of intersecting horizontal and vertical fractures charac-
terizes the oxidized upper 2 m of the till near the landfill. Fractures
and root tubes whose surfaces are bordered by firm oxidized till were
recognizable to depths of3mto4.5min cores from several test holes
next to the landfill (3). Some vertical fractures extend downward into
the unweathered till. Dana and others (7) noted that the number of
fractures and their degree of openness decreased with depth in excava-
tions near the landfill. Both manganese and calcite deposits have been
observed along several fractures in excavations and also in cores from
test holes next to the landfill. Results from a laboratory study (8)
suggests that the theoretical limit to which a fracture could penetrate
in the till is about 15 m.
-------
425-1
Waste-burial
trenches
D
November 1976-April 1980
Piezometer^ I
375
LOCATION MAP
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LS.LY
SG.etc.
LY/L/S
T(L3%)
Gravel
Silt
Sand
EXPLANATION
Y Clay
T Till:50%clay,27%silt,
10% sand, 13% grave I
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Base from U.S. Geological Survey
Ashford Hollow, N.Y., 1 964
Mixed materials with increasing abundance
from left to right (example - LS is a si Ity sand)
Interbedded layers; commonly deformed and
dipping at angle to core
Till incorporating torn, discontinuous deformed
layers of silt in percentage indicated
Dip of interbedded layers as identified in cores;
orientation not known
Log inferred from geophysical data and other holes
Unsaturated; based on examination of core samples
and from neutron-moisture logs
D Test hole (See figure 1;
Interval not cored, or hole
penetrated reworked material
(cover or backfill)
Continuous or frequent cores
of inplace material
Infrequent cores of inplace
material
Figure 2. Generalized cross section from test hole G to Buttermilk Creek. Spring
may be perched water of local origin; flow is probably less than
1 liter per minute. (Modified from Prudic and Randall, 1979.)
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221
GROUNDWATER FLOW NEAR THE NORTH TRENCHES
Much of the precipitation on the landfill either runs off to nearby
streams and gullies or returns to the atmosphere by evapotranspiration
(3). The small amount that infiltrates below the roots of plants moves
generally downward, even beneath small valleys bordering the site, as
indicated by the distribution of hydraulic head along a section perpen-
dicular to the north burial trenches in February 1976 (fig. 3). At that
time, most of the north trenches were nearly filled with water (3) that
was generally moving outward and downward from the trenches. A small
flow component toward trench 5 may be inferred from the hydraulic head
near land surface within 10 m west of the trench (fig. 3).
Pressure head within the till ranged from near zero to more than 5 m.
Low-pressure heads generally occurred beneath steep and(or) smooth
slopes or where the upper layers of soil were scraped to allow for
rapid and complete runoff. Pressure heads in the till were generally
greater where water had accumulated above the till, as in the trenches,
in natural depressions in land surface (as near holes J and G in fig.
1), and in remnants of late-glacial fluvial gravel atop the till (3).
Response of Water Levels in Piezometers to
Pumpout of Water from Trenches 2-5
Water was pumped from trenches 3, 4, and 5 on several dates between
July and November 1976 to lower .the water levels and was pumped again
between August and November 1977, including trench 2. The piezometers
next to trenches 5 and 2 responded to the pumpout (figs. 4 and 5), but
those farther away showed no response (fig. 6).
The decline of water levels in piezometers near trench 5 was most
rapid in those finished in undisturbed material near the reported alti-
tude of the trench floor (D-l, E-2, and F-2) or in a sandy backfill next
to the trench (E-l) (3).
The sudden water-level increases in piezometers D-2 and F-3 (fig. 4)
in May and June 1976 were caused by attempts to sample the piezometers,
whereby nitrogen was injected at pressures as high as 4.2 kg/cm2 down
the larger of two tubes that extended to the piezometer screen* which
forced water up the smaller tube to the surface. Apparently pressures
were large enough to deform the till and spread it away from the column
of hardened cement grout above the piezometer, in effect increasing the
length of piezometer. Because total head in the till declines with depth,
the water level rose when water from shallower depth entered the piezo-
meter. The subsequent decline in water level reflects partial resealing
of till against the grout column as well as a response to trench pumpout.
Other piezometers were sampled by suction lift in combination
with gas pressures less than 0.5 kg/cm2.
-------
430-i
r\\) Backfill
Weathered till
V/X Till with oxidized fractures
Unweathered till
• • • • • * • • •
Assumed water-table
Water level m trench
414
Equipotential line, altitude, in meters
ro
ro
390
Figure 3. Cross section A-A1 through north trenches showing head distribution,
February 1976. (Location of section is indicated in fig. 1.)
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420
I I I I I I I I I I | I I I I I I I I 1 I I I I I I I
I_J _ L_l _ \ — I
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406
Figure 4. Water-level trends in piezometers finished next to trench 5.
(Letter and number identifies piezometer, sump refers to well in trench.)
-------
418
416
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406
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(dry)
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I I I I I I I I I I I I I I I I I I I I 1 I I I I I
JFMAMJJASONDJ FMAMJJASONDj FMAM
1976 1977 1978
Figure 5. Water-level trends in piezometers finished next to trench 2.
(Letter and number identifies piezometer, sump refers to well in trench.)
ro
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-------
420
418
I I 1 1 1 1 1 1 1 1
I I 1 1 1 1 1 1 1
CM
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1976 1977 1978
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Figure 6. Water-level trends 1n piezometers finished distant from trenches 2 through 5.
(Letter and number identifies piezometers.)
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226
Although the two effects cannot be separately measured, for purposes of
model calibration, the water-level trends in piezometers D-2 and F-3
caused by the pumpouts were estimated from measurements in E-3, which
was finished at similar depth along the west side of trench 5.
The decline in water level in piezometers next to trench 2 was less
pronounced than that next to trench 5. Because water level in trench 2
was lower than in trench 5,'the decrease in water level was less. Of
the piezometers finished next to trench 2, A2-1, finished in a lens of
layered silt and fine sand that dips from piezometer R-l toward trench 2,
responded most rapidly (fig. 5). Piezometers in test hole R, 6 m east
of those in test holes A and A2 (fig. 1), did not show a decline beyond
the normal yearly fluctuations (fig. 4).
Test holes I, 12, 13, and 14 are near the south end of trench 2,
about 70 m from the sump from which water was pumped out of trench 2
(fig. 1). Well 2-1A, driven into trench 2 in June 1976 near test hole
I, had nearly the same water level as the sump until the sump water was
lowered below an altitude of 414.5 m (fig. 4); however, continued
pumping from the sump did not cause a decrease in well 2-1A. After the
pumpout of trench 2, the difference between water-level trends in piezo-
meters at the north end and those at the south end probably reflects
differences in water level within trench 2 itself.
Hydraulic Properties of Saturated and Unsaturated Till
Hydraulic conductivity of the saturated unfractured till was
obtained from both field and laboratory studies. Field tests were done
by withdrawing 200 to 500 ml of water out of piezometers finished at
depths from 5 to 16 m, then measuring the water-level rise in the piezo-
meters until they had stabilized.
Two methods were used to analyze the data. The first assumed hori-
zontal flow to the piezometer (9); the second assumed spherical flow
(10). Average hydraulic conductivity calculated from 12 tests was 6 x
10'8 cm/s by the first method and 2 x 10'8 cm/s by the second.
Hydraulic conductivity measured in laboratory tests at confining
pressures on seven core samples from depths of 5 to 16 m produced an
average vertical hydraulic conductivity of 2 x 1Q-8 cm/s, which compares
well with values obtained from field tests that assumed spherical flow.
Fickies and others (8) reported anistropy with 40 times greater horizon-
tal than vertical conductivity in two core samples, although laboratory
tests of nine core samples suggest little or no anistropy. Analyses of
thin sections of 18 till samples do not suggest preferential horizontal
layering of the clay grains within the till.
Consolidation tests were run on four samples to determine the
effects of increased confining pressures on hydraulic conductivity. In
general, the vertical hydraulic conductivity decreased by about 40 per-
cent as confining pressures increased from near atmospheric to 7
kg/cm2 (equivalent to pressure at a depth of 30 m), which suggests that
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227
increased overburden pressures may reduce hydraulic conductivity of the
till (fig. 7). However, much of this decrease occurred at confining
pressures between near atmospheric and 1 kg/cm2. A decrease of less
than 10 percent was observed between pressures of 1 to 3.5 kg/cm2
(equivalent to pressure at a depth of 5 to 16 m), and at pressures from
3.5 to 7 kg/cm2, the decrease averaged 13 percent.
Field tests of seven piezometers finished in small lenses of silt
and sand yielded an average hydraulic conductivity of 2 x 10'6 cm/s;
their range of values (7 x 10~8 to 6 x 10-6 cm/s) was considerably
greater than the range for till. (Weighted average hydraulic conduc-
tivity of the till and sorted materials was calculated to be 8 x
10-8 cm/s.j
Porosity of the till was determined by measuring dry unit weight and
specific gravity of 28 core samples as described by Morris and Johnson
(11). Average porosity was 32.4 percent with a standard deviation of
4.7 percent.
85
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Hole E - 10.35to 10.5m
Hole M- 15.8to 16.0m
Hole D - 5.8to6.0m
Hole C2- 12.35to 12.5m
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VERTICAL HYDRAULIC CONDUCTIVITY, IN CENTIMETERS PER SECOND X 10
12
,-8
Figure 7. Relationship of hydraulic conductivity to confining
pressures as determined from consolidation tests.
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228
Because the model simulates unsaturated as well as saturated con-
ditions, functions relating volumetric moisture content and hydraulic
conductivity to soil-moisture tension were needed. Both functions were
assumed to be nonhysteretic and were calculated from pore-size-
distribution curves derived from mercury porosimeter tests (12) of seven
core samples of till. The curves relating soil-moisture content to
soil-moisture tension were calculated on the assumption that the
capillary pressure of mercury is 5.2 times that of water in air (12).
Curves relating hydraulic conductivity to soil-moisture tension were
calculated from the soil-moisture curves with an equation by Millington
and Quirk (13); both curves used in the model are depicted in figure 8.
Saturated hydraulic-conductivity values calculated from the soil-
moisture tension curve and an equation by Marshall (14) averaged
6 x 10-8 cm/s, which is reasonably close to measured values.
125
u
-20.000 -15.000 -10.000 -5000
PRESSURE HEAD, IN CENTIMETERS OF WATER
Figure 8. Relationship of hydraulic conductivity and volumetric soil
moisture content to soil-moisture tension, as calculated
from seven mercury porosimeter tests of the till.
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229
It was not considered necessary to then relate the computed curves to
measured values, as is often required, because the till was nearly
saturated, even with soil-moisture tensions exceeding 1,000 cm.
To simulate transient conditions of groundwater flow at the burial
site, data on storage within the geologic materials are needed. The
storage equation used in the model uses the modified terms of compressi-
bility for water and the solid matrix (5); the modified term of
compressibility of water was obtained by multiplying the compressibility
of water at 10°C (15) by the density of water and by gravity. The value
used in model simulations was 4.3 x 10~8 cnr*.
Specific storage, which is the sum of the modified terms of
compressibility of the water and the till, was calculated from the four
consolidation tests described previously. The specific storage value of
the till, which averaged 8 x 10-° cm'1, consistently decreased with
increasing confining pressures, similar to the hydraulic-conductivity
values of the till. Specific-storage values ranged from 16 x 10~6 cm'1
at a depth of 5.8 m to 2 x 10~° cnr* at a depth of 16 m. These values
are essentially the modified term of compressibility of the matrix
because the value of water compressibility is much lower.
The specific storage values obtained from the consolidation tests
are in the lower range of values typical of a medium-hard clay as pre-
sented by Domenico and Mifflin (16) and an order of magnitude less than
intergranular values presented by Grisak and Cherry (17) for a clay-rich
till in Manitoba, Canada.
COMPUTER MODELING OF GROUND-WATER FLOW PERPENDICULAR
TO NORTH TRENCHES
The groundwater flow model developed by Reeves and Duguid (5) was
chosen for the study at West Valley because it is capable of simulating
(a) saturated and unsaturated flow with a moving water table, in cross
sections, (b) groundwater outflow to land surface, and (c) temporal
variation in inflow from precipitation. Other reasons include (d) no
restrictions on the number of units with varying hydraulic properties;
(e) no maximum difference between hydraulic-conductivity values of model
units; (f) use of constant infiltration rate from land surface during
steady-state simulations; and (g) availability of a solute model that
was developed and coupled to the flow model (18).
Reeves and Duguid (5) stated that the equations representing flow in
saturated and unsaturated porous media consist of equations of (a) con-
tinuity of the water; (b) continuity of the matrix; (c) motion of the
water; (d) consolidation of the matrix, and (e) compressibility of the
water and combined these equations into one:
(1)
-------
230
Where e = volumetric moisture content, dimension!ess ratio;
n = porosity, dimensionless ratio;
3' = modified coefficient of compressibility of the water, L"1;
a1 = modified coefficient of compressibility of medium, L'1;
•4& = change in moisture content with respect to pressure head, L"1
3J1 = change in pressure head with respect to time, LI"*;
3t
]( = hydraulic conductivity tensor, LT'1;
h = pressure head, L;
z = vertical distance, L;
x = horizontal distance, L.
This equation reduces to Richard's equation (19) for unsaturated flow
and reduces to the elastic storage equation for saturated flow (5).
Because the model solves for total head, both equations are automati-
cally satisfied.
The hydraulic-conductivity tensor is dependent on pressure head, as
shown in the following equation:
K = Ks • Kr(h) (2)
Where K = hydraulic-conductivity tensor, IT'1;
Ks = saturated-hydraulic conductivity tensor, LT'1;
Kr(h) = relative hydraulic conductivity as function of
pressure head, dimensionless.
Steady-State Simulations
A finite-element grid was developed to represent the cross section
A-A' from hole G east to Franks Creek (fig. 1). The following condi-
tions were established for steady-state simulation of flow in February
1976 and February 1978:
(a) No-flow boundaries were assigned to both sides of the section
because hydraulic gradient is predominantly vertical.
-------
231
(b) The silt was modeled at a pressure head of zero at the base of
the till because the silt underlying the till is generally
unsaturated.
(c) Infiltration from precipitation was at a steady rate sufficient
to saturate the till to land surface; however, the rate was
decreased in some model runs to create local unsaturated conditions.
(d) A horizontal hydraulic conductivity of 8 x 10~8 cm/s was assigned
to the unweathered till unit; this is a weighted average hydraulic
conductivity of the till and the sorted material.
(e) Constant head within the trenches was made equal to observed head
in February 1976 or February 1978. (This assumption of constant
head in the trenches is reasonable for periods of 2 to 3 months
because the water level in these trenches did not generally change
more than 5 cm in a month.)
Several steady-state simulations were run to evaluate the relative
significance of am'sotropy, variations in hydraulic conductivity caused
by fracturing and increased confining pressures within the till, and
infiltration from precipitation. Simulations became progressively more
complicated, starting with a single isotropic unit but eventually con-
sidering as many as five units with varying degrees of am'sotropy. The
mean absolute departure of simulated heads from heads observed at 17
piezometers in the plane of the section was calculated for each model
run as a general index of degree of ,fit; the results are given in table 1.
Simulations that incorporated a single isotropic unit yielded the
same pressure heads regardless of hydraulic conductivity (table 1, case
1); however, changes in computed inflow and outflow rates were directly
proportional to any change in hydraulic conductivity. Simulations that
incorporated a single anisotropic unit produced better fits (decreased
the mean absolute departure), and best results were obtained when hori-
zontal hydraulic conductivity was 100 times the vertical value (table 1,
case 2). Simulating several shallow isotropic layer(s) at a hydraulic
conductivity greater than 8 x 10~° cm/s to represent the abundantly
fractured weathered till and the fractured unweathered till also pro-
duced better fits than a single isotropic unit, and best results were
obtained when the hydraulic conductivity was between 10 and 100 times
higher in the weathered till than in the unweathered till (table 1, cases
3 and 7). Simulating am'sotropy of the layered units reduced the mean
absolute departures (table 1, cases 4, 5, 6, and 8).
Most simulations incorporated an infiltration rate large enough to
saturate the till to land surface, but infiltration rates in a few simu-
lations were reduced enough to cause the uppermost nodes of the model to
become unsaturated. When am'sotropy was included, the result was a
slight lowering of heads in most nodes in the section. In isotropic
simulations, large changes in heads occurred in the nodes beneath points
where infiltration was the only source of water. For example, reducing
infiltration rate near model holes D and I did not greatly affect heads
-------
232
Table 1. Differences between heads observed in February 1976
and simulated heads for assumed conditions.
" Mean absolute
departure from
observed heads
of February 1976
Case Conditions (centimeters)
one isotropic unit
(a) Kx = Kz = 3.0x10-8 cm/s 201
(b) Kx = Kz = 3.0x10-7 Cm/s 201
one unit with anisotropy
(a) Kz = 0.1 Kx 161
(b) Kz = 0.01 Kx 114
(c) Kz = 0.001 Kx 116
two isotropic units (weathered and
unweathered till)
(a) K (weathered) = 10 K (unweathered) 123
(b) K (weathered) = 100 K (unweathered) 127
two anisotropic units (weathered and
unweathered till)
Kx (weathered) = 10 Kx (unweathered)
(a) Kz = 0.1 Kx in each unit 93
(b) Kz = 0.01 Kx in each unit 74
two anisotropic units (weathered and
unweathered till)
Kx (weathered) = 100 Kx (unweathered)
(a) Kz = 0.1 Kx in each unit 90
(b) Kz = 0.1 Kx (weathered)
Kz = Kx (unweathered) 109
(c) Kz = 0.01 Kx (weathered)
Kz = 0.01 Kx (unweathered) 92
two anisotropic units (weathered and
unweathered till)
Kx (weathered) = 10 Kx {unweathered)
Kz = 0.01 Kx in each unit
(a) simulated sand and silt lenses near
D, I, and J 65
(b) as above plus increased heads at
base of model near G 60
-------
233
Table 1. Differences between heads observed in February 1976
and simulated heads for assumed conditions. (Continued)
Mean absolute
departure from
observed heads
of February 1976
Case Conditions (centimeters)
7 Three isotropic units (weathered, unweathered
with fractures and unweathered till)
(a) K (weathered) = 10 K (unweathered)
K (unweathered w/fractures) = 5 K (unweathered) 116
(b) K (weathered) = 100 K (unweathered)
K (unweathered w/fractures) = 10 K (unweathered) 121
8 Three anisotropic units (weathered, unweathered
with fractures and unweathered till) with Kx the
same as noted in case 7(a) above
(a) Kz = 0.1 Kx in each unit 96
(b) Kz = 0.01 Kx in each unit 93
(c) Kz = 0.01 Kx in each unit plus simulated
sand and silt lenses near D, I, and J;
increased head at base of model near G,
and lower infiltration near holes D and I 57
(d) same as 8(c) except Kz = 0.1 Kx for each unit 53
9 Four isotropic units (weathered, unweathered with
fractures, and 2 unweathered till units to
compensate for overburden pressures)
K (weathered) = 10 K (unweathered)
K (unweathered w/fractures) = 5 K (unweathered)
(a) K (lowest unweathered unit) = 0.5 K (unweathered) 120
(b) K (lowest unweathered unit) = 0.7 K (unweathered) 104
(c) K (lowest unweathered unit) = 0.85 K (unweathered) 107
(d) K (lowest unweathered unit) = 0.7 K (unweathered)
plus simulated sand and silt lenses near D, I
and J; and reduced infiltration near D and I 31
KSHowest unweathered unit) = 0.85 K (unweathered)
except beneath hole G where
K (lowest unweathered unit) = 0.7 K (unweathered) 21
-------
234
In nodes below the bottom of the trenches, whereas it did near holes H
and J. Infiltration rates near swampy areas were adjusted to saturate
the till barely to land surface. The final infiltration rate used in
the computer simulations was about 3.8 cm/yr near swampy areas, whereas
the model infiltration rate near hole I was 1.5 cm/yr.
In all model runs, simulated heads in the deepest piezometer at hole
G were lower than observed heads. This would be expected if the
lacustrine silt or sand beneath the till south and east of the trenches
became thinner to the west, pinching out near hole G, or that the unit
became finer toward the west. Because this unit seems to be unsaturated
beneath the trenches, such a thinning out or fining of the sediments
would indicate saturated conditions near hole G at the altitude of 391
m. In this case the model boundary below hole G would be represented by
pressure heads greater than zero. Although this hypothesis is unproved,
it seems plausible from records of test holes in the area. Alternatively,
the till at depth beneath hole G could have a slightly lower hydraulic
conductivity. Pressure head in the piezometers in hole G was reasonably
simulated assuming isotropic conditions and assuming hydraulic conduc-
tivity of the till below 400 m altitude beneath hole G to be 30 percent
less than that of the unweathered till. Although neither assumption has
been proved, the latter seems more reasonable than assuming slightly
positive heads at the base of the till near hole G.
The best simulations with anisotropy included three units (the
weathered zone, the unweathered till with fractures, and the unweathered
till), and also included localized silt and sand lenses near piezometers
D-l, J-l, and 12-1. In these tests the weathered till unit had a horizon-
tal hydraulic conductivity 10 times higher than the unweathered till
with oxidized fractures, whereas the localized silt and sand units had
values 5 times higher than the unweathered till. The horizontal hydrau-
lic conductivity in all three till units was 10 times higher than the
vertical; the localized silts and sand lenses were assumed to be isotro-
pic. The mean absolute departure in this run was 53 cm (table 1, case 8d).
The best simulations assumed four isotropic till units with the
horizontal hydraulic conductivities described above, plus an additional
unit below the 400-m altitude beneath the trenches and 396 m beneath
Franks Creek. The hydraulic conductivity of this unit was 6.9 x 1Q-8
cm/s, or 15 percent less than that of the unweathered till except below
hole G, where the value was 5.6 x 10~8 cm/s. The purpose of this lower
unit was to represent the reduction in hydraulic conductivity with
depth, either from greater overburden pressure or the lower percentage
of disturbed silt and sand layers below 400 m, as suggested by some of
the test borings. A four-unit model was also hypothesized by field
observations in test trenches dug near the burial ground (7).
The above simulations included localized silt and sand lenses near
piezometers D-l, J-l, and 12-1. The mean absolute departure in this run
was 21 cm. Figure 9 depicts the nodes, boundary conditions, and the
relative hydraulic-conductivitiy values used in the best-fit model simu-
lations.
-------
430—1
Trenches: Fixed head for nodes below
trench water level; seepage
nodes above water level
Constant infiltration rate
along land surface
{ 1 i j | LLK=5 0-00-0 O
' 1 I ! ! 90 o o oo _
_-d o-__A 6—6 • °
!>—A o-
o o o o
oo oo
oooo
Node associated with piezometer9
•"o oo oo o o'ooo
ro
to
en
390
Figure 9. Cross section A-A' through north trenches showing arrangement of nodes,
boundary conditions, and relative hydraulic conductivities used to best-fit
model simulation.
-------
236
Table 2 compares simulated values for best computer simulations
with observed values for both anisotropic and isotropic conditions for
both February 1976 and February 1978. The isotropic four-layer simula-
tions produced smaller average departures than the anisotropic three-
layer simulations and also were qualitatively more reasonable in at
least two respects: (a) they yielded negative heads at piezometers 1-1
and 14-1, which were dry on both dates, whereas the best anisotropic
simulations yielded positive heads, and (b) they also yielded about the
same heads on both dates in the piezometers at hole G, in agreement with
observed conditions, whereas the best anisotropic simulations yielded
significantly lower heads at hole G in February 1978 in response to the
lower water level simulated in the trenches.
The distribution of heads within the till during February 1976 and
1978, based on the best steady-state computer simulations, are shown in
figures 10 and 11. From the simulation of February 1976, water moving
from trench 5 would flow outward only 2 to 3 m before it would move
downward; the same pattern Is indicated east of trench 2. In February
1978, the outward movement of water was less than 2 cm/yr. Average com-
puted vertical flow rate beneath the trenches was 3 cm/yr in February
1976 and less than 2 cm/yr in February 1978. These model simulations
indicated a groundwater discharge to land surface west of hole G and on
the lowest part of the slope near the east side of the model; then model
discharges averaged 100 cm^/d (0.05 gallons per day) per m^ of surface
but occurred only over small areas and were derived largely from flow
through the weathered till. These rates could not be verified in the
field because the streams bordering the landfill are small and during
summer are dry much of the time. In addition, streamflow could not be
measured to the accuracy required to determine such small seepage rates.
However, near the computed areas of discharge are marshy areas that
could be the result of shallow groundwater seepage.
Although the steady-state model simulations reproduced observed
heads in the till from hydraulic properties obtained from field and
laboratory measurements, the simulations could not independently define
the hydraulic conductivity of the till because the same heads could be
simulated by changing all values of hydraulic conductivity and infiltra-
tion by a proportionate amount.
Transient-State Simulations
Transient conditions were simulated in an attempt to reproduce the
observed drawdowns in piezometers finished in the till after the pumpout
of water from trenches 3-5 in the summer of 1976. Initial calibration
of the flow model was obtained by simulating the drawdown in piezometer
D-l after the pumpout of trench 5 from July 7-17, 1976.
Figure 12 compares observed water levels in piezometer D-l with
simulated values resulting from different specific-storage values. The
best fit was obtained from a specific-storage value of 16 x 10~6 cm~l.
-------
237
Table 2. Heads observed in February 1976 and February 1978
compared with results of best-fitting computer
simulations for isotropic and anisotropic conditions
[All values are in centimeters]
Piezometers
February 1976-
6-3
6-2
G-l
H-l
D2-1
0-2
D-l
13-1
1-2
12-1
1-1
14-1
J4-1
J5-1
J-l
J2-1
Sl-6
S2-1
February 19/8-
G-3
6-2
6-1
H-l
D2-1
D-2
0-1
13-1
1-2
12-1
1-1
14-1
J4-1
J5-1
J-l
J2-1
Sl-1
S2-1
Mean absolute
Observed
heads
382.
374.
320.
178.
47.
348.
562.
68.
169.
248.
dry
dry
214.
84.
545.
235.
260.
178.
387.
386.
327.
225.
47.
246.
285.
39.
137.
224.
dry
dry
221.
75.
561.
260.
261.
181.
departure
Calculated
heads from
best
computer
simulation
with
anisotropy
(table 1:
case 8,
part d)
254.
276.
291.
187.
46.
308.
629.
142.
218.
220.
91.
144.
202.
45.
537.
166.
221.
140.
179.
221.
277.
177.
26.
172.
322.
94.
155.
152.
67.
32.
200.
45.
526.
158.
221.
138.
Standard deviation
Absolute
departure
from
observed
heads
128.
98.
29.
9.
1.
40.
67
74.
49.
28.
>91.
>144.
12.
39.
8.
69.
39.
38.
208.
165.
50.
48.
21.
74.
37.
55.
18.
72.
>67.
>32.
21.
30.
35.
102.
45.
43.
1976=53':
1978=62.
1976=41.
1978=51.
Calculated
heads from
best
computer
simulation
with
isotropic
conditions
(table 1:
case 9,
part e)
359.
330.
307.
187.
47.
359.
560.
89.
173.
180.
-79.
-96.
219.
46.
524.
288.
262.
152.
360.
329.
304.
188.
15.
220.
268.
73.
152.
158.
-22.
-154.
220.
48.
520.
286.
262.
152.
Absol ute
departure
from
observed
heads
23.
44.
13.
9.
0.
11.
2.
21.
4.
68.
™ •
-.
5.
38.
13.
57.
4.
25.
27.
57.
23.
37.
32.
26.
17.
34.
15.
66.
—
-
1.
27.
41.
26.
1.
29.
1976=21.
1978=29.
1976=27.
1978=17.
-------
430-i
WEST
A
-3.8-
EAST
A1
< 05 420-
5/2 < 410-
-
Q LU
DQ
I- O
O400 —
390-
-2.9-
-1.5-t
Infiltration rate, in centimeters per year
-1.5 \-3.Q4 3.8 1 1.5-
Trenches
I
I I I
VL-J i I i—
• 414.0
Observed head in piezometer
D
Backfill
EvX-1 Weathered till
V/J Till with oxidized fractures
I I Unweathered till
• • • • •% • • *
Computed water-table
Water level in trench
414
Equipotential line, altitude, in meters
50 METERS
Figure 10. Cross section A-A' through north trenches showing computer-simulated
distribution of head in February 1976.
-------
WEST
A
EAST
A1
430-1
-3.8-
< 0,420-
02
I- LL
<£<410-
uj U
LLI JZ
Qoi
DQ
to
l_ UJ
-JO400-
390-
-2.9-
-1.5H
Infiltration rate, in centimeters per year
-1.5 H3.0H 3.8 1 1.5-
-3.8-
Trenches
-J I I L..J L..J
-406
-404
-402
Backfill
1<1^ Weathered till
V/\ Till with oxidized fractures
1 | Unweathered till
• 414.0
Observed head in piezometer
D
Test hole
• • • • y* • • • •
Computed water-table
-402-
Water level in trench
414
Equipotential line, altitude, in meters
2.5
50 METERS
PC
u
u
Figure 11. Cross section A-A1 through north trenches showing computer-simulated
distribution of head in February 1978.
-------
240
However, the simulation assumed that water was instantaneously withdrawn
from trench 5 to lower the water-surface altitude from 418.5 m to 416.5 m,
whereas this decrease actually occurred over a 10-day period. An
instantaneous drop in trench 5 water level would have caused the water
level in piezometer D-l to decline more rapidly than it did (see curve
in fig. 8)); thus, a more representative value of specific storage would
be between 8 x 10'6 and 12 x 10'6 cnr1, which agrees well with the
average value of specific storage obtained from laboratory tests. In
simulations over longer periods, the best fit to the water-level de-
clines in the deeper piezometers (D-2, F-3, E-3) were obtained when the
specific-storage value of the till was near 12 x 10~6 cm-1. Simulated
water levels in piezometers D2-1, D-l, and D-2, based on a specific-
storage value of 12 x 10~6 cm"1, are compared in figure 13 to values
observed during the 8 months after the pumpout of trenches 3-5. Con-
sistent with field observations, water levels in piezometers distant
from trench 5 showed no substantial declines.
As discussed previously, more than one combination of hydraulic con-
ductivity, specific storage, and infiltration value can be used to
correctly simulate head in the till after the pumpout of the trenches.
However, the simulations must approximate actual flows within the till
because (a) simulated and observed drawdowns in piezometers next to
trench 5 are controlled more by the trench water level than by infiltra-
tion from land surface, and (b) the values of hydraulic conductivity and
specific storage tested by model calibration are close to the values
determined in the field and laboratory.
i—i—i—i—i—i i i
Observed water level in piezometer D-1
6 8
10 12 14 16 18 20 22 24 26 28 30 1 3
JULY DATE' 1976
7 9 11 13 15 17 19 21 23
AUGUST
Figure 12. Comparison of observed water level in piezometer D-l,
July-August 1976, with computer-generated water levels
produced by changing values of specific storage.
-------
241
No attempt was made to calibrate the unsaturated functions of
hydraulic conductivity and volumetric soil-moisture content in relation
to soil-moisture tension because the entire section is simulated with
saturated or nearly saturated conditions.
After the pumpout of water from trench 5 in September 1976, water
level resumed the rising trend observed before the pumpout (fig.13).
Transient-state model results of the situation 25 days after the pumpout
was complete showed flow into trench 5 through the till to be 0.2 cm3/s
and flow out of the trench to be 0.4 cm-Vs. (As a verification, flow
out of trench 5 was manually calculated from Darcy's law to be 0.8 cm3/s
based on a hydraulic conductivity of 8 x 10~8 cm/s, a trench-floor area
of 1 x 10? cm^, and a gradient of 1:1.) The transient-state simulation
also indicated that, 6 months after the pumpout, flow from the till into
trench 5 had decreased to 0.1 cm3/s. Thus, the continued rise in trench
5 water level after the pumpout was probably not caused by flow of water
from the surrounding till. If the rise were caused by the slow drainage
of water from less permeable zones within trench 5 that were saturated
before the pumpout, the rise should have slowed with time, which was not
the case after the first few days (fig. 4). Therefore, it seems likely
that the rise was caused by continued infiltration of precipitation
through the cover.
COMPUTER MODELING OF GROUNDWATER FLOW TO NEAREST STREAM
In general, water from the trenches moves downward through the till
beneath the trench floor until it reaches the lacustrine unit 30 m below
land surface; from there it should move eastward to Buttermilk Creek.
However, if some of the trench water could flow to any of Buttermilk
Creek's tributaries that surround the trenches, both the length of the
flow path and travel time of trench water to reach land surface would be
much less. The shortest distance between a trench and an adjacent
stream, and also the greatest vertical relief, is at the north end of
trenches 3 and 4 (fig. 1). To evaluate this possible flow path, piezo-
meters were installed along the slope from trench 4 to the small stream
north of trench 4, in line with holes C, C2, and an older test hole (2C)
that had been installed by the site operator (fig. 1).
A finite-element grid (fig. 14) was designed to represent the cross
section through these piezometers, and the model was used to simulate
conditions observed in February 1978. The boundary conditions were the
same as those used for the section A-A1, perpendicular to the north
trenches (fig. 9), and values of hydraulic conductivity and specific
storage were obtained from the same section. The best simulations used
infiltration rates of 2 and 2.5 cm/yr along the steep slope and the
smooth surface near hole 2C and C, respectively, and these rates
approximate those rates used for scraped or sloping surfaces in the other
section. The average absolute deviation from observed heads in seven
piezometers was 14 cm. The shallow piezometer in test hole C (C-l) was
simulated as dry. Computed average head value near hole 2C compares
well with the observed head value, although the exact screened interval
in hole 2C is uncertain.
-------
<
O
420
419
418 -
417
65
> 416
y
§415
2
C3
_i 414
i= 413
> 412
<
Observed water level in trench 5
oi
Z
LU
I
411
410
409
< 408
D2-1
Observed water level in piezometer
Simulated water level in piezometer
ro
4*
ro
D-2
JUNE
JULY
AUG
SEPT
1976
OCT
NOV
DEC
JAN
FEB MAR
1977
APR
MAY
Figure 13.
Comparison of computer-generated water levels with observed water levels
1n piezometers D-l, D-2, and D2-1 during 8 months after pumpout of water
from trench 5.
-------
E
425-i
420-
0)415-
00
-
S410-
'•
Q
<405-
-y400-
yj P
D uj
r; LU
LJ0395H
390-
385
Fixed head for nodes below trench
water level, seepage nodes above
water level
Constant infiltration rate
along land surface
OO O
o o
o-o-o-
associated with piezometer
o o o o
o o
-O-
K- 0.85
Node
-O O-
Fixed head (zero pressure)
EXPLANATION
Backfill I I
Weathered till
Unweathered till with
oxidized fractures
-o-
Unweathered till
Alluvium
10 METERS
Figure 14.
Cross section B-B1 from trench 4 to small stream showing arrangement of nodes,
boundary conditions, and relative hydraulic conductivities used in the best-fit
model simulations. (Location of section is indicated in fig. 1.)
i
-------
244
Simulation of heads for February 1976, when water level was near the
top of trench 4, produced slightly positive heads in piezometer C-l,
although it had been dry at this time, and simulated heads in piezome-
ters C2-1 and C2-2 increased to almost 110 cm, or about 70 cm more than
the observed increase. Increasing the hydraulic conductivity of the
weathered unit in the model lowered the model heads near test holes C
and C2 and increased them along the slope, but, when the lower trench-
water levels of February 1978 were simulated, the decrease in piezome-
ters C2-1 and C2-2 was still 70 cm more than the observed change.
The change in simulated heads in piezometers C2-1 and C2-2 that
resulted from the lowering of model trench-water level approximated the
observed change when the model trench boundary was moved to 4 m south of
the monument that reportedly marks the end of the trench. (Exact loca-
tion of the trench boundaries is not known; the monuments that mark the
ends of trenches 1-4 were not placed immediately after completion of the
trenches but after the cover over trenches 1-4 had been modified into
individual mounds. Also, the ends of the trenches are sloped to allow
for bulldozer access (20), but the slopes may differ from their design.)
Moving the model boundary away from test holes C and C2 seemed logical
1n that the observed head decline in piezometers C2-1 and C2-2 was
considerably less than that observed in piezometers next to trench 5.
Piezometers in test hole B, finished 3 m north of the monuments on
trenches 2 and 3 (fig. 1), did not show any appreciable decline in water
levels after the pumpout of water from the sumps in those trenches.
The change in trench 4 water level between February 1976 and 1978
was reflected In wells 80 m and 175 m south of the north end; it is
possible that the water level in the extreme north end of trench 4 did
not fully respond to the pumpout of water from the sump 135 m to the
south. A similar lack of response was noted In the north end of trench
5 (18), although Kelleher (2) reported that water had seeped out along
the north end of trench 4 when the water level was at an altitude of 421
m. A simulation of the October 1975 water level in trench 4 (altitude
419 m) produced a slight pressure head (13 cm) in piezometer C-l, which
is consistent with Indications during the drilling of test holes C and C2
that a sand interval 2 cm thick, encountered at the depth of piezometer
C-l, was saturated.
Simulations with the trench boundary moved 4 m south produced an
average absolute deviation of 11 cm for February 1978 and 14 cm for
February 1976. In both simulations, heads did not change along the
slope toward the stream. The simulation of February 1978 suggests that
water leaving trench 4 at that time would not move to the nearby stream
and that there was a slight gradient in the water table toward trench 4
from holes 2C, C, and C2. Even when the water level approached the top
of trench 4 (altitude 418 m) as In February 1976, water from the trench
would move only about 8 m through the weathered till before it would
move downward through the unweathered till (fig. 15).
Discharge Into the stream was about 100 (cnP/di/m2, which is similar
to the values calculated for section A-A1, perpendicular to the north
-------
B
425-i
:<>
400
Infiltration rate, in centimeters per year
2.5
.............
Backfill
Weathered till
Unyveathered till with
oxidized fractures
Unweathered till
Alluvium
EXPLANATION
Observed head in piezometer
Test hole
Water level in trench
/
Flow path
Computed water-table
- 414 -
Equipotential line, altitude, in meters
10 METERS
i. ,
.! ,
• n
Figure 15. Computer-simulated heads along section B-B1 from
trench 4 to a small stream during February 1976.
-------
246
burial trenches. Most of this flow was from infiltration of precipita-
tion along the slope. The discharge rate increased by only 0.25
(cm-Vdi/np when the simulated water level in trench 4 was increased to
match the level for February 1976; this increase is attributed to the
slightly higher water levels near test hole 2C.
MODEL LIMITATIONS
In general, simulated heads closely matched observed heads when silt
and sand lenses of limited extent were represented. Although no con-
tinuous stratigraphic unit within the till is known that is more per-
meable than the till, a lens of silt and sand a few meters in length in
one direction would locally alter the flow system, as conceptualized
from simple modeling based on average till properties.
The simulations were based on the assumption (a) that the unweathered
till has a uniform hydraulic conductivity along the sections described
except for variations caused by known sand and silt lenses and by con-
solidation of the till from overburden, and (b) infiltration rates of
swamp areas differ from those of smooth, sloping surfaces. Although the
modeling results would be the same if infiltration rates were made
constant along the entire section and hydraulic conductivity were
increased throughout areas of lower heads. Such a change in hydraulic
conductivity does not seem correct, though, because cores from swampy
areas did not differ significantly from those taken from smooth, sloping
surfaces and because piezometers along smooth, sloping surfaces did not
consistently show higher hydraulic conductivity than those near swamps.
SUMMARY
Groundwater flow at the burial site is predominantly downward at
an average rate of generally less than 3 cm/yr. Results of computer
simulations support conclusions from laboratory tests and visual exami-
nation of cores that the till is generally isotropic but becomes less
permeable with depth as a result of two factors: (a) fractures caused by
weathering and a corresponding increase in hydraulic conductivity of the
till near land surface (these fractures become less abundant with depth
and are absent below 5 m), and (b) consolidation of the till by overbur-
den pressure, which reduces intrinsic permeability progressively with
depth. Simulation of four isotropic layers of differing hydraulic con-
ductivity was an adequate representation of these factors. Hydraulic
conductivity of the upper two layers (fractured till) was determined by
model calibration only.
The specific storage value of the till derived from computer simula-
tions of water-level trends in piezometers next to trench 5 was about 12
x 10-6 cm-l. Th^ is a factor of only 1.5 more than the average value
obtained from four consolidation tests of core samples.
-------
247
Some of the local variations in pressure head can be explained by
areal differences in infiltration rates. Adequate simulations were
obtained by assuming that infiltration rates on areas of smooth, sloping
surfaces were about 50 percent less than the rate of 3.8 cm/yr near
swampy areas.
A principal conclusion from the computer model runs is that water
leaving the trenches will not intersect the nearby streams unless the
trench-water levels rise to intersect the reworked till that forms the
trench covers. This conclusion assumes that none of the pods of layered
sediments encountered at various depths within the till are large enough
to extend from a trench to a nearby stream.
REFERENCES CITED
1. N. M. Fenneman, Physiography of Eastern United States (1st ed.), New
York, McGraw-Hill Book Company, Inc., 714 p. (1938).
2. W. J. Kelleher, "Water Problems at the West Valley Site," in M. W.
Carter, Bernd Kahn, and A. A. Moghissi, (eds.), Management of
Low-Level Radioactive Waste. V. 2, New York, Pergamon Press, p.
843-851 (1979).
3. D. E. Prudic, and A. D. Randall, "Groundwater Hydrology and
Subsurface Migration of Radioisotopes at a Low-Level, Solid
Radioactive-Waste Disposal Site, West Valley, New York, in M. W.
Carter, Bernd Kahn, and A. A. Moghissi, (eds.), Management of
Radioactive Waste. V. 2, New York, Pergamon Press, p. 853-882 (1979)
4. D. E. Prudic, Core Sampling Beneath Low-Level Radioactive-Waste
Burial Trenches 11-14, West Valley, Cattaraugus County, New York,
U.S. Geological Survey Open-File Report 79-1532, 55 p. (1979).
5. Mark Reeves, and J. 0. Duguid, Water Movement Through Saturated-
Unsaturated Porous Media—A Finite Element Galerkin Model, Oak
Ridge, Tenn., Oak Ridge Natl. Lab, ORNL-4927, 232 p. (1975).
6. R. G. LaFleur, Glacial Geology and Stratigraphy of Western New York
Nuclear Service~Center and Vicinity, Cattaraugus and Erie Counties,~
New York:U.S. Geological Survey Open-File Report 79-789, 17 p.
(1979).
7. R. H. Dana, Jr., R. H. Fakundiny, R. G. LaFleur, S. A. Molello, and
P. R. Whitney, Geologic Study of the Burial Medium at a Low-Level
Radioactive Waste~Burial Site at West Valley, New York. New York
State Geological Survey Open-File Report 79-2411, 70 p. (1979).
8. R. H. Fickies, R. H. Fakundiny, and E. T. Mosley, Geotechnical
Analysis of Soil Samples from Test Trench at Western New York
Nuclear Service Center, West Valley, New York, New York State
Geological Survey, NUREG/CR-0644, 21 p. (1979).
-------
248
REFERENCES CITED (Continued)
9. H. H. Cooper, Jr., J. D. Bredehoeft, and S. S. Papadopulos,
"Response of a Finite-Diameter Well to an Instantaneous Change Of
Water," Water Resources Research, v. 3, p. 263-269 (1967).
10. M. J. Hvorslev, Time Lag and Soil Permeability in Groundwater
Observations, Vicksburg, Miss., U.S. Army Corps Of Engrs., Exp.
Station, Bull. 36, 50 p. (1951).
11. D. A. Morris, and A. T. Johnson, Summary of Hydrologic and Physical
Properties of Rock and Soil Materials, as Analyzed by the Hydrologic
Laboratory of the U.S. Geological Survey, 1948-60, U.S. Geological
Survey Water Supply Paper 1839-D, 42 p. (1966).
12. W. R. Purcell, "Capillary Pressures—Their Measurement Using Mercury
and the Calculation of Permeability," Am. Inst. of Mining and
Metalurgical Engineers, Transactions, v. 186, Petroleum Development
and Technology, p. 39-48 (1949).
13. R. J. Millington, and J. P. Quirk, "Permeability of Porous Solids,"
Trans. Faraday Soc., v. 57, p. 1200-1207 (1961).
14. T. J. Marshall, "A Relation Between Permeability and Size
Distribution of Pores," Journal of Soil Science, v. 9, no. 1, p. 1-8
(1958).
15. S. N. Davis, and R. J. DeWiest, Hydrogeology, New York, John Wiley
and Sons, Inc., 463 p. (1966).
16. P. A. Domenico, and M. D. Mifflin, "Water from Low-Permeability
Sediments and Land Subsidence," Water Resources Research, v. 1, p.
536-576 (1965).
17. G. E. Grisak, and J. A. Cherry, "Hydrologic Characteristics and
Response of Fractured Till and Clay Confining a Shallow Aquifer,"
Canadian Geotech. Jour., v. 12, no. 1, p. 23-43 (1975).
18. J. 0. Duguid, and Mark Reeves, Material Transport Through Porous
Media—A Finite-Element Galerkin Model, Oak Ridge, TennT, Oak Ridge
Natl. Lab., ORNL-4928, 201 p. 11976).
19. R. A. Freeze, and J. A. Cherry, Groundwater. Englewood Cliffs, N.J.,
Prentice-Hall, Inc., 604 p. (197571
20. D. E. Prudic, Installation of Water-and Gas-Sampling Wells into
Low-Level Radioactive-Waste Burial Trenches. West Valley, New York,
U.S. Geological Survey Open-File Report 78-718, 70 p. (1978).
-------
STATE-OF-THE-ART IN MODELING SOLUTE AND
SEDIMENT TRANSPORT IN RIVERS
William W. Sayre
U.S. Geological Survey
Box 25046, Mail Stop 413
Denver Federal Center
Denver, Colorado 80225
ABSTRACT
This overview is structured around a comprehensive
general model based on the conservation of mass principle
as applied to dissolved and particulate constituents in
rivers, with a few restricted but more specific examples
that illustrate the state-of-the-art in modeling typical
physical, chemical, and biological processes undergone by
selected constituents in rivers. These examples include:
simplified one- and two-dimensional formulations focusing
on the hydrodynamic advection and dispersion mechanisms; a
two-dimensional biochemical oxygen demand-dissolved oxygen
model; a one-dimensional polychlorinated biphenyl model that
includes uptake and release of constituent by suspended sedi-
ment, and deposition and erosion of contaminated particles;
and a one-dimensional sediment transport model that accounts
for interactions between the flow and the bed, and is capable
of tracking dispersing slugs of sediment through cycles of
erosion, entrainment, transport in suspension and as bed
load, and burial and storage in the bed.
INTRODUCTION
There are many models in various stages of development that simu-
late one aspect or another of solute and sediment transport in rivers.
I will not attempt to discuss them all or even a representative sample.
There are a number of summaries, symposia, and bibliographies in the
literature that do this, for example [1, 2, 3, 4, 5, 6, 7]. Rather, I
will present a somewhat subjective overview that is structured first
around a fairly comprehensive but nonspecific general model, and then
some restricted but more specific examples that illustrate the state-of-
the-art in modeling typical physical, chemical, and biological processes
that are undergone by selected constituents in rivers.
Almost all mathematical models of river-transport processes are
essentially accounting procedures. Nearly always they are based on the
principle of conservation of one or more of the following entities:
249
-------
250
(1) Mass of dissolved or participate constituents, and transporting
media (water, sediment particles!; C2) momentum of the flowing water,
which fixes the basic flow properties, including the velocity, shear
stress, and pressure distributions; C3) energy, for example, thermal
energy in waste-heat loading and ice-related phenomena, or kinetic and
potential energy associated with fluid motion; and (4) probability, for
example, accounting for the sums of probabilities in a stochastic model.
A GENERAL MODEL
The starting point for modeling solute and sediment transport
processes in rivers is usually the conservation of mass principle. The
other conservation prtnctples are more likely to enter in supporting
roles. The basic three-dimensional p-D) conservation of mass equation
for a dissolved or particulate constituent that is being transported by
the flowing stream is:
fed*) - fcUx £} - fefcy |£) - fj(.z jf) - is,
Advection Turbulent diffusion Internal
••*——^————^—— sources
Hydrodynamic and
sinks
CD
where
C = concentration of solute or particulate constituent,
x, y, z = distances in longitudinal, vertical and transverse
directions,
u, v, w = time-averaged local velocities of solute or particulate
constituent in x, y, z directions,
e , e , EZ = local coefficients of turbulent diffusion for solute
y or particulate constituent in x, y, z directions,
ZS. = sum of internal source and sink terms,
t = time.
The advection and turbulent diffusion terms on the left-hand side
of Eq. 0) represent the hydrodynamic transport and mixing mechanisms
that occur within the river channel. They must be known if Eq. (1) is
to be solved. Usually they are estimated by some empirically based
computational method or determined by measurement. It should be noted
that particulate constituents, unless they consist of or behave like
fine suspended sediments, cannot be assumed to have the same local
velocities and turbulent diffusion coefficients as the transporting
fluid. In particular, if deposition, burial, scour and re-entrainment
of particulate constituents occur, an additional conservation equation
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251
should be introduced for the material stored in the bed. Except for
determination of water-surface elevation and the cross-sectional average
velocity in the downstream direction, methods for solving the complete
momentum and energy equations to obtain local velocities and diffusion
coefficients in natural rivers are still in their infancy. This is
mainly because of complications associated with irregularities of
channel shape and alinement, and with the nonlinearity of the equations.
The internal source and sink terms on the right-hand side represent
time rates of increase Cor decrease) of solute or particulate concen-
tration due to physical, chemical, or biological reactions and transfer
processes occurring within the flow cross section. Examples of these
mechanisms are radioactive decay, uptake and release of constituent by
suspended sediment, photolysis, hydrolysis, oxidation-reduction reac-
tions, bacterial degradation, and accumulation and depuration by
biological organisms. The rates of these processes depend on factors
such as concentrations of the constituent and substances with which it
reacts, solar radiation, and variables which define the state of the
aquatic environment, such as temperature, turbidity, pH, and turbulence
level. External source and sink mechanisms, such as evaporative trans-
fer, uptake and release of constituents by bottom sediments, scour and
deposition of contaminated sediment, are represented as boundary con-
ditions at the water surface or river bed and banks. The rates of these
processes may depend on variables that define the state of adjacent
environments, for example, wind velocity, air temperature and humidity,
dynamics of flow-sediment interaction at the bed, in addition to the
factors listed previously for the internal sources and sinks. Fortu-
nately, many source/sink mechanisms can be simulated acceptably by
first-order-reaction kinetic models.
If there are reactions between different constituents, or if
exchange of a constituent occurs between different phases of the aquatic
environment (for example, uptake and release of dissolved constituent by
suspended or bottom sediments), a set of coupled equations, with one
equation for each constituent and (or) phase, may be formulated. These
equations, each similar to Eq. 1, are coupled by source/sink terms that
specify the rates of reaction or exchange. To obtain spatial and tem-
poral concentration distributions for the different constituents and
phases, the entire system of equations must be solved. Formulations of
coupled-system models for biochemical oxygen demand (BOD) and dissolved
oxygen (DO), and polychlorinated biphenyls (PCB's) are presented in the
next section.
SELECTED RESTRICTED FORMS OF GENERAL MODEL
Two-Dimensional and One-Dimensional Formulations
Except in the initial mixing region just downstream from a con-
centrated source of constituents, for example, a simple pipe outfall,
adoption of 3-D models like Eq. (1) for use in rivers is rarely either
justified or feasible. Firstly, substances transported by rivers tend
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252
to become mixed, first over the depth of flow Cor possibly across the
channel 1n situations where there is persistent density stratification),
and then throughout the entfre channel cross section. Thus, after
distances that are typically on the order of 50 to 100 times the depth
and 100 to 500 times the vridth, the mixing can be modeled first as a
two-dimensional (2-D) and eventually as a one-dimensional (1-D) process.
Secondly, neither 3-D modeling of the momentum or energy equations nor
any other computational method 1s far enough advanced to provide suf-
ficient information about the local velocities, particularly the com-
ponents normal to the mean flow, and diffusion coefficients. Thirdly,
even with advances in modeling and increased computer storage capacity,
complete solutions for 3-D models are likely to remain prohibitively
expensive and time-consuming, except for the most idealized and
simplified cases, such as uniform flow in a prismatic channel. Using
integral transform methods, Cleary and Adrian 18] have obtained analyt-
ical solutions for the 3-D advection-diffusion equation with constant
longitudinal velocity, zero secondary velocities, and constant diffusion
coefficients, for a solute in a rectangular channel with uniform flow.
In reducing general 3-D conservation equations to simplified 2-D
and 1-D forms, integrating the equations over the depth of flow and (or)
across the width of the channel , with the use of appropriate boundary
conditions, is preferred rather than simply neglecting the advection and
diffusion terms for whatever dimension is being eliminated, and assuming
for convenience that the remaining velocities and diffusion coefficients
are constant throughout the depth or cross section. Following the
appropriate integration procedure, one finds that some terms are elim-
inated naturally due to boundary conditions, and that other terms may
adopt a new form 1n the reduced equation. Also, one is more likely to
obtain correctly formulated external source/sink terms when they arise
naturally from applying boundary conditions to the integrated 3-D
equations. Following the concepts of Fischer 19], integration of Eq.
0) over the depth of flow leads to the 2-D conservation of mass equation:
Advection Dispersion
* S5* + rS* . (2)
Internal External
Sources and sinks
Similarly, Integration over the cross- sectional area leads to the 1-D
formulation:
•* * • <3>
Advection Dispersion Internal External
Sources and sinks
-------
253
—d —A
In Eqs. C2) and (3) C ) and ( } represent averages taken over the
depth, d, and the cross-sectional area, A, respectively. Also, the new
coefficients E , E , and K no longer represent turbulent diffusion
alone. They now represent longitudinal and transverse dispersion from
the combined action of vertical and cross-sectional variation of the
local velocities u and w and turbulent diffusion.
Formulating Eq. (2) and the depth-integrated continuity equation
for water in a 2-D orthogonal curvilinear coordinate system, and re-
placing the transverse distance z by the transverse cumulative discharge
measured from one bank
qc = / d Od mz dz (4)
ZL
leads to the stream-tube version 110] of the 2-D equation:
3t m
A
A definition sketch for the coordinate system is shown in Fig. 1. Eq.
(5) is more suitable than Eq. (2) for solute-transport modeling appli-
cations in natural rivers. The transverse advection term, which does
not appear in Eq. (5), has been formally eliminated by introducing the
continuity equation, and not simply neglected. Another advantage of Eq.
(5) is that the new transverse coordinate variable, q , varies within
a fixed range from zero to the total river discharge, Q, irrespective
of whether the river meanders, or contracts and expands in width.
In Eqs. (4) and (5), m and m are metric coefficients to correct
for differences in transverse and longitudinal distances resulting from
curvature in channel alinement. They can be evaluated from the stream
tube geometry. Except in abruptly converging or diverging sections and
in sharply curved reaches, little error results from assuming that they
are equal to unity. The longitudinal dispersion term containing E is
not included In Eq. (5), because it contributes little to longitudinal
dispersion in comparison to the advection term. However, there is some
evidence [11, 12] that retaining the term and assigning small values of
E helps to control troublesome overshoot problems in numerical solu-
tions of Eqs. (2) and (5).
Finite-difference models based on Eq. (2) [11] and Eq. 5 [12]
have been used successfully to simulate both unsteady and steady-state
dispersion of dye and waste heat in the Missouri River. The alter-
nating-direction implicit (ADI) method was used in both cases. Analyt-
ical solutions of a simplified steady-state form of Eq. (5) for a con-
O J
servative solute, wherein the quantity m d u E is assumed to be a
J\ £
constant or longitudinally varying diffusion factor, have been used with
good results In a number of cases [10, 13, 14]. Analytical solutions of
-------
254
Horizontal distances
along coordinate surfaces
LOA= x
OB
z axis
Transverse
coordinate surfaces
x axis
Longitudinal
coordinate surfaces
Right bank
PLAN VIEW
Longitudinal
coordinate surfaces
y = VB (x. z. t)
y = ys (x, z. t)
y =0
SECTION D - D
Fig. 1. Coordinate system for natural channels,
-------
255
Eq. (5) also have been obtained for transient cases that have been
simplified still further by assuming the channel to be straight and
prismatic, and that u and E are constants 115].
Analytical and numerical solutions of the 1-D model, both with and
without source/sink terms, have been widely and successfully used, espe-
cially in thermal energy (temperature} and BOD-DO models 116, 17, 18, 19,
20]. However, in modeling instantaneous concentrated-source dye experi-
ments that simulate accidental releases of slugs of conservative soluble
pollutants, significantly improved results commonly are obtained by cou-
pling the 1-D advection-dlffusion equation (Eq. (3) with no source/sink
terms) for the main stream to a dead-zone equation 121, 22] as follows:
m . nA m
at
aC,
The subscripts m and d denote main stream and dead zone respectively;
K = coefficient for mass transfer between main stream and dead zone (in
units of velocity); and P = average length of interface, in plane of
cross section, between the subareas A_ and Ad. Coupling with the dead-
zone equation simulates the effect of^separation zones caused by bank
and bed irregularities, which trap some of the dispersing material and
release it slowly back into the main stream, adding to the tails, and
consequently the skewness of the concentration-distribution curves, as
observed in many dye experiments 123, 24]. In thermal energy and BOD-DO
applications, temperature and concentrations usually change slowly
enough that addition of the dead-zone equation, and often even inclusion
of the longitudinal dispersion term, Is unnecessary.
Two-Dimensional Biochemical Oxygen Demand-Dissolved Oxygen Model
The BOD-DO model is selected for illustration because it exem-
plifies a coupled-system model that includes both internal and external
source/sink terms. Also, its 1-D form has a long history of successful
application dating back more than 50 years to the Streeter-Phelps equa-
tions [25], the most rudimentary form of the model. The stream-tube
version of the 2-D model formulated here has not yet been applied.
However, the ADI finite-difference techniques used in solving Eqs. (2)
and (.5) should work, after modification to include the source/sink
terms. The 2-D model would be appropriate for use in wide rivers, where
the time scale for mixing across the channel equals or exceeds the time
scales for BOD assimilation and DO replenishment.
The model consists of a coupled set of three equations for first-
stage carbonaceous BOD (CBOD), nitrogenous BOD CNBOD), and dissolved
oxygen (DO), in most respects, it follows Thomann's 126] 1-D formu-
lation. The dependent variables are:
-------
256
L = depth-averaged CBOD concentration, in mg/A;
L = depth-averaged NBOD concentration, in mg/A;
C = depth-averaged DO concentration, in mg/Ji.
The three equations are:
(8)
Bacterial oxidation;
Sedimentation, Adsorption
3t
— t qu _ Q
m
3X
} = -K
Bacterial
oxidation
(9)
Bacterial
oxidation
Surface
reaeration
(10)
SpCx,p,t) - SR(x,p,t)
Removal by
benthal
demand
Production by
plant
photosynthesi s
Removal by
plant
respiration
On the left-hand side of the equations, p = qc/Q is the cumulative
discharge normalized by the total river discharge Q, and D is the
p J p
transverse diffusion factor, taken as the average value of m d u E /Q
/» £
and assumed to be constant across, but not necessarily along, the
channel. On the right-hand side, KJ, K , 1C, and K are all first-
order rate coefficients, K. and K., are biological-rate coefficients
that depend on the properties of tne oxygen-demanding pollutants and the
river water. K relates to settling (or resuspension) and adsorption,
and, consequently, to the properties of the pollutant and the turbulence
in the river. The reaeration coefficient, K , depends on the rate of
surface renewal and water temperature; it hai been related to river
depth and velocity, as well as temperature, by several empirical equa-
tions. C is the saturation concentration of DO. The functional forms
of the soDrce/sink terms SB, Sp, and SR are not well understood. The
benthal oxygen-demand rate, SB» relates to the aerobic decomposition of
organic matter in the surface layer of bottom deposits. The net photo-
synthetic oxygen production rate, Sp-SR, is highly variable and depends
primarily on the amount of solar raaiant energy received, and its
-------
257
distribution over the river depth, nutrient concentrations, dissolved
oxygen, and water temperature. Point and distributed sources of BOD,
from waste outfalls, tributary Inflows, runoff, and so forth, can be
represented by boundary conditions at and along the banks (p=0,l).
As yet there are no operational 2-D BOD-DO computer models. How-
ever, some of the better-known computer models that are available for
1-D modeling of BOD and DO In rivers are DOSAG-1 127]; SNOSCI and SRMSCI
[28, 29]; QUAL-I 130, 31]; QUAL-II 132]; and RECEIV [33]. With the
exception of DOSAG-1, these programs also can model other selected
water-quality constituents.
One-Dimensional Polychlorinated Biphenyl Model
Although this model has not yet been developed beyond the Initial
formulation stage, It 1s presented here because it includes uptake and
release of constituent by suspended sediment, and deposition and erosion
of contaminated sediment particles. Further investigation may well
indicate that some revisions, additions, or deletion of terms, and so
forth, should be made. However, for the purpose of this presentation, a
hypothetical chemical, say ABC, would serve about as well as PCB.
Because PCB in both dissolved -and adsorbed form, together with
suspended and bottom sediments, are known to be important in determining
the distribution of PCB in the aquatic environment, it is appropriate to
formulate a coupled-system model consisting of four equations. The
dependent variables for the four equations are:
C(x,t) = concentration of dissolved PCB, yg/Jl, assumed to be
constant throughout the cross section;
M(x,t) = concentration of suspended sediment, mg/*,, assumed
to be constant throughout the cross section;
rM = C (x,t) = concentration of PCB adsorbed by suspended
p sediment, ng/fc, assumed to be constant throughout
the cross section;
rhM.=C.(x,t) = concentration of PCB adsorbed by bottom sediment,
vg/l, assumed to be constant throughout the active
bottom layer of thickness, h(x), that is involved in
the erosion-deposition process;
where
r(x,t) e adsorption ratio for suspended sediment, (iig adsorbed
PCB)/(nig suspended sediment), assumed to be constant
throughout the cross section;
Mh = concentration of bottom sediment in active bottom
layer g/A, assumed to be constant;
rh(x,t) = adsorption ratio for bottom sediment, (yg adsorbed
PCB)/(g bottom sediment), assumed to be constant
throughout active bottom layer.
-------
258
The governing equations are:
iC + n l£ _ 1 3_ r*v iCi
3t 3x ~ A~ 3X
= -K, CM(rc-r) + l^rM + W - Sb (11)
(Hi) - U |^rM) - Jfe [AK, fj (rM)J
V. .
- K, CM(rc-r) - K2rM + Wp - / rM + Y £ rbMb (12)
ft W=M-^bV (14)
The variables t, x, U, A, and K , were defined previously. The
remaining quantities to be defined are:
K, = adsorption coefficient, Ji/pg PCB-day;
r = limiting adsorption ratio at full adsorptive capacity,
tug adsorbed PCB)/(mg suspended sediment);
Kp = desorption coefficient, day" ;
W(x,t) = rate of addition of dissolved PCB from nonpoint external
sources (for example, runoff and atmospheric fallout),
ug/ji-day, assumed to be instantaneously mixed throughout
the cross section;
Sb(x,t) = net rate of removal of dissolved PCB due to accumulation
and depuration by biological organisms, "«/»-^"-
W_(x,t) • rate of addition of particulate PCB from nonpoint
" external sources, vg/fc-day, assumed to be instantaneously
mixed throughout the cross section;
YS = settling velocity of suspended sediment, m/day;
d(x) = mean depth of channel, m;
Y = entrainment rate coefficient for bottom sediment, day" ;
h(x) = mean thickness of active bottom layer involved in
erosion-deposition process, m.
A definition sketch for the model is shown in Fig. 2.
Eqs. (11) through (14) are written for a particular size fraction
of fine sediment, for which it is assumed that: (1) Concentration
distribution of suspended sediment throughout the cross section is
-------
11
h
U
W. W,
(Additions from nonpoint
external sources)
C - +~rM
(Release and uptake
by suspended sediment)
Vs
rM (Settling)
d
^(Temporary storage in the bed)
Fig. 2. Definition sketch for one-dimensional PCB model.
U
(Entrainment)
ro
un
-------
260
uniform; C2) mean velocity and longitudinal dispersion coefficient are
the same as for water; and (3) settling velocity, V_, limiting adsorption
coefficient, r , and rate coefficients (KI.KO.Y) are all constants. If
all size fractions are to be considered, Eqs. (12)-(14) are written
separately for each size fraction, and the terms, K-,CM(r -r) and K2rM, in
Eq. (11) are summed over all size fractions. No provisions were made in
Eqs. (11) through 04) for either adsorption or desorption to occur
directly between the water column and the bottom sediment, or for bed-
load transport. Also, an additional set of equations could be written
to simulate exchange mechanisms for PCB within the bed. Such equations
would be required to forecast long-term Cseveral years) changes in PCB
concentration following curtailment of its use.
Modeling of coupled sediment-contaminant systems (such as the PCB
model) Is still very new. The only operating models of which the writer
is aware are a group of models developed by Onishi and his colleagues
134, 35, 36] at BatteH e Pacific Northwest Laboratories.* These consist
of 1-D and 2-D models that include equations for dissolved constituent,
suspended sediment, adsorbed constituent, and exchange mechanisms that
differ somewhat from those postulated in the PCB model. These models,
which include both finite-element and finite-difference methods, have
been used for radionuclides In the Columbia and Clinch Rivers, and for
Kepone in the James River. The development of this class of models is
hindered by the lack of reliable physically-based relationships and
coefficients for computing rates of exchange of sediment between the
active bottom layer, the bedload zone, and suspension in the flow. In
addition, existing bedload transport formulas are not sufficiently
reliable, except possibly for steady flow in straight uniform channels.
One-Dimensional Sediment Transport Model
The model formulated 1n this section exemplifies the type of
sediment-transport model that needs to be developed further before the
modeling of sediment-contaminant systems can come of age. In addition
to predicting transport rates and hydraulic resistance, such a model
would account for Interactions between the flow and the bed for non-
uniform and slowly-varying unsteady flow, including the development of
bed forms; erosion and deposition by particle size class, possibly
leading to degradation or aggradation of the bed; and changes in sed-
iment size gradation and armoring of the bed. The model also needs to
be capable of tracking dispersing slugs of sediment through cycles of
erosion, entrainment, transport in suspension and as bedload, deposi-
tion, and burial and storage in the bed.
The use of trade name is for descriptive purposes only and does
not imply endorsement by the U.S. Geological Survey.
-------
261
A definition sketch for a model that meets the above requirements
is shown in Fig. 3. The model consists of the following system of
equations.
Equation of motion for water:
-------
u
QM
Suspended load zone
Qb
Inactive deposition layer
Original bed material
Datum
Fig. 3. Definition sketch for one-dimensional sediment transport model
ro
en
IN3
-------
263
Continuity equation for water:
H^ v ^iv \ /
0 A 0 A
Continuity equation for sediment in suspended load zone:
1M. + n — - ! — /A" aMx - W " M ^ KW
at ax A ax
Settling Diffusive (17)
transfer
Continuity equation for sediment in bedload layer:
aMD. an.
Wa -^ + -=§. = WVJ^ -
o U pA b O,
Settling Diffusive
transfer
YWhM (18)
Settling Entrainment
Continuity equation for sediment in active bed layer:
Wr\FYi
C/IH _ i ii| ij
P,
- YWhMb , (19)
Settling Entrainment
Bedload transport:
Qb = f (U,d,D,...) (20)
Eqs. (17) -(20), the sediment equations, are all written separately for
each size fraction; following solution, the results are summed over all
size fractions. The sum of the rates of change of masses, m . , of the
different size ranges of sediment in the active bed layer from Eq. (19),
is related to the rate of change of bed elevation, z^, by the equation:
az. - am.
where n = porosity of the bed, and p.. = mass density of particles in the
i-th size range.
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264
Quantities in Eqs. (15)-(20) not heretofore defined, or with new
definitions, are:
g = acceleration of gravity; a_
eJZu
S = longitudinal bed slope - ^-;
Sf = friction slope determined from a hydraulic resistance
equation;
M = average volumetric sediment concentration; and the subscripts
a, BL and b denote, respectively: at the level a above the
bed, in the bedload layer, and in the active bed layer;
W = active bed width, over which deposition or entrainment
1s occurring;
K = coefficient governing rate of diffusive transfer of sediment
between suspended and bedload zones, in units of velocity;
Y = coefficient governing rate of entrainment of sediment from
active bed layer into bedload layer, in units of reciprocal
time;
D = characteristic sediment diameter.
Eq. (20) is a symbolically stated formula for bedload discharge.
Any one of a number of formulas 137], or even an empirical rating rela-
tionship developed for the reach being modeled, could be used.
In addition to the system of equations (15)-(21), the model also
needs to include an accounting procedure for computing and updating the
size distribution of the bed material. Selective net removal of finer
size fractions by erosion tends to coarsen the bed, leading to an
increased resistance to erosion, or armoring, of the bed.
Probably the weakest link in the model described in the preceding
paragraphs is the lack of Information on the transfer and entrainment
coefficients, K and y Capability for predicting a and h, the thick-
nesses of the bedload and active layers, 1s not much better.
The model as proposed in Eqs. (15)-(21), with the bed material
accounting procedure, is a modification of the sediment transport and
armoring model developed by Bennett and Nordln [38], which has been
applied to reaches of the East Fork River 1n Wyoming, the East Fork of
the Carson River In Nevada 139], and the Rio Grande River downstream
from Cochiti Dam in New Mexico [40]. Significant modifications include
the replacement of a single bedload-zone equation by the bedload-layer
and active bed-layer equations, and modifications of the settling,
diffusive transfer, and entrainment terms on the righthand sides of Eqs,
(17)-(19). The Bennett-Nordin model is in some ways similar to the U.S.
Army Corps of Engineers HEC-6 model [41] for simulating scour and depo-
sition in reservoirs and rivers. However, HEC-6 neither explicitly
separates the bedload from the suspended-load transport nor simulates
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265
the exchange of sediment between suspension in the flow and storage in
the bed, so that dispersing slugs of sediment can be tracked through
repeated transport-deposition-burial-resuspension cycles. The Bennett-
Nordin and HEC-6 models apply only to cohesionless sediment; erosion and
deposition mechanisms for cohesive sediments are somewhat different.
Transport models for cohesive soils have been developed by Ariathurai
and Krone 142] , and Odd and Owen [43] .
In most instances, significant simplifications of the sediment
transport model are possible. For gradually varying subcritical flow,
bed transients are known to propagate at a much slower rate than either
water-surface transients 12] or concentration transients for suspended
or dissolved matter. If one's interest is restricted to long-term
phenomena, a quasi-steady state for the liquid phase, suspended-load
zone, and bedload layer can be assumed, wherein A, U, M, and MB, are
taken to be invariant throughout the time interval At = x/U, wnere At
and AX are the time and distance steps in the numerical computation.
Eqs. 05} and 06) then reduce to the steady-state non-uniform flow
equation for computing water surface profile 144]:
dd _ o f
-
where Fr is the Froude number, and the time derivative terms in Eqs. (17)
and 08) are eliminated. Also, in long-term studies, the longitudinal
dispersion term in Eq. (17) can be eliminated, because longitudinal
dispersion in the suspended load zone is negligible in comparison to
that due to storage in and resuspension from the stream bed. In sit-
uations where the changing bed geometry is of much more concern than
exchanges between and distinguishing among the modes of transport in the
different zones, further simplification can be achieved by combining Eq.
(18) with Eq. (17) and (or) Eq. (19). This results in an equation
involving transport of the total sediment load: Qs = QM + Qfa.
CONCLUSIONS
This review of solute- and sediment- transport modeling in rivers
concludes with an attempt at rating the state-of-the-art for various
classes of models. State-of-the-art is taken here to mean mainly
overall reliability in application. A steady or quasi-steady flow, for
a period that is not less than the duration of one time step, throughout
the reach that is being modeled, is assumed in all cases. State-of-the-
art ratings on a scale of 1 to 10 (in order of improving state),
assigned subjectively by the writer to various classes of models, are
listed in Table 1. The classes are defined by groups of the categories
of characteristics and phenomena, Identified by capital letter in the
listing beneath the table.
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266
Table 1 .—Subjective Rating of State-of-the-Art, on 1 to 10 Scale
(in order of improving state), for Various Classes of
River Transport Models
Categories^ of Characteristics
and Phenomena included in Model
A, C, E
B, C, E
B, C, E, G
B, C, F, G
B, C, F, H
B, C, F, G, H
A, D, E
B, D, E
B, D, E, G
B, D, F, G
B, D, F, H
B, D, F, G, H
A, D, F, J
B, D, F, I, J
B, D, F, H, I, J
B, D, F, G, H, I, J
1-D
Model
8
8
7
6-7
6-7
5-7
6
6
5
4-5
4-5
3-5
5
3-4
2-3
2
2-D
Model
7
7
6
4-5
3-5
3-5
4
4
3
2-4
2-4
2-3
3
1-2
1-2
1
Listing of categories
A. Conservative constituents
B. Non-conservative constituents
C. No density (buoyancy) effects
D. Density (buoyancy) effects
E. Single system
F. Coupled system
G. Gas or heat transfer at water surface
H. Chemical/biological reactions
I. Adsorption/desorption by sediment
J. Deposition and entrainment of sediment
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267
In the listing of categories, the designations single system and
coupled system are used in the sense defined by Thomann 126]. In a
single system, usually simulated by a differential equation for a
single constituent, there is a functional relationship between an input
(discharge of constituent at source) and a single output (concentration
of constituent in river). Coupled systems are simulated by two or more
differential equations, with interacting source/sink terms.
Neither the categories nor the groupings listed in the table
purport to be exhaustive. However, they do represent a large proportion
of present river-transport models.
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Board, Simulation of Water Quality in Streams and Canals: Theory
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Ml
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(January, 1976).
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Transport in Surface Waters-'- Transport of Sediment and Radionuclides
in the Clinch River, BNWL-2227, Battelle Pacific Northwest
Laboratories, Richland, Washington (July, 1977).
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River" (Chapter VII), and "Mathematical Simulation of Sediment
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River Basin, Appendix A to the EPA Kepone Mitigation Project Report ,
Battell e Pacific Northwest Laboratories, Richland, Washington
(June, 1978).
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Discharge Formulas, p. 190-230, ASCE Task Committee for the
Preparation of the Manual on Sedimentation of the Sedimentation
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Engineering Practice-No. 54 (1975).
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Armouring," Hydro! ogtcal
555-569 (December, 1977).
and Armouring," Hydro! ogtcal Sciences-Bulletin, Vol. 22, No. 4,
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271
39. T. Katzer and J. P. Bennett, Sediment Transport Model for the East
Fork of the Carson River, Carson Valley, Nevada, Open File Report
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M.S. Thesis, Colorado School of Mines (May, 1980).
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Sediment Transport," Journal of the Hydraulics Division, ASCE,
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43. N. V. M. Odd and M. W. Owen, "A Two-Layer Model for Mud Transport
in the Thames Estuary," Proceedings, Institution of Civil
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44. F. M. Henderson, Open Channel Flow, The Macmillan Co. (1966).
-------
AN OVERVIEW OF RESUSPENSION MODELS:
APPLICATION TO LOW-LEVEL WASTE MANAGEMENT
J. W. Healy
Los Alamos Scientific Laboratory
P. 0. Box 1663
Los Alamos, New Mexico 87545
ABSTRACT
Resuspension is one of the potential pathways to man for
radioactive or chemical contaminants that are in the
biosphere. In waste management, spills or other surface
contamination can serve as a source for resuspension during
the operational phase. After the low-level waste disposal
area is closed, radioactive materials can be brought to the
surface by animals or insects or, in the long term, the
surface can be removed by erosion. Any of these methods
expose the material to resuspension in the atmosphere.
Intrusion into the waste mass can produce resuspension of
potential hazard to the intruder. Removal of items from the
waste mass by scavengers or archeologists can result in
potential resuspension exposure to others handling or working
with the object. The ways in which resuspension can occur are
wind resuspension, mechanical resuspension and local
resuspension. While methods of predicting exposure are not
accurate, they include the use'of the resuspension factor, the
resuspension rate and mass loading of the air.
INTRODUCTION
Resuspension and inhalation of the resuspended material is a
potential pathway whereby radioactive materials can be transferred from
the ground, or other surface, to man. In general, it is of greatest
importance for those radionuclides that are strongly discriminated
against in the food pathway. If the radionuclides are strongly taken up
by foods, and food from the contaminated area is used, the contribution
to the dose from resuspension and inhalation is usually small.
In this paper, we discuss the application of this pathway to the
management of low level wastes and describe briefly some of the methods
of estimating air concentrations from resuspended material. More
detailed discussions of the technical details are available in the
literature [1,2].
273
-------
274
RESUSPENSION
Before entertaining the question of how resuspension is involved in
low-level waste management, let us consider what resuspension is.
Technically, it is the suspension of a material in a fluid after this
material has once been deposited on a surface from a previous
suspension. Thus, j^esuspension. This fluid can be either air or water
and resuspension in both can be of importance in analyzing potential
problems from low-level wastes. Air suspension is used in estimating
quantities inhaled and water suspension is important in the movement of
radioactive particles in streams or other bodies of water. However, for
this discussion, we will confine our attention to the problem of
suspension in air.
It should be apparent that not all sources meet the technical
definition of resuspension given above. For example, if a radioactive
material is spilled on the surface or if it moves upward to the surface,
it has not been deposited by a previous suspension and should not really
be considered as £esuspension. However, because the mechanisms for
suspension are similar regardless of the method of contamination, it is
convenient to change the definition to include all cases in which the
material becomes airborne from the surface.
It should also be made clear that the suspensions that we are
concerned with are not the true colloidal suspensions that the chemists
are used to but. rather, are particles kept in suspension by the
turbulent movement of the fluid in which they are carried. Thus, an
aerosol resulting from resuspended materials can contain much larger
particles than could be indefinitely suspended in still air. In fact,
those of you who have been caught in sand storms with strong winds will
realize that large particles that can sting when they hit can be carried
under conditions of highly turbulent winds.
RESUSPENSION AND LOW-LEVEL WASTES
Now, how does resuspension fit into the analysis of low-level
waste? Obviously, if we do our jobs perfectly it will not become a
factor. There are. however, complications such as accidents or spills
that must be considered and accounted for as well as a few potential
complications that could arise with the long-term behaviour of the
burial grounds.
In figure 1, we find an elementary schematic drawing of a few ways
that people can be exposed from low level waste operations. A number of
the pathways to man have no connection with resuspension but I have
included some of these as dotted boxes and lines for the sake of
completeness.
The transport accidents and spills can result in contaminated
surfaces that can give rise to resuspension and inhalation of
-------
Direct
radiation
• —••• <^^W —••• ^"W ~~H
Milk I
•— r —
I
^ Animal
L-i-
Suseensionj Plant i
I I I
ICQD sport _,
Fig. 1. Schematic drawing of pathways to man from low-level waste
operations. Solid lines are related to resuspension while dotted lines
are other pathways.
ro
^j
en
-------
276
radioactive materials by man. These, however, should be of relatively
minor concern, from the resuspension point of view, because they are
usually noted and the area cleaned before serious resuspension can
occur. There may be some long-term build-up of activity at places such
as loading docks where the material could be resuspended, but this
problem can be controlled by monitoring. These potential problems are
encountered in the operational phase of the disposal area and can be
handled by "institutional" methods because the workers are already
there.
However, there are later opportunities for movement of
radioactivity to surfaces where resuspension could be a factor. Some of
these are shown in Fig. 1. For example, transport of the activity to
the surface by water, ants, burrowing mammals or other means can produce
a source exposed to winds or other disturbances, that could result in
resuspension. The possibility that could expose the largest number of
people to small quantities starts with transport to an aquifer. The
water from this aquifer could be withdrawn from wells and used for
irrigation, with the radioactive materials adsorbing on clays or humus
to gradually produce large contaminated areas. If the aquifer empties
into a stream, radioactive materials will accumulate in sediments which
can be deposited on flood plains or the water can be used for irrigation
with the same result as the use of well water. One aspect of
resuspension in these cases which leads to the food pathway is the
resuspension and deposition on food plants. Experience with nuclides
that are only poorly taken up by plants indicates that this can be an
important factor.
Another series of pathways involving resuspension occurs in
disposal areas containing long-lived radionuclides. Here we can
postulate eventual removal of the cover material by erosion, either by
wind or water. When this occurs, we are left with a contaminated area
to depths of several meters, augmented by surface contamination in the
surrounding area caused by the materials removed and carried downwind or
downstream. This radioactive material can be resuspended by winds or by
movement in the area. This is a futuristic scenario depending upon the
particular site of disposal. With average erosion rates, it could occur
some thousands or tens of thousands of years from now although gully
erosion could occur much faster but on a limited scale. In some areas,
the net erosion pattern is such that the disposal area will be simply
buried deeper, but the present tendency to look for areas that are "high
and dry" to site disposal areas could eliminate this possibility because
the high sites with a long distance to the water table are just those
that will have a negative-gain erosion potential.
The intrusion case, i.e. where an individual digs into an area,
either knowingly or unknowingly, results in resuspension caused by his
activities. In an enclosed space this can result in high concentrations
of dust and associated radioactive materials with the individual
inhaling these materials. However, this exposure will be limited to the
-------
277
individual or individuals concerned unless contaminated items or
materials are removed and taken to a place where others can be involved.
This possibility brings up the case in which an archeologist.
either amateur or professional, investigates these disposal areas for
"treasures" of the past. Old burial grounds are the source of much of
our knowledge of the prehistoric past and they serve as magnets to draw
those interested in history. We hope that the level of knowledge in the
future will be such that proper precautions are taken when this event
does occur, but this is just the type of point that we cannot assure
without surveillance of disposal areas until the radionuclides have
decayed. Obviously any contamination on any objects removed could serve
as a source of resuspension and inhalation by any individual who handles
or works with these objects.
One special case in this category results from the current practice
of using non-degredable plastics to contain the contamination during
handling of the wastes en route to the disposal area. Current
information gives little reason to believe that these plastics will not
preserve their contents and remain intact under the conditions of burial
in the soil. This raises the possibility of recovery of such packages
with consequent exposure when the package is opened.
CLASSES OF RESUSPENSION
We have divided the overall resuspension problem into three classes
defined by both their characteristics and the information needed to
solve them.
The first class is wind resuspension. This occurs when the wind
blowing over a field resuspends particles and moves them downwind. Much
is known about the mechanics of this class through the many studies of
wind erosion of agricultural fields. In essence, there is little
suspension of soil particles unless a phenomenon called "saltation"
occurs. Here, particles on the order of 100-200 ym in diameter start
rolling over the surface and then jump into the air under the pressure
of the wind. They, then, travel forward with the wind, gaining energy
and meanwhile falling under the force of gravity, until they impact on
the ground. This impact dislodges other smaller particles that are
suspended in the air. Such a process will start at the edge of a field
and increase in intensity downwind until a maximum is reached. One
other process for wind resuspension, that has not been well investigated
could result from the deposition of an aerosol on vegetation.
Presumably any of this material that is not bound to the foliage by
adsorption or absorption could be dislodged by movement of the foliage
under the action of the wind.
The second class of resuspension is that caused by mechanical
disturbance of the soil with the resuspended particles moving downwind
to expose people. This could be, for example, a farmer plowing a field
-------
27S
or people walking or working and disturbing the soil. The familiar dust
cloud raised by such activities in dry weather is a sign of such
resuspension. Here the forces required for resuspension come from the
activity but the movement, once the material is in the air. is dependent
upon the winds. Many such disturbances are point sources at the point
of disturbance rather than a broad area source as with wind
resuspension. Another difference between these two classes is the depth
to which the disturbance reaches. In wind resuspension the depth from
which resuspension can occur is dependent upon the depth reached by the
seltating particle or fairly shallow. In mechanical disturbance the
depth can be inches or even feet depending upon the type of disturbance.
Thus a source at greater depth can be resuspended than occurs with
winds.
The final class is local resuspension. This is the resuspension
leading to a cloud in the immediate vicinity of an individual causing
the disturbance. It can apply to an individual handling a contaminated
object, to an individual digging a hole in contaminated soil or to the
dust cloud surrounding a farmer plowing a field. Whereas the concern
with the previous two classes was with people downwind from the source
of the resuspension. the concern with this class is with the individual
causing the disturbance.
EVALUATION OF RESUSPENSION
Now that we have pointed out some of the ways in which the
resuspension pathway to men can be of interest to low-level waste
disposal, how can we evaluate its seriousness? In other words, if we
can define a source term how can we evaluate the potential exposure to
man? The only proper answer at this time is that we do it with
considerable uncertainty and with many assumptions. Unfortunately, the
studies required to better understand these problems and to obtain
adequate coefficients for use are not well funded.
There are three methods, or models, for estimating resuspension:
the resuspension factor, the resuspension rate and the mass loading of
dust in the air.
The resuspension factor is defined as the ratio of the air
concentration resulting from resuspension to the quantity of contaminant
per unit area at the location where the ay sample was taken. Usually,
the air concentration is in activity per m and the ground contamination
is.in activity per nr. Thus, the units of the resuspension factor are
nf . There are many values of the resuspension factor in the literature
because it has been the most widely used method for many years and
measurements are relatively simple. Typical of these values are those
in a table by Mishina [3\ In this table, the values vary over about 10
order of magnitude. However, closer examination of this table shows
that wind resuspensiqn with freshly deposited material is in the range
of 1C~B to 2xlO"b m'1 and for aged deposits from 6xlO~10 to 10~13 m~r.
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279
Fo£ mechanical disturbance the range seems to be from 2xlO~6 to 7xlO~5
m~ . However, many of these tests were done in arid country and may not
be typical of other areas.
A decrease in the resuspension with time can be expected following
deposition from the atmosphere. This decrease is caused by penetration
of the contaminant into the soil or by other means of fixation, such as
adsorption on large soil particles. Initially, rather crude
measurements at the Nevada Test Site indicated a decrease with a half
life of about one month [4j. However, as information has accumulated,
it is apparent that any decrease with time is over a much longer period
of time than one month. Anspaugh-CSjjhas formulated a relation in which
the resuspension is j^nitially 10" m~ with a decrease over a period of
several years to 10" m" . It should be stressed, however, that such a
relation is based upon meager data obtained in only one area. It may
also be noted that such a decrease with time will probably not occur for
other types of deposition such as liquid spills or contaminants brought
to the surface from a below grade disposal site because these actions
result in an initial mixing of the contaminant with the soil.
The resuspension factor has conceptual problems when it is used for
describing wind or mechanical resuspension. For example, it does not
account for resuspension that occurs upwind. It is primarily an
empirical concept with little hope of eventually describing the
important variables involved in resuspension. However, from a practical
standpoint, most of the past measurements have been made using this
method of interpretation so that some data are available. The
resuspension factor, or some variation of it. may also be a likely
choice in describing local resuspension because the receptor and the
source of contamination in the air are at the same place.
The second method of describing resuspension is through the use of
the resuspension rate. The resuspension rate is the fraction of the
contaminant in the ground that is resuspended per unit time. This rate
describes a source term that can be used with the known correlations for
meterological dispersion to estimate the air concentrations of the
contaminant downwind. There are not many measurements of the
resuspension rate available. Sehemel,,^] using, a nonradioactive
contaminant, measured values of 10~ to 10" sec" for wind
resuspension in,& lightly vegetated area. Anspaugh et al. [7] measured
values of 2x10 to 10 for wind resuspension in an area,at the Nevad^
Test, Site. For mechanical resuspension values of lxlO~ to 1x10"
sec" for an individual walking in a contaminated asphalt highway have
been inferred [2] from measurements made by Sehmel [8]. For a truck
driving through a tracer ongan asphalt highway Sehmel [9j reports values
equivalent to rates of 1C to 8xlC sec with the resuspension rate
varying with the speed of the vehicle.
The resuspension rate is plagued with the difficulty that too few
measurements have been made to allow use for types of soils and in areas
different than those in which measurements have been made. It also
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280
requires knowledge of the pattern of contamination on or in the soils of
the contaminated area. However, it does provide additional detail on
the meaning of the measurements and. as information increases, may well
be the method of choice for specific areas. It s value in studying the
phenomenon of resuspension is illustrated by the findings in the past
few years that the rate of resuspension increases with the wind speed at
rates up to the 8th power of the wind speed depending upon the type of
soil [10].
The third method for calculating resuspension is the mass loading
approach. Here it is assumed that the dust in the air has the same
concentration of contaminants as occurs in the soils. Anspaugh et al
[11] have collected information from a number of sources and have shown
that reasonable agreement between measured and calculated concentrations
can be had if one assumes, for the Calculation, that the concentration
of dust in the air is 100 ug per m . Since it is doubtful that the
actual concentration is this high, this finding should be regarded as a
correlation rather than as a true relation between actual dust and
concentration of contaminant.
There are problems with this concept also in that the measured mass
loading at a given point depends upon the dust which could have arisen
from a large distance upwind and would serve to dilute the dust from the
contaminated area, particularly if the contaminated area is small.
There have also been questions raised as to the source of the dust from
the soil. That is, if the soil contains a large fraction of smaller
particles which contain all of the activity, the concentration in this
fraction is the one that should be used in the calculations. One
investigator [12] has gone so far as to destroy all of the natural
aggregates in the soil and to use the concentration in the remaining
small particles which do not exist in nature, as the basis of the
calculation. However, all correction factors suggested for this effect
using the particle size distribution in natural soil have been
comparatively small, less than a factor of two. The use of the mass
loading has certain value, however, particularly in view of the
correlation mentioned earlier. It should be noted that this correlation
was obtained on ambient air with no disturbances in the vicinity. Thus,
if it is to be used, the mass loading should be increased by some factor
to allow for the possibility of mechanical disturbance. We believe that
the mass loading may be the best method for generic studies not tied to
a specific site.
DISCUSSION
This has been a very brief discussion of the role of resuspension
in low level waste management. Other references are available that will
provide additional detail [1,2]. However, it is apparent that the
information available is not really adequate to do more than make a
crude estimate of the inhalation problem from any of the three classes
of resuspension. Additional studies, providing both basic
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281
understandings of the mechanisms of wind resuspension and empirical data
tied to specific types of areas and actions are needed to provide data
for better estimates.
REFERENCES
]. G. A. Sehmel. "Transuranic and Tracer Simulant Resuspension."
Transuram'c Elements in the Environment, W. C. Hanson Ed., Tech.
Info. Center, U.S. Dept. of Energy (1980).
2. J. W. Healy, "Review of Resuspension Models." Trensuram'c Elements
in the Environment. W. C. Hanson Ed., Tech. Info. Center, U.S.
Dept. of Energy (1980].
2. J. Mishima. "A Review of Research on Plutonium Releases During
Overheating and Fires." Hanford Laboratories Report HW-83668
(1964).
4. R. H. Wilson. R. G. Thomas, and J. N. Stannard, "Biomedical and
Aerosol Studies Associated with a Field Release of Plutonium."
University of Rochester Atomic Energy Project Report WT-1511
(November I960).
5. L. R. Anspaugh. J. H. Shinn, and D. W. Wilson. "Evaluation of the
Resuspension Pathway Toward Protective Guidelines for Soil
Contamination with Radioactivity," Population Dose Evaluation and
Standards for Man and his Environment. Symposium Proceedings,
Pcrtcroz. STI/PUB/375. International Atomic Energy Agency, Vienna
T1974).
6. G. A. Sehmel, "Plutonium and Tracer Particle Resuspension: An
Overview of Selected Battelle-Northwest Experiments." Transuranics
in Natural Environments. Symposium Proceedings, M. G. White and P.
B. Dunaway, Eds.. ERbA Report NVO-m, pp. 181-210, Nevada
Operations Office, NT1S.
7. L. R. Anspaugh, P. L. Phelps, N. C. Kennedy, J. H. Shinn, and J. M.
Reichmann. "Experimental Studies on the Resuspension of Plutonium
from Aged' Sources at the Nevada Test Site, Atmosphere - Surface
Exchange of Particulate and Gaseous Pollutants (1974)," ERDA
Symposium Series. No. 36. R. J. Enclemann and G. A. Sehmel
Coordinators, pp. 'M-M, CONF-740921,NHS (19/b).
8. G. A. Sehmel, "Tracer Particle Resuspension Caused by Wind Forces
Upon an Asphalt Surface." Pacific Northwest Laboratory Annual
Report for 1976 to the USAEC Division of Biolocy and Medicine, Part
1 Atmospheric Sciences. Battelle Pacific Northwest Laboratory
Report BNKl-1651 (Ptl]; pp. 136-138, NllS (19/ZJ.
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282
9. G. A. Sehmel, "Particle Resuspension from an Asphalt Road Caused by
Car and Truck Traffic," Atmos. Environ.. 7:291-309 (1973).
10. D. A. Gillette, and I. H. Blifford, Jr., "Measurements of Aerosol
Size Distribution and Vertical Fluxes of Aerosols on Land Subject
to Wind Erosion," J. Appl. Meteorol., 11, 977-987 (1972}.
11. L. R. Anspaugh, J. H. Shinn, P. L. Philps, and N. C. Kennedy,
"Resuspension and Redistribution of Plutonium in Soils." Health
Phys., 29, 571-582 (1975).
12. C. J. Johnson. R. R. Tidball, and R. C. Severson. "Plutonium Hazard
in Respirable Dust on the Surface of Soil," Science. 193, 488
(1970). '
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AN OVERVIEW OF EPA'S HEALTH RISK ASSESSMENT MODEL
FOR THE SHALLOW LAND DISPOSAL OF LLW
G. Lewis Meyer
Cheng Y. Hung
Criteria and Standards Division
Office of Radiation Programs
U. S. Environmental Protection Agency
Washington, D. C. 20460
ABSTRACT
EPA is developing a generally applicable environmental standard
for managing and disposing of LLW as part of the National Radioactive
Waste Management Program. To support the LLW standard, EPA will
assess potential impacts from disposing of LLW by a number of
alternatives including estimating potential releases to the
environment, doses, health risks and costs from eight land disposal
methods. This paper presents an overview of EPA's assessment model
for shallow land disposal (SLD) of LLW. It describes model design
considerations} conceptual design of model and approach to developing
model. The SLD model has data preparation and main calculational
sections. The main calculational section has release, transport and
dose/health submodels including leaching, release, groundwater
transport, surface water transport, atmospheric transport,
resuspension, irrigation, operational spillage, human intrusion,
biological uptake by man, dose and health risk submodels. Where
possible, existing models/submodels were used. However, new
submodels were developed for infiltration, leaching, release and
groundwater transport. Basic system equations and brief explanations
of these four submodels are given.
INTRODUCTION
The Environmental Protection Agency (EPA) is developing a
generally applicable environmental standard for the management and
disposal of low-level radioactive wastes (LLW) as part of the U.S.
National Radioactive Waste Management Program.(1,2,3) The President
has directed EPA to accelerate development of its standard.(3) The
current schedule calls for EPA to issue a proposed LLW standard in
September, 1982, and to publish a final LLW standard in January, 1984.
283
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284
The EPA's program to develop the LLW standard includes assessing
the health risks from disposing of LLW by shallow land disposal (SLD)
and by other alternative disposal methods. The potential costs and
benefits (reductions in health risks) from using each of these
alternatives will also be compared. This information will provide
input to and support for EPA's decisions concerning the standard.
The -analyses will also be used to fulfill the requirements of NEPA,
the Environmental Impact Statement and the Regulatory Analysis.
The EPA has guidance from a number of sources on how to develop
the LLW standard. These include NEPA (4), EPA's Improving Government
Regulations (5), BIER II (6), Decision Making in EPA (7), ICRP 26
(8), and EPA's proposed Criteria for Radioactive Waste Management
(9). These guidelines advise EPA to:
o compare the health risk and cost of reasonable alternatives;
o ensure that the health risk from the activity is acceptable;
o ensure that the health risk is as low as reasonably
achievable, taking into account social and economic factors
(ALMA); and
o use predetermined model(s) to make the above assessments.
The above guides clearly establish EPA's use of models to support
the development of standards. Models for estimating the health
risks, making the benefit/cost analyses and comparing the
alternatives will be necessary for the LLW standard. The purpose of
these models is to assess and predict the potential future impacts
from managing and disposing of LLW by a number of realistic
alternatives. These assessments will include estimating the
potential releases of radionuclides to the environment and the
subsequent doses, health risks and associated costs.
The focus of this paper is on EPA's environmental assessment
model to simulate the health risks from disposing of LLW by shallow
land disposal methods (SLD model). The general design and layout of
EPA's SLD model was done in-house and a preliminary report on it was
presented in February, 1979.(10) This report included a general
description of the model's data input requirements, the pathways,
scenarios and land disposal alternatives to be considered, and its
potential applications. Specific guidance for development of the SLD
model, with particular attention to the infiltration, leaching,
release and groundwater transport submodels, were developed in-house
by Hung and other ORP staff. (11) Since then, Hung has developed a
groundwater transport submodel for the SLD model (12), and a method
for characterizing the long term leachability of solidified
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285
radioactive waste.(13) The SLD model is currently being developed by
the Oak Ridge National Laboratory under an interagency agreement with
the U.S. Department of Energy. It is expected that the assessment
model for other land disposal alternatives can be adapted from the
basic SLD model without major modifications.
CONCEPTUAL MODEL DESIGN AND DEVELOPMENT
Model Design Considerations
The following considerations influenced the design of EPA's SLD
assessment model and its submodels.
o A generic model is required because EPA will be analyzing
the potential impact of disposing of a wide variety of waste
types by different disposal methods under sharply different
hydrogeologic and climatic conditions.
o A land disposal facility should be modeled as a complete
disposal system which includes the wastes, the site geology,
hydrology, and climate, and all the pathways from it because
the performance of the system will change if changes are
made to any of its parts.
o A systems approach is needed to evaluate the effectiveness
of each disposal system and proposed changes in its design
(i.e., modifying the waste form or packaging).
o The model should be modular in construction to allow
substitution or modification of submodels when it is used to
analyze different disposal systems and sufficiently flexible
to analyze disposal systems with a wide range of
hydrogeologic and climatic conditions.
o A simple, one-dimensional analytical approach would be
adequate and, in fact, preferred for EPA's risk assessment
modeling needs to keep computer time and costs within
reasonable bounds, to avoid potential errors in the
numerical approach, to stay within the quality of the input
data currently available, and to utilize and stay within the
state-of-the-art modeling for many pathways.
Conceptual Model Design
The conceptual model for the environmental assessment of SLD
consists of a group of preparation submodels group and the main
calculational model. The main calculational model includes the
leaching, release and environmental transport submodels and
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286
simplified individual and population dose and health risk submodels.
A block diagram of the general layout of the SLD model is given in
Figure 1.
The preparation submodels prepare secondary input data for use in
the main calculational model. These secondary input data are
independent of the time of simulation and can be processed prior to
being integrated into the main calculational model (i.e., annual rate
of infiltration, health effects conversion factors, unit response of
region to a release). The purpose of the preparation submodels is to
prepare files of these data (i.e., annualized synthetic hydrological
or meteorological conditions or other appropriate basis) as
efficiently as possible, for input into the calculational submodels,
thus reducing central processing unit time and costs.
The submodels within the main calculational model simulate the
release and transport of radionuclides from the waste by a driving
force due either to natural processes or to human activities. The
transport submodels include: leaching; release; groundwater
transport; surface water transport; atmospheric transport;
resuspension; irrigation; operational spillage; human intrusion,
including farming on the site; and biological uptake by man including
food chain, drinking water, inhalation, and direct irradiation.
Water is considered to be the most important driving force for
releasing radionuclides from the waste. The predominant release
modes are expected to be leaching, erosion, gas generation, and human
intrusion. When the radionculides are released from a burial trench,
they will be available for transport to environmental receptors. The
receptors considered are groundwater, surface water, air, and land
surface. The main model simulates the transport of the radionuclides
released among these receptors. The resultant accumulation of
radionuclides in the receptors are then available for biological
uptake by humans through the food chain, drinking water, inhalation,
and direct irradiation pathways.
The results of these simulations furnish input into the dose and
health risk submodels. The dose submodel integrates the total doses
from the total radionculide uptake by humans from all critical
pathways to provide an estimate of potential individual dose. The
health risk submodel evaluates the corresponding health effects and
integrates them over the affected population to obtain the potential
total fatal health effects and number of years of life lost.
Approach to Model Development
Development of our conceptual SLD assessment model into a working
model was guided by the considerations discussed earlier and the need
to develop it within a relatively short time. To meet our schedule,
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287
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Block Diagram of EPA Environmental Assessment
Model for the Shallow Land Disposal of LLW.
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288
we took the following steps; we, (1) reviewed existing assessment
models, (2) selected viable models from those reviewed, (3) utilized
viable submodels from the models selected, and (4) filled in any gaps
with new or improved submodels.
A recent review sponsored by ORNL identified over 350
codes or models that are available which might have application to
LLW management and disposal.(14) More recently, R. W. Nelson
identified more than one hundred groundwater models which might be
useful for LLW management.(15) These reviews clearly point out the
size of the task of reviewing "all available" models and of culling
and selecting appropriate models and submodels. We selected the
following models for in depth review:
o NRG model for waste classification (NRC-FBD)(16);
o NRG model for LLW EIS (NRC-D&M)(17) ;
o DOE model for SLD disposal site at Savannah River
(DOE-SRLX18)
o AIF SLD assessment model (AIF-D&M) (19) ;
o EPA model for assessing deep geological disposal of HLW
(EPA-LOGX20);
o EPA model for HLW standard (EPA-HLW)(21); and
o EPA models for Clean Air Act (EPA-CAA)(22).
We are continuing to review new models coming on the scene to
take advantage of new and improved approaches. For example, the new
NRG model for systems analysis of SLD (23) looks very promising and
useful. It has several submodels which could be very useful to our
SLD model when they are developed. However, the approach of NRC's
systems is different than EPA's on several points. Also, it may not
be completed in time for use in developing our standard. In any
case, we tried not to duplicate the modeling efforts of others and,
where possible, to use the best (for EPA's purposes) of others.
Findings of Model Review
From our review of the above models, we found that none of them
met EPA's requirements without additions or modifications. We did
find, however, that some existing submodels developed by EPA and
others could be reasonable combined together without major
modifications. The infiltration, leaching, release, and groundwater
transport submodels were an exception. In some cases, EPA models and
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289
submodels were available for adaption and EPA models were selected
for the SLD model to maintain consistency with current EPA policies
and calculational practices. Figure 2 summarizes the comparison of
the risk assessment models which were reviewed in detail prior to
design of EPA's LLW model. An estimate of the relative suitability
of submodels within these models for use in EPA's risk assessment
model for SLD are indicated. A description of key differences
between EPA submodels and approaches and other assessment models
follows.
Among the risk assessment models reviewed, the infiltration,
leaching and release processes had been lumped together into a
combined simple model represented by a leaching rate which is
proportional to the remaining inventory of radionuclides. This is
too simple for realistic health risk and benefit/cost assessments.
Such a combined model assumes a constant leaching rate. This rate
must be guessed at over an infinite period of time.
A significant problem with this type of model is that it can only
be validated through extensive field and laboratory studies. It is
doubtful that even this approach can successfully accomodate all of
the boundary conditions and factors involved. For example, the
leaching rate of radionuclides from the wastes is affected by many
parameters such as trench cap conditons, the amount of infiltration,
the accumulation of water in the trenches, waste form
characteristics, and the properties of the radionuclides. Therefore,
the reliability of estimates of leaching rates will be extremely
variable, even if they can be determined with field data.
In addition, leaching and release represent two different
processes. Predominant factors governing these processes include the
rate of infiltration, accumulation of water in the trench, waste
form, and disposal site characteristics. The processes may be
controlled through changes in waste preparation and disposal
operations such as trench engineering, waste pretreatment and site
selection. Use of the existing combined models for reasonably
accurate predictions at a site would require expensive and time
consuming pilot studies before this type of model would have a valid
application.
A more realistic yet simple way of modeling these processes is to
subdivide them into three separate components, using dynamic
simulation models for the infiltration, leaching and release
processes. A major advantage of using individual dynamic simulation
models for each of the processes is that it allows one to simulate
changes in waste treatment and disposal procedures, which further
allows an analysis of the costs and benefits of these changes.
Another advantage is that the basic input parameters for these
-------
SUBMODELS
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Figure 2. The availability of submodels from existing risk assessment models and those
selected for use in the EPA risk assessment model.
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291
dynamic models can be obtained from simple field measurements and
laboratory tests without great delays or expense for lengthy field
and laboratory studies.
Some of the groundwater transport which were models reviewed
utilized an analytical approach which is economical to compute but is
limited by a specific boundary condition. The boundary condition
commonly used for these models, except the D&M model, is an
exponentially depleted release rate. However, the output function of
radionuclides released from a realistic physical model is normally an
irregular function of time and may not reasonably be approximated by
the above boundary condition. Therefore, those existing models which
used an analytical approach, were not applicable to our needs.
The groundwater transport models, which used a numerical
approach, are generally not economical to operate if the numerical
error is to be controlled to an acceptable level. This is because
improper division of time and space increments may introduce an
unacceptable level of numerical error. Therefore, the numerical
groundwater transport model is, in general, considered to be
undesirable for use in a risk assessment model. In addition
computational times and costs of numerical models are generally too
high for risk assessment models where it intended to make a large
number of calcualtions.
INFILTRATION, LEACHING AND RELEASE SUBMODELS
The release of radionuclides from the waste disposal trench to
the geosphere is a complicated series of physical processes which
includes infiltration, leaching and release from the trenches. The
rate of infiltration of water through the trench cover is a function
of meteorological conditions such as precipitation, temperature, wind
speed and air humidity; the characteristics of the trench cover; and
drainage conditions at the site. The leaching of radionuclides from
the waste is a function of the form and sorption characteristics of
the waste and the duration of its inundation in trench water. The
rate of release of radionuclides from the trench is a function of the
transmisivity of the host formation, its natural sorptive and
retentive properties and the pressure head of water in the trench.
The basic system equations for each of these processes are
briefly described in the following sections. It should be noted that
the basic system equations presented herein cannot be directly
employed as basic equations for risk assessment modeling applications
because they are too sophisticated mathematically and will be
simplified further during development of the EPA model.
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292
Infiltration Submodel
The infiltration submodel is intended to simulate the average
annual rate of infiltration through the trench cover during and
subsequent to a period of precipitation. Figure 3 is a schematic
diagram of the hydrological system associated with the earthen trench
cover. It includes overland flow and subsurface flow systems. The
momentum and continuity equations for each of these subsystems are
° *S" s(t) " E(t> " v(t> w
for the overland flow subsystem and
v . -K »<* * y> (3)
*y
3s _ 30 ,,x
3t ^F
for the subsurface flow system. In the above equations, u and v are
the velocities of water along and perpendicular to the slope of the
trench cover, respectively; h is the depth of overland flow; D is the
thickness of the trench cover; K is the permeability of the trench
cover; n is the Manning coefficient of roughness of the surface of
the trench cover; R is the rate of rainfall: E is the rate of
evaporation; ¥ is the soil moisture tension potential of the trench
cover; S is the water content of the trench cover: x and y are the
coordinates along and perpendicular to the slope of the trench cover
repectively; and t is the time.
Since the above system equations are extremely difficult to solve
and our primary interest is determining the average infiltration rate
v, one may simplify the momentum equation by neglecting the
convective acceleration and local acceleration terms appearing in
Equation 1. When this is done, Equation 1 can be simplified and it
yields
q - JL_JtSiSH_ (5)
and Equation 2, the continuity equation, can be rewritten as
R(t) - E(t) - v(t) —2- {6)
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293
Free surface of water
Diversion
Radioactive waste
in a LLW trench
Figure 3. Schematic Drawing of a Two-Dimensional
Infiltration Model for a LLW Burial Trench
-------
294
where q is the overland flow and H is the average depth of flow.
Equations 5, 6, 3 and 4 are the basic equations for solving the
infiltration of precipitation through the trench cover having
overland runoff q, depth of overland flow H and the moisture content
of the trench cover S as dependent variables. These system equations
can be further simplified and solved by a linearized equation for a
tank model.(24)
Leaching and Release Submodels
Some LLW, such as bulky solids and trash, are disposed of
directly into the trench in their original form. Other LLW, which
were originally liquids or slurries, are solidified to immobilize the
radionuclides in suspension or solution before disposing of them into
the trenches. EPA's risk assessment model is designed to simulate
the leaching and release of both types of waste. The following
discussion uses as its general case, a 55-gallon drum (or larger
container) filled with solidified liquid waste such as evaporator
concentrates solidified in cement and is applicable for the leaching
and release of one radionuclide.
The basic mass balance equation governing the radionuclide
concentration within the solidified liquid waste has been derived by
Hung (13) as
D_ 32C,
where C^ is the radionuclide concentration in the solidified waste;
De is the effective dispersivity: R is the retardation factor; £ is
the porosity: X^ is the decay constant; x is a coordinate; and t is
time. The initial conditions used to solve this equation are
(8)
Cl * C0, for 0 < x £ X.
c2-o,
and the boundary conditions are
CL - C2(t) at x - 0
(9)
dCl
— - " at x = L
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295
In these equations, C2 is the concentration of the radionuclide in
the trench water: L is the leaching length, which is computed from
the ratio of the volume of the waste container to its surface area;
and C0 is the initial radionuclide concentration in the solidified
waste.
The computations for solving Equation 7 for solidified wastes are
too complicated to integrate into EPA's risk assessment model. It
was determined, however, that the processes described by this
submodel can further be approximated and simplified by eliminating
the space independent variable. Thus Equation 7 can be simplified
from a partial differential equation into an ordinary differential
equation.
Despite the appearance of adding complexity to modeling the
infiltration, leaching and release processes (i.e., going from a
simple declining percentage model to three dynamic simulation
models), in our opinion, it is a necessary step if the effects of
changes and improvements in radionuclide retention due to waste
solidification and engineering changes to the trench cap are to be
evaluated. Further, such detailed analyses are necessary to provide
data for our cost/benefit analyses to support our standard.
The dynamic equation for the exfiltration of water from the
trench can be approximated by
Qe » 0, for Hw
Where At is the bottom area of the trench; K^ is the hydraulic
conductivity of the host formation; and H,^ is the depth of water in
trench. The rate of radionuclide release can thereafter be computed
from the rate of water leaving the trench and the radionuclide
concentration G£ in the trench water.
The above basic equations consider the water which has been
solidified to improve radionuclide retention in the waste. For
wastes which were originally solid or are liquid, the diffusivity
process is sufficiently large to maintain C]^=C2 at all times.
Therefore, computation of leaching and release for these types of
waste is much simpler than for wastes which have been solidified.
GROUNDWATER TRANSPORT SUBMODEL
To date, more than one hundred and eight groundwater transport
models have been developed.(15) However, some of these models fail
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296
to consider the process of radioactive decay and/or sorption and are,
therefore, not suitable for application to health effects
evaluations. The rest of the models which may be suitable for health
effects evaluation can be subdivided into two groups, the analytical
model and the numerical model. In general, analytical models are
limited by specific mathematical boundary conditions which require
some approximation of actual physical conditions. As a result, these
models suffer considerable error in simulation due to these
approximations.
A numerical model requires, in general, many more computations
than the analytical model does. In addition the accuracy of
simulation depends greatly on the adjustments of time and space
increments. Severe errors may result if they are not properly
adjusted. On the other hand, obtaining proper adjustment of these
increments may, in some cases, result in excessive computer time for
making the simulation. Thus, calculational costs may become
prohibitive when the model is applied to long-term simulations such
as health effects evaluations.
For example, Hung (12) reviewed the use of a finite element model
developed by Duguid and Reeves (25) to simulate the vertical
migration of tritium from a burial trench towards the groundwater
table at the West Valley (NY) disposal site.(26) This application
required seven minutes of central computer process time on a
UNIVAC-110 to simulate 200 years of real time. If it is further
assumed that 1000 years of real time would be required to simulate
the vertical migration of a radionuclide to the nearest underlying
aquifer and its horizontal migration in the groundwater to the
nearest discharge point, 700 minutes of processing time would be
required for simulating ten radionuclides. This would be equivalant
to $10,000 per run on a commercial computer, a cost which is
unquestionably excessive when a number of runs may be required to
test the effects of changing several parameters.
A one-dimensional numerical model developed for EPA by INTERA
Environmental Consultants, Inc., (27) employs the finite difference
method. In a review of this model, Hung (12) found that it had a
computer processing cost of $200 for simulating the migration of a
single radionuclide in an aquifer one mile long for a real time of
100,000 years. This, also, is too costly when many other
radionuclides and pathways must be considered.
Several analytical models, for example those by Lester, Jansen,
and Burkholder (28), Ford Bacon, and Davis, Utah, Inc. (16), and
Dames and Moore (17), have made considerable contributions to
groundwater transport modeling by simplifying the procedures for
simulation. However, the application of these models to health
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297
effects assessments is limited because it requires converting
approximations of physical boundary conditions into mathematical
forms which meet the requirements of an analytical model. This may
result in considerable error in some cases.
For the reasons outlined above, the existing numerical and
analytical models seemed either too costly to process or contained
the possibility of introducing unexpected errors of analysis when
applied to health risk calculations. A more economical and, we
believe, more suitable groundwater transport model for health risk
assessment has been developed by Hung. (12)
The basic equation for the groundwater transport model is
expressed by
Q(t>
in which Q and QQ are the rate of radionuclide release at the
discharge end and at the trench; r| is the correction factor for
health effects; R is the retardation factor: L is the length of the
aquifer; A^ is the decay constant for the radionuclide; and V is the
interstitial velocity of groundwater flow. Because this is an
algebraic equation, it is simple and economical to process when
integrated into a health effects assessment model.
The correction factor closely approximates and corrects for the
effects of dispersion. The correction "factor »; was expressed as
/" (1/2) /(RP/ir 93 ) Exp{ -Nd 9- {P 9/4R) (R/6-1)2 }dO
» -fl - - — (12)
Derivation of 17 is described in detail in a separate paper by
Hung. (12)-
SUMMARY
EPA is developing an environmental assessment model to support
its environmental standard for the management and disposal of LLW.
This model will be used to analyze the potential transport and
release of radionuclides to the biosphere and to estimate the
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298
subsequent doses, health risks, and costs from disposing of LLW by
shallow land disposal.
The SLD model is a generic, analytical systems model for
analyzing a disposal facility as a complete "disposal system" which
includes the waste, the site geology, hydrology and climate, and all
of the pathways from it. It can be used for analyzing the
performance of planned and existing disposal facilities, and is
particularly suited to evaluating the effects of changes to a
disposal facility (i.e., changes in waste form and packaging, trench
cap design or water control engineering). The SLD model is
sufficiently flexible to use for analyzing sites having disposal
media with a wide range of permeabilities and located in either humid
or arid climates.
Improvements have been made in EPA's SLD model for modeling the
influx of water into the trenches, the leaching of radionuclides from
the waste, and the release of radionuclides from the trenches. These
features are particularly important for evaluating LLW management and
disposal practices because they are the principle parts of the SLD
disposal system which man can change to increase the retention
capability of a disposal facility.
Individual dynamic simulation models have been developed for
infiltration, leaching and release which can use information from
simple laboratory tests and field measurements for input data. These
separate submodels allow one to simulate changes in the performance
of a disposal facility caused by changes in waste treatment and
packaging, trench cover modification and site water control
engineering. This further allows one to analyze the costs and
benefits of any changes. The SLD model also has a new groundwater
transport submodel which includes a correction factor for
dispersivity, even though the model is an analytical, one-dimensional
model. The basic transport equations for these four submodels are
presented. These basic equations will be simplified further during
coding of the model. The other submodels are, for the most part,
being adapted from existing models with minor modifications.
It is believed that the basic features of EPA's SLD model, such
as economical operation, realistic simulation, use of generally
available data, and capability of evaluating changes to a disposal
facility, will make it a potentially useful model for other
government agencies and industry who are interested in assessing the
impact of proposed or existing disposal facilities. When completed,
coding and documentation on it will be made available to interested
parties.
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299
REFERENCES
1. U.S. Department of Energy, Report to the President by the
Interagency Review Group on Nuclear Waste Management, TID-29442,
(March 1979).
2. U.S. Department of Energy, National Plan for Radioactive Waste
Management, Draft No. 4 (Jan. 1981).
3. Office of the U.S. President, Comprehensive Radioactive Waste
Management Program - Message from the President of the United
States. (Feb. 12, 1980).
4. U.S. Code of Federal Regulations, National Environmental Policy
Act of 1969. (1969).
5. U.S. Environmental Protection Agency, Improving Government
Regulations, Final Report Implementing Executive Order 12044,
Federal Register, Vol. 44, No. 104, pp. 30988-30998, May 29, 1979.
6. U.S. Environmental Protection Agency, Considerations of Health
Benefit - Cost Analysis for Activities Involving Ionizing
Radiation Exposure and Alternatives, EPA 520/4-77-003, (1977).
7. U.S. National Research Council, Volume II Decision Making in the
Environmental PRotection Agency, A National Research Council
Report to EPA on Environmental Decision Making, (1977).
8. International Commission on Radiation Protection, Publication
26; Recommendations of the International Commission on Radiation
Protection, (Jan. 1977).
9. U.S. Environmental Protection Agency, Criteria for Radioactive
Wastes, Recommendations for Federal Radiation Guidance,
(Nov. 15 , 1978).
10. G.L. Meyer, S.T. Bard, C.Y. Hung and J. Neiheisel, "Modeling and
Environmental Assessment of Land Disposal Methods for Low-Level
Radioactive Waste," Proceedings of Health Physics Society Twelvth
Midyear Topical Symposium on Low-Level Radioactive Waste
Management, Williamsburg, Viriginia, Feb. 11-15, 1979.
11. U.S. Environmental Protection Agency, Technical Specifications
for Environmental Asssessment Model^ Development, unpublished
TAD/ORP document, (1979).
12. C.Y. Hung, "An Optimum Model to Predict Radionuclide Transport in
an Aquifer for Application to Health Risk Evaluations,"
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300
Proceedings of DOE/EPA/NRC/USGS Interagency Workshop on Modeling
and Low-Level Waste Management, Denver, Colorado, (Dec. 1980).
13. C.Y. Hung, "Prediction of Long-Term Leachability of a Solidified
Radioactive Waste from a Short-Term Test by a Similitude Law for
Leaching Systems," Proceedings of ORNL Conference on the
Leachability of Radioactive Solids, Gatlinburg, Tennessee,
(Dec. 1980).
14. Science Applications, Inc., A Compilation of Models and
Monitoring Techniques, Report prepared for DOE,
ORNL/SUB-79/13617/2, SAI/OR-56-2.
15. R.W. Nelson, "Subsurface Modeling: Goals, Expectations and
Performance", U.S. Department of Energy Workshop on Low-Level
Waste Management Environmental Modeling, Bethesda, Maryland
(July 30-31, 1980).
16. Ford, Bacon, & Davis Utah, Inc., Screening of Alternative Methods
for the Disposal of Low-Level Radioactive Waste, Report prepared
for U. S. NRC, NUREG/CR-0308, UC 209-02, 1978.
17. U.S. Nuclear Regulatory Commission, Report on Waste
Classification Methodology, unpublished report by Dames and
Moore, Inc., 1980.
18. E.L. Wilhite, Dose to Man Model for Buried Solid Waste,
DPST-78-296, Savannah River Laboratory (June 1978).
19. Atomic Industrial Forum, Generic Methodology for Assessment of
Radiation Doses from Groundwater Migration of Radionuclides in
LWR Wastes at Shallow Land Burial Trenches, a report by Dames &
Moore, Inc., June, 1978.
20. U.S. Environmental Protection Agency, Development and Application
of a Risk Assessment Method for Radioactive Waste Management,
EPA-520/6-78-005 (June 1978).
21. U.S. Environmental Protection Agency, Population Risks from
Disposal of High-Level Radioactive Wastes in Geologic
Repositories. EPA 520/3-80-006 (December, 1980).
22. U.S. Environmental Protection Agency, Nemos (Atmospheric), TERRA
(Terrestrial) and ANDROS, DARTAB and RADRISK (Dose and Risk
Factor) Codes, under development by ORNL.
23. U.S. Nuclear Regulatory Commission, Systems Analysis of Shallow
Land Burial, unpublished report by Science Applications, Inc.
(1980).
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301
24. M.H. Diskin., A Basic Study of the Linearity of the
Rainfall-Runoff Process in Watersheds, Ph.D. Thieses Univ. of
111. (1964).
25. J. 0. Duguid and M. Reeves, "Material Transport Through Porous
Media: A Finite-Element Galerkin Model: Oak Ridge National
Laboratory Report, ORNL-4929, (1976).
26. INTERA Environmental Consultants, Inc., "Development of
Radioactive Waste Migration MOdel," Report prepared for Sandia
Laboratories, (Aug, 1977).
27. D. H. Lester, George Jansen, and H.C. Burkholder, "Migration of
Radionuclide Chains through an Adsorbing Medium," AICHE Symp.
Series, 71, 202, (1975).
-------
RISK ASSESSMENT AND RADIOACTIVE WASTE MANAGEMENT*
Jerry J. Cohen
Craig F. Smith
Lawrence Livermore National Laboratory, University of California
Livermore, California 94550
ABSTRACT
In order to evaluate the hazard of toxic material buried under-
ground, the concept of geotoxicity is presented. This concept per-
mits the development of a tool for quantifying the degree of hazard
of such material, and this tool is formulated as the geotoxicity
hazard index. The components of the geotoxicity hazard index are
described and discussed. Finally, some sample results are offered.
A more detailed description can be found in the report "A Hazard
Index for Underground Toxic Material," UCRL-52889, June 1980.
INTRODUCTION
The waste management hazard assessment program at LLNL has as its
objective the development and assessment of rationale for radioactive
waste management programs. The bulk of previous research in waste
management programs has been devoted to the development of methodologies
for handling and disposing of waste. Our program deals with the develop-
ment of rationale. We are more concerned with the "Why?", than the "How
to" aspects of the problem. Obviously, predictive modeling plays an
important role in this work. In this paper we will cover our overall
program and in particular, our efforts in developing a hazard index for
underground toxic material which can perhaps provide an alternative
approach toward defining risks. A hazard index might also provide some
insight for waste management programs in determining "how good is good
enough."
GEOTOXICITY
The overall approach to problem definition which we apply is the
geotoxicity study. This study involves the characterization and asses-
sment of the harmful effects of hazardous material buried in the earth(s
*This work was performed under the auspices of the U. S. Department of
Energy by the Lawrence Livermore Laboratory under contract No.
W-7405-Eng-48.
303
-------
304
crust by either man or nature. It should be recognized that incorpora-
tion of toxic materials in the earths crust via radioactive or toxic
waste burial does not present unique or unprecedented risks to health and
the environment. Nature has done essentially the same throughout the
entire history of the earth in toxic mineral deposits. The geotoxicity
concept utilizes this perspective to provide useful insights on the fate
of these materials in the biosphere. B. L. Cohen has characterized the
approach as using the earth itself as a large analog computer. In any
case, it is most useful to have a measure or "yardstick" with which one
may characterize the hazard of toxic material buried underground. With
this objective in mind we are developing the Geotoxicity Hazard Index
(GHI). This index may be used:
• to provide an easy to use comparative tool
• to gain perspective on geotoxicity
• to enable screening of concepts for the disposal of toxic material
• to provide insight on the relative hazard of toxic components
• to permit examination of the source of toxic hazard
It should be noted that the GHI is not intended to substitute for
detailed systems modeling used for prediction of specific consequences
and their magnitude. It can however be used to provide additional
insight on the severity of the general problem (or lack of it). Also,
the results of systems analysis can provide valuable input to hazard
index determinations.
REVIEW OF PREVIOUS WORK
The approach that was taken in our work was to review and evaluate
previous efforts in the safety index area to either select an existing
index, or identify the requirements for a new index if, as it turned out,
a new index would be required.
Evaluation of previous work resulted in a classification of indices
as shown on the first portion of SLIDE fl. The first category consists
of the simple indices which are measures of quantity such as mass, number
of curies, volume, heat output, and so on. Somewhat more complicated are
the toxic indices which incorporate specific toxicity in addition to the
quantity of material. Some examples of this second category are the
dilution volume hazard index or the volume of water required in dilution
to drinking water levels, and the number of lethal doses index. These
indices are widely used in the literature. However, they are measures of
toxicity and not hazard. Clearly, a material can be toxic but not par-
ticularly hazardous if there does not exist a means for exposure to the
material. The next level of complexity involves indices which, in addi-
tion to toxicity, incorporate availability of the toxic material at the
end of the transport process. Such indices may incorporate transport and
radioactive decay effects. The final category consists of complex
indices which include value judgement or decision analysis. This ap-
proach was rejected for the geotoxicity hazard index. The incorporation
-------
Slide 1
Our Review of Previous Work Included:
• Evaluation of previous indices for radioactive and stable toxic
materials
simple indices (mass, volume, heat, etc.)
toxic indices (toxicity and quantity)
complex indices (incorporating availability)
complex indices (including value judgment/decision theory)
• Determination of desirable features for our index
simple, easy to use
containing enough information to be meaningful
applicable to both radioactive and stable toxic materials
emplaced by man or nature
• Identification of factors that could be included
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306
of value judgement greatly increases the level of subjectivity and there-
fore, such indices would be difficult to defend.
Since it was determined from our review that none of the previous
indices met our objectives for a geotoxicity index, we used the review to
focus on the desirable features and factors that could be included in the
new index. The desirable features include simplicity in form to promote
easy use, and, as a corresponding trade-off, sufficient complexity to
provide for a meaningful measure of hazard. Other desired features are
the capability to handle radioactive as well as stable toxic material,
and natural deposition as well as emplacement by man.
There are several factors which could be included in an index of
hazard. Some of these are shown on SLIDE #2. Of the parameters listed,
the four on the left were considered basic requirements for our index;
toxicity, availability, longevity or persistence, and transformation
product effects are key parameters and have been explicitly incorporated
into the Geotoxicity Hazard Index.
Chemical and physical form, and the effects of biological concen-
tration are factors which are implicitly included in the previous four
factors. Perceived harm as opposed to actual detriment could be included
in an index, but, as stated earlier, this approach was rejected.
THE GEOTOXICITY HAZARD INDEX
Based upon the previously discussed considerations, a Geotoxicity
Hazard Index (GHI) was developed in the following form.
GHI =TIi • Pi
where
TI = toxicity index
P = persistence
A = availability
C = buildup correction
i = material index
For a given material i, GHI is determined as a product of four
factors which represent toxicity, persistence, availability, and decay
product buildup. The index can be viewed as a toxicity index which is
modified by three dimensionless parameters to successively incorporate
time, transport, and decay effects. For a mixture of toxic materials,
the total GHI is obtained through summation of the individual indices for
the toxic components. Each of the factors of the GHI will be discussed
in turn.
-------
Slide 2
POTENTIAL PARAMETERS FOR A HAZARD MEASURE INCLUDE:
• TOXICITY • CHEMICAL FORM
• AVAILABILITY (TRANSPORT) • PHYSICAL FORM
• LONGEVITY x • BIOAGCUMULATION
• TRANSFORMATION PRODUCTS • PERCEIVED HARM
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308
Toxicity Index
The first term in GHI is the toxicity index, TI (SLIDE #3). The
basic toxicity index which was chosen is the public drinking water dilu-
tion volume. For radionuclides, it is the volume of water required for
dilution to MPC level, while for stable toxic materials, it is the
analagous volume relative to the EPA drinking water standards, or DWS.
Both apply to public drinking water supplies. This choice of a toxicity
index permits the intercomparison of radioactive and nonradioactive
materials. While not based on a rigorous equivalence of effects, the
comparability of MPC and DWS has been studied previously and they are
probably the best available measures of relative toxicity, for chronic,
low level exposures. SLIDE #3 shows some examples of specific toxicity
values for several materials. To produce a toxicity index, the specific
toxicity values would be multiplied by the appropriate material inven-
tory. It is interesting to note the values for 1-129 versus elemental
mercury. On a unit mass basis, these materials are of comparable
toxicity.
Persistence
The second factor in the GHI is the persistence factor, P (SLIDE #4),
For stable materials, the persistence factor has a value of one; indica-
tive of maximum persistence. For materials which decay, P has a value
less than one. Persistence is determined as the average fraction of the
original material which is present in undecayed form, over a reference
300 year time frame. The lower part of this slide illustrates the rela-
tionship between P and half-life. Also shown on SLIDE #4 are some
examples of materials with half-lives running from infinity down to the
24 day half-life of Th-234.
Availability
The third term in the GHI is the availability factor, A (SLIDE #5),
that relates the ingestion of a material to its concentration in the
earth's crust. The availability factor, A, is durther defined in terms
of a modification factor, m, and AQ which is the average availability
of the natural analog of the material under consideration. Ag is the
ratio of the ingestion rate of material i to its crustal abundance,
divided by the same ratio for naturally occurring radium-226. The modi-
fication factor, m, is intended to account for differences between the
actual burial conditions and the average conditions for the natural
analog. For example, if a material is significantly less available than
its natural analog because of isolation barriers, siting, waste form,
depth, or other condition, the m factor would be significantly less than
one. For availability no better or worse than the average natural con-
ditions, m would be given a value of one. Some examples of the
natural availability factor for several elements are also shown SLIDE #5.
Note that radium, by definition, has a value of one. Other elements are
-------
Slide 3
GHI = TI • P • A • C
TI = TOXICITY INDEX: A MEASURE OF INNATE TOXICITY
• FOR RADIONUCLIDES: TI = Q/MPC
• FOR STABLE MATERIALS: TI = M/DWS
• PERMITS INTERCOMPARISON OF RADIOACTIVE AND NONRADIOACTIVE MATERIALS
• EXAMPLES:
MATERIAL SPECIFIC TOXICITY SPECIFIC TOXICITY
(per Curie) (per gram)
70 ! 9 3
SR-90 10' m per Curie 1.4 x Kr m per gram
PU-239 2 x lO'' m3 per Curie 1.2 x 10 m3 per gram
O o /£ *j
H-3 3 x 10 nr per Curie 2.9 x 10 nT per gram
6 °i 23
1-129 2.5 x 10 nr per Curie ^.^ x 10 nr per gram
2 3
MERCURY 5.0 x 10 m per gram
LEAD 2.0 x 10 m per gram
-------
Slide 4
GIU = TI * P * A . C
P » PERSISTENCE FACTOR
• FOR STABLE MATERIALS, P = 1.0
• FOR DECAYING MATERIALS, P^l.O
• PERSISTENCE IS JUDGED RELATIVE TO A 300 YEAR TIME PERIOD
EXAMPLES:
1,0
0.8
: 0.6
I
0,4
0.2
MATERIAL
10'' 1.0 10 102 ID3
Half-Life, years
104 105
HALF-LIFE
STABLE LEAD
U -
PU
SR
TH
238
- 238
- 90
- 231*
00
4.5 x 109
88 YEARS
28 YEARS
2k DAYS
1.0
1.0
.38
.14
3.2 x 10"1*
CO
o
-------
Slide 5
GHI = TI • P • A • C
A = AVAILABILITY FACTOR
A IS THE FACTOR THAT RELATES THE INGESTION RATE OF A MATERIAL
TO ITS CONCENTRATION IN THE EARTH'S CRUST
INGESTION RATE. / CRUSTAL ABUNDANCE-
:_ / Vsf\uo i ni. nuuiiunnuEr,
Kfi K3
INGESTION RATE.. / CKUblAL
A * m AQ , WHERE AQ = INGESTION RATED / CRUSTAL
- AQ IS THE AVAILABILITY OF THE NATURAL ANALOG
- m IS A MODIFICATION FACTOR THAT ACCOUNTS FOR DIFFERENCES
BETWEEN ACTUAL BURIAL CONDITIONS AND THE NATURAL ANALOG
- EXAMPLES:
ELEMENT
RADIUM
LEAD
URANIUM
IODINE
A0
1.0
9.7
.27
54.8
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312
either more or less available in nature than radium, and these differ-
ences are reflected in their values for AQ.
Buildup Correction
The last component of GHI is C, the buildup correction factor
(SLIDE #6). This factor is used to accommodate the buildup of decay
progeny more toxic than the parent. Two examples are the decay of U-238
through Ra-226 to stable lead, and the transformation of elemental
mercury into the more toxic methylated form. SLIDE #6 shows graphically
how C is determined. On the right side of SLIDE #6 is a table which
gives some examples of the buildup correction factors for several
materials. It should be noted that in the majority of cases, C is equal
to one. Materials with a significant buildup of decay toxicity are the
exception rather than the rule. U-238 is the most extreme case which we
have identified.
SAMPLE RESULTS
In order to demonstrate the use of the Geotoxicity Hazard Index,
several representative toxic material deposits were considered. SLIDE #7
shows the results of the GHI calculations for these sample cases. While
they involve several assumptions and simplifications, they are shown to
illustrate the nature of the calculational results and to indicate the
types of applications for which the GHI could be used. Future effort on
the GHI will include converting the index into dimensionless form, and
normalization to avoid the large powers of ten seen in these results.
Another goal toward which future effort will be devoted is improve-
ment in estimation of the availability factor (A). This work is seen as
an iterative process by incorporating new information into improved esti-
mates. Further refinement of the calculational parameters will provide
increased confidence in the results.
DISCLAIMER
This document was prepared as an account of work sponsored by an agency of
the United States Government. Neither the United States Government nor the
University of California nor any of their employees, makes any warranty, ex-
press or implied, or assumes any legal liability or responsibility for the ac-
curacy, completeness, or usefulness of any information, apparatus, product, or
process disclosed, or represents that its use would not infringe privately owned
rights. Reference herein to any specific commercial products, process, or service
by trade name, trademark, manufacturer, or otherwise, does not necessarily
constitute or imply its endorsement, recommendation, or favoring by the United
States Government or the University of California. The views and opinions of
authors expressed herein do not necessarily state or reflect those of the United
Stales Government thereof, and shall not be used for advertising or product en-
dorsement purposes.
-------
Slide 6
GHI = TI • P • A • C
C = BUILDUP CORRECTION FACTOR
C IS THE CORRECTION FACTOR TO ACCOUNT FOR THE BUILDUP OF DECAY
PROGENY MORE TOXIC THAN THE PARENT
,226
•PB
206
HG
EXAMPLES:
CO
CO
TI
max
TOTAL
TOXICITY
INDEX
TI
C =
max
TI.
MATERIAL
U - 238
NP - 23T
PU - 239
ARSENIC
LEAD
> x ]
T9
1.0
1.0
1.0
TIME
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Slide 7
Results of Sample GHI Applications
Site GHI
Low Level Waste Facility 2,7 v 1011
13
High Level Waste Repository 4.4 x 10
12
Uranium Ore Deposit 3.0 x 10
13
Mercury Ore Deposit 1.3 x 10
12
Lead Ore Deposit 9.7 x 10
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MODELING IN THE DETERMINATION OF USE CONDITIONS FOR A
DECOMMISSIONED SHALLOW-LAND BURIAL SITE
E. S. Murphy
Energy Systems Department
Pacific Northwest Laboratory
Richland, Washington 99352
ABSTRACT
Because site stabilization results in smaller dollar costs
and lower occupational doses than does waste removal, it is the
preferred alternative for the decommissioning of shallow-land
burial grounds. Release of a burial ground for public use fol-
lowing site stabilization would probably be on a conditional-use
basis. A pathway modeling methodology has been developed for
the evaluation of use conditions for a decommissioned site.
The methodology consists of estimating the maximum annual dose
to the maximum-exposed individual resulting from a specific use
scenario. The methodology is sensitive to site parameters, to
radionuclide inventories, and to uncertainties that exist in
nuclide transport models and in the parameters used in the
models to represent specific sites. Several research needs
are identified that would result in improved predictions of
radionuclide migration from shallow-land burial grounds.
INTRODUCTION
A study, ' sponsored by the Nuclear Regulatory Commission, was per-
formed to estimate dollar costs and public and occupational radiation doses
resulting from decommissioning reference shallow-land burial grounds by
the decommissioning alternatives of 1) site stabilization plus long-term
care and 2) waste removal. On the basis of minimum cost and occupational
radiation dose, site stabilization is the preferred decommissioning alter-
native.
Site stabilization implies that all or some portion of the waste is
left in place, and release of the site would probably be on a conditional-
use basis. A methodology was developed for evaluating scenarios for the
conditional or unrestricted release of a burial ground after burial opera-
tions cease. The methodology uses pathway modeling techniques to estimate
the maximum annual dose to the maximum-exposed individual for various site
use scenarios. The maximum annual doses for the given use scenarios are
compared to determine which scenario is more appropriate.
Results and conclusions derived from this pathway analysis approach
depend on the site characteristics, on the burial ground radionuclide
inventory, on the mathematical models used to evaluate potential exposure
pathways and estimate radiation doses, and on the parameters used in the
models. Uncertainties exist in the models used to estimate radionuclide
315
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316
transport and in the parameters and boundary conditions used in the models
to represent specific sites. These uncertainties can affect the validity
and seriously limit the usefulness of the pathway methodology approach.
DECOMMISSIONING ALTERNATIVES
Decommissioning is defined as the measures taken at the end of a
facility's operating life to ensure that future risk to public safety from
the facility is within acceptable bounds. For a shallow-land burial site,
the basic decommissioning alternatives are 1) site stabilization followed
by a period of long-term care and 2) waste removal.
Site stabilization involves the use of engineered procedures to reduce
the rate and extent of radionuclide migration from buried wastes left in
place in a decommissioned shallow-land burial ground. Potential site stabi-
lization activities include:
• engineered routing/flow control of ground and surface water
• modification of trench caps to minimize water infiltration into the
trenches
• stabilization of the land surface and erosion control
• grouting and/or use of chemical additives to reduce the mobility of
the waste
• control of plants and animals that might disrupt surface stabilization
measures or transport radioactivity from the trenches
• erection of physical barriers to control human activities at the site.
Site stabilization implies that all or some portion of the waste is
left in place, and onsite public activities are restricted to land uses
that do not result in excessive public exposure to radiation and that do
not compromise stabilization procedures or waste confinement.
Site stabilization is followed by a period of long-term care of the
site. Long-term care activities include site surveillance and maintenance,
environmental monitoring, and administrative procedures for control of the
site. Long-term care continues until it is determined that the radioactivity
at the site has decayed to the point where the waste no longer poses a
significant radiological hazard, or until additional actions are taken to
reach this point.
Waste removal involves the exhumation of buried waste and contaminated
soil, packaging of the waste and soil, transportation, and reburial of the
waste at another disposal site. Because of the potential for significant
radiation dose to decommissioning workers and the high dollar costs, waste
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317
removal would likely be considered only in situations where site stabi-
lization and long-term care are insufficient to ensure the continued
capability of the site to provide adequate containment of the buried waste.
At a particular site, combinations of decommissioning activities may
be necessary to ensure that future risk from buried radioactive material
is within acceptable bounds. Combinations of decommissioning alternatives
may be necessary when individual burial trenches are known to contain high
concentrations of transuranic waste or waste mixed with organic complexing
agents. In these instances, the removal of the waste from part or all of
a particular trench or trenches may be a requirement in conjunction with
stabilization of the rest of the site. Partial waste relocation may also
be required if burial trenches are located in an area with geologic or
hydrologic characteristics that make stabilization a costly or technically
unfeasible alternative. In this case, the waste removed from one trench
might be reburied in another onsite trench or it might be transported to
another disposal site.
DECOMMISSIONING OF REFERENCE SITES
Two generic burial grounds, one located on an arid western site and
the other located on a humid eastern site, were used as reference facilities.
The two burial grounds were assumed to have the same site capacity for waste,
the same radioactive waste inventory, and similar trench characteristics and
operating procedures. The climate, geology, and hydrology of the two sites
were chosen to be typical of actual western and eastern sites. Each site
description provided a basis for evaluating decommissioning methods and
costs, and for estimating possible environmental impacts.
Costs of stabilization of the reference sites were estimated to range
from $0.5 million to $8 million depending on the site and on the stabiliza-
tion option chosen. Annual long-term care costs were estimated to range
from $80,000 to $400,000.
Stabilization of a burial ground involves modification of and addition
to surface soils, but no intentional uncovering or exhumation of buried
waste. There is no transportation of radioactive waste. Therefore, routine
site stabilization operations are not expected to result in any significant
radiation dose to the general population. Modest exposure of decommission-
ing workers is expected; however, the dose rates would be less than those
experienced during waste burial operations.
Waste removal was estimated to cost about $1.4 billion and to require
approximately 1000 man-years of effort extending over a period of more than
20 years. Approximately 93% of the cost of waste removal was found to be
associated with the packaging, shipment, and offsite disposal of the
exhumed waste.
The 50-year committed dose equivalent to the affected population from
routine waste removal operations was estimated to be very small compared to
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318
the 50-year dose from natural background radiation. However, the occupa-
tional dose from waste removal was estimated to be approximately 30,000
man-rem, indicating that this decommissioning alternative can be very costly
in terms of worker exposure to radiation.
ANALYSIS OF RELEASE CONDITIONS
Waste removal implies the complete removal of the buried waste and
contaminated soil and can result in release of the site for unrestricted
public use. Site stabilization implies that all or some portion of the
buried waste is left in place. Release of a stabilized site would presumably
be on a conditional-use basis, with public activities restricted to those
that limit exposure to radiation and that do not compromise stabilization
procedures or waste confinement. An important factor in determining the
appropriate decommissioning alternative for a particular site is the esti-
mated risk to the public from the decommissioned site.
A methodology was developed for predicting conditions for the conditional
or unrestricted release of a burial ground after burial operations cease.
The methodology consists of estimating the calculated maximum annual dose
to the maximum-exposed individual resulting from a specific release scenario.
The maximum annual doses from different scenarios are compared to determine
which scenario is more appropriate. Doses to the maximum-exposed individual
are calculated for all important exposure pathways using state-of-the-art
modeling techniques. For the western site the important pathways are
inhalation, direct external exposure, and ingestion of foods grown on the
released decommissioned burial ground. For the eastern site, additional
exposure pathways of importance are ingestion of aquatic foods from a nearby
river and ingestion of water from a well drilled into an aquifer beneath the
site.
Land-use scenarios are evaluated to determine release conditions
following stabilization of the reference burial grounds. Examples of path-
way analysis results are shown in Table 1. The dose numbers in the table
are calculated maximum annual total body doses to an individual who lives
and works on a conditionally released burial site having various use
restrictions.
The first two values in the table are estimated maximum annual total
body doses at the western and eastern sites during the first 50 years after
site release assuming that individuals living on a site do not engage in
excavation or drink water from onsite wells. The major contributors to
dose are 63ni (assumed to be present in reactor decommissioning waste
buried on the site) and 210Pb (a radioactive daughter of 226Ra). The third
value is the estimated maximum annual total body dose at the western site
assuming that excavation is permitted. Most of the dose from site excavation
is from external exposure to the gamma radiation from 137cs.
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319
TABLE 1. Maximum Annual Total Body Doses for Conditional
Release of the Reference Burial Grounds With
Different Use Restrictions(a)
Maximum Annual
Total Body Dose
Release Scenario (mrem)
No excavation and no drinking of water 4
from onsite wells - Western Site
No excavation and no drinking of water 23
from onsite wells - Eastern Site
Excavation permitted - Western Site 380
Total erosion of trench overburden^ ' 1 300
- Western Site
Drinking permitted from contaminated well 120 000
- Eastern Site
(a) Conditional release assumed to occur 200 years after
site closure.
(b) Calculated at 450 years after site release (650 years
after site closure).
The fourth value in the table is the estimated maximum annual total
body dose at the western site assuming total erosion of the trench over-
burden. Total erosion is estimated to occur approximately 450 years after
site release, based on the assumed erosion rate at the western site.
Radionuclides that contribute most of the dose are 230Th, 226Ra, 210pD, 235u,
238u, 239pu, 240pu and 24lAm. While it is unlikely that site conditions
would result in total overburden removal, this scenario demonstrates the
importance of maintaining an adequate layer of overburden to prevent both
crop root penetration into the waste and radionuclide dispersal by human
activities or wind action. Institutional controls may be necessary to
restrict human activities at a conditionally released site and to maintain
an adequate depth of overburden.
The last value in the table is the estimated maximum annual total body
dose at the eastern site assuming that the site resident obtains all of his
drinking water from a shallow well drilled into the contaminated aquifer
beneath the site. The dominant radionuclides contributing to the large
annual dose are 14C, 63N1, 90$r, 137Cs, and 226Ra.
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320
For the reference burial sites, the pathway modeling approach indicates
some restrictions that should be placed on a conditionally released site.
Conditional release of the western site following site stabilization would
be possible provided that the following actions are enforced:
• stabilize the ground surface to minimize surface erosion
• control the type of farming or other land use to prevent the growth
of deep-rooted plants
• restrict activities that result in excavation of the site.
The conditional release of the decommissioned eastern site would include
enforcement of the restrictions described for the western site, plus the
following additional restrictions:
• prohibit the use of water from shallow wells drilled on or near the
site
• maintain site drainage features to control surface water runoff and
to prevent inundation of burial trenches with water
• stabilize the waste to minimize leaching to the aquifer, or control
the use of aquatic organisms and water from nearby streams.
For the site stabilization alternative, unconditional release of the
reference sites, even after a long-term care period of 200 years, would
not be feasible because of the significant inventory of long-lived radio-
nuclides in the buried waste.
The methodology developed for defining site use conditions is both
site- and inventory-specific. The use restrictions listed above for the
reference sites are given only as examples of results to be expected from
the application of the modeling technique. The methodology must be
reapplied and the radiation doses recalculated for each burial ground having
a different radionuclide inventory and different site characteristics.
In addition to a dependence on site characteristics and radionuclide
inventories, the results and conclusions of the pathway-modeling approach
depend on the mathematical models used and on the parameters and boundary
conditions employed with the models to represent specific site character-
istics. Many uncertainties exist in the radionuclide transport models and
in the parameters (e.g., distribution coefficients, dispersion coefficients,
leach rates) used with the models. Because of these uncertainties, a
generally conservative approach was attempted that may result in conservative
(high) estimates of doses to the maximum-exposed individual.
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321
TRANSPORT MODEL UNCERTAINTIES
Radionuclide migration via the groundwater pathway was simulated by
use of the MMT (Multicomponent Mass Transport) model.(2) There are no
established models for treating overland flow. To account for overland flow,
the conservative approach is taken that the burial trenches are saturated
with water. All of the water flowing through the trenches arrives at the
surface and flows overland to the nearby river. No significant sorption is
assumed during overland flow. These factors result in a conservatively
high estimate of the radioactivity leached to the river by this pathway.
Although there are some uncertainties in nuclide transport models, the
major uncertainties are in the parameters and boundary conditions fed into
the models to represent specific sites. Order of magnitude uncertainties
exist in values for soil permeability, dispersion coefficients, distribution
coefficients (Kd), and leach rates. Some of these parameter values
(e.g., Kd's and leach rates) are relatively easy to measure in the labora-
tory, but extremely difficult to measure under field conditions.
Because of the great difficulty of direct field measurement of Kd,
values determined by laboratory measurement are normally used to calculate
water transport of radionuclides. However, the validity of applying labora-
tory values to field situations is questionable because of the strong
dependence of measured values of Kd on the physical and chemical conditions
of measurement. Variables such as mineralogy, particle size, nature of
solution, and chemical nature of the radioactive species can strongly affect
the measured values. Table 2 gives examples of published values of Kd used
in several studies of radionuclide migration. It is apparent that, for some
radionuclides, significant differences exist between values reported by dif-
ferent authors.
Leach rates are influenced by many factors, including the character-
istics of the radionuclide and of the waste material, the properties of the
leachant, frequency of leachant changing, leaching time, and temperature.
Specific field data on the Teachability of radionuclides from waste buried
in shallow-land burial sites are not available. The effects of chelates on
the mobility of the radionuclides in buried waste are just beginning to be
studied. Published leach-rate data come mainly from laboratory measurements
in which small samples are leached by distilled water or by actual or simula-
ted disposal-environment water. Laboratory leach-rate data are summarized
in a recent Brookhaven National Laboratory report.(3) This report gives
leach-rate values for cement monoliths that range from 10-1 to TO"9 g/cm2-day.
An example of the effect of a change in leach rate on radionuclide con-
centration in a surface stream located 1 km from the reference eastern burial
ground is shown in Figure 1 for a long-lived radionuclide with a small Kd
value. The nuclide chosen is "Tc (Kd = 0). Increasing the assumed leach
time* by an order of magnitude results in almost an order of magnitude decrease
in the maximum radionuclide concentration in the ground water at the point
The leach time is the time required for the radioactive material to migrate
from the burial ground at a constant leach rate.
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TABLE 2. Reported Values of Distribution Coefficients for Selected Elements
Element
Co
Sr
I
Cs
Pu
Am
. .
ARHCO
5-38
12-200
200
1200
NECO NECO/ \
May(b) June^ '
138-593 700-800
3-17 3-6
4000-9000 27-50
Dames. &.
Moore^'
75
20
0
200
2000
2000
Kd U/kg)
Leddicotte^e;
Dry
Site
1000
10
5
1000
1000
1000
Leddicottetej ,^
Humid
Site
2500
50
25
2500
2500
2500
Staleyv"
Sand
100
2
0.1
20
200
70
Staley^t;
Silt &
Clay
1000
20
1
200
2000
700
(a) R. C. Arnett, D. J. Brown and R. G. Baca, Hanford Grpundwater Transport Estimates for Hypo-
thetical Radioactive Waste Incidents. ARH-LD-162, Atlantic Richfield Hanford Company,
Richland, Washington, June 1977.
(b) Sheffield. Low-Level Radioactive Waste Disposal Site - Report of Additional Investigations
Conducted During February and March. 1978. Nuclear Engineering Company. Inc. Louisville.
Kentucky, Revised May 5. 1978.(Distribution coefficients for sands.)
(c) Sheffield. Illinois Low-Level Radioactive Waste Disposal Site - Responses to Questions,
Nuclear Engineering Company, Inc., Washington, DC, June 29, 1978.(Distribution coefficients
for loess, till, and shale.)
(d) A. E. Aikens, Jr., R. E. Berlin, J. Clancy and 0. I. Oztunali, Generic Methodology for Assess-
ment of Radiation Doses from Groundwater Migration of Radionuclides in LWR Wastes in Shallow-^
Land Burial Trenches, AIF/NESP-013, Prepared by Dames and Moore for Atomic Industrial Forum.
Inc., January 1979. (Estimated values for a typical desert soil.)
(e) G. W. Leddicotte and W. A. Rodger, "Suggested Quantity and Concentration Limits to be Applied to
Key Isotopes in Shallow Land Burial," in M. W. Carter, et al., Management of Low-Level Radioactive
Waste, Vol. 2, New York, Pergamon Press, 1979.
(f) G. B. Staley, G. P. Turi and D. L. Schreiber, "Radionuclide Migration from Low-Level Waste: A
Generic Overview," in M. W. Carter, et al., Management of Low-Level Radioactive Waste. Vol. 2,
New York, Pergamon Press, 1979. (Distribution coefficients for sand were selected to be near the
low end of the range of values reported in NUREG-0140. Kd values were increased by a factor of 10
for slit, clay, loess, and till.)
CO
ro
r>o
-------
8.000E-05
UJ
ce.
=)
g 6.000E-05
o
4.000E-05
2.000E-05
100 YEAR LEACH TIME
1000 YEAR LEACH TIME.
— --^ — —
300
FIGURE 1.
320
340
360
380 400 420
TIME (YEARS)
440
460 480 1500
Predicted Technetium-99 Concentration in the Ground Water at the Point of
Discharge to the Surface Stream
CO
ro
CO
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324
of discharge to the surface stream. It also results in a significant post-
ponement of the time when this concentration attains its maximum value. For
a radionuclide with a large K^ value, such as 230ih, neither the maximum
radionuclide concentration nor the time when the concentration attains its
maximum value are significantly affected by an order of magnitude increase
in leach time.
To improve the reliability of modeling studies of radionuclide trans-
port, the need exists to demonstrate (verify) models on actual field sites
and to develop better methods for measuring critical parameters such as
distribution coefficients and leach rates under actual field conditions.
CONCLUSIONS
Decommissioning of reference shallow-land burial grounds by the alter-
native of waste removal is estimated to be much more expensive and to
result in a much larger dose to decommissioning workers than site stabili-
zation. Therefore, site stabilization is assumed to be the preferred decom-
missioning alternative. Site stabilization would probably result in the
conditional release of a decommissioned site with use restrictions that
would prevent excessive public exposure to radiation and that would ensure
the adequate confinement of the buried waste.
A methodology has been developed for evaluating use restrictions for
the conditional use of a burial ground after burial operations cease. The
methodology uses pathway modeling techniques to estimate the calculated
maximum annual dose to the maximum-exposed individual. Results and con-
clusions of the pathway modeling approach depend on the site characteristics,
on the radionuclide inventories, and on the mathematical models and the
parameters employed with the models to represent specific site character-
istics.
Some uncertainties exist in nuclide transport models and major uncer-
tainties exist in parameters and boundary conditions used in the models to
represent a specific site. Several research needs can be identified that
would result in improved predictions of radionuclide migration from
shallow-land burial grounds. These research needs include:
1. development of transport models for overland flow
2. verification of models by comparison of predicted results with experi-
mental results for real sites
3. identification and quantification of differences between field- and
laboratory-measured values of distribution coefficients
4. determination of leach rates for specific radionuclides under field
conditions
5. quantification of the effect of complexing agents on the mobility and
transport of radionuclides from shallow-land burial grounds.
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325
REFERENCES
1. E. S. Murphy and G. M. Holter, Technology, Safety and Costs of Decom-
missioning a Reference Low-Level Waste Burial Ground. NUREG/CR-0570,
Pacific Northwest Laboratory for U.S. Nuclear Regulatory Commission,
June 1980.
2. S. W. Ahlstrom, et al., Multicomponent Mass Transport Model: Theory
and Numerical Implementation (Discrete-Parcel-Random-Walk Version).
BNWL-2127, Pacific Northwest Laboratory, Richland, Washington,
May 1977.
3. P. Colombo and R. M. Neilson, Critical Review of Properties of Solidified
Radioactive Waste Packages Generated at Nuclear Power Stations. BNL-
NUREG-50591, Brookhaven National Laboratory, Upton, New York, 1976.
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A SYSTEMS MODEL FOR EVALUATING POPULATION DOSE
FROM LOW-LEVEL WASTE ACTIVITIES
J. A. Stoddard D. H. Lester
Nuclear Technology Division
Science Applications, Inc.
1200 Prospect St.
LaJolla, California 92038
ABSTRACT
This paper briefly describes a computer code
developed to evaluate population dose or maximum individual
dose from the various activities associated with shallow land
burial of low-level wastes. The code, called BURYIT, treats
a wide variety of scenarios to determine the effects of
various parameters associated with siting, packaging and
burial procedures on potential public exposure. In the
analysis of radionuclide dispersal, BURYIT treats unsaturated
zone seepage, aquifer transport, wind erosion and atmospheric
transport. Population exposure pathways considered include
direct exposure to undispersed wastes, direct exposure to
contaminated air, direct exposure to contaminated ground,
inhalation of contaminated air, and ingestion of contaminated
food, water and milk.
INTRODUCTION
As part of an effort to develop analysis tools to support
rulemaking and evaluation of license alternatives, the Nuclear
Regulatory Commission* has sponsored the development of a systems level
computer code (BURYIT) for use in assessing potential population dose
from the various activities associated with low-level waste disposal.
This code is currently under development by Science Applications, Inc.
(SAI). When completed, it will derive either population dose or
maximum individual dose for a wide variety of shallow land burial
problems and conditions and will be used to evaluate the effects of
several parameters including site location, site design, package
integrity and operational procedures.
*NMSS, Low-Level Waste Branch, J. D. Thomas, Project Officer.
327
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328
PROBLEM DESCRIPTION
The problems which must be analyzed are varied and broad in
scope. Many different release scenarios, ranging from short term to
long term and involving different pathways, must be analyzed.
The scope of the problems was established early in the code
development project. At that time, we visited five shallow-land burial
sites (Maxey Flats, Barnwell, Rich!and (NECO), Han ford (DOE) and
Beatty) and reviewed available literature to determine the nature of
low-level wastes, their sources, and the most likely pathways which
would lead to public exposure. We identified over fifty release
scenarios for the various phases of low-level waste disposal. These
scenarios ranged widely in characteristics and indicated the need for
a very versatile code. For example, we wanted to be able to evaluate
direct radiation exposure from wastes in transport or in an open pit,
population exposure from contamination released by explosion or fire,
and long-term exposure from nuclides leached from the site over periods
of thousands of years.
CODE DESCRIPTION
Our approach to solving the variety of release scenarios was
to develop a modular code which is software linked within the
executive. This software linking enables BURYIT to call the
appropriate analysis modules in the proper order for the particular
scenario. For example, for a scenario involving leaching of nuclides
to the surface followed by wind erosion and atmospheric transport, the
scenario-related input data includes a specification for successive
calls to the transport module for unsaturated soil, the wind erosion
module, the air transport module and the dose evaluation module.
The executive, in solving a problem, may call some or all of
the following modules:
PREDOS - This module loads required data for the nuclides to be
analyzed. These data include half lives, dose factors, and transfer
coefficients (for example, from grass to milk via cow). The data are
extracted by direct access reads from a disk file containing data for
about 350 radionuclides.
UNSAT - This module, which was derived from the HYDRO code, calculates
the transport of radionuclides in the unsaturated soil zone. Given an
isotope distribution in layered soil of specified characteristics, it
calculates the nuclide distribution as a function of time, treating
nuclide discharge to an aquifer and surfacing by evapotranspiration.
The HYDRO code, from which UNSAT was derived, was developed in 1979 by
SAI[1] based on an irrigation control model developed for the U.S.
Environmental Protection Agency.[2]
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329
AQUIFR - This module, which was developed from the PNL GETOUT code,[3]
calculates nuclide transport in the saturated soil zone. An aquifer is
treated as an one-dimensional flowing body with axial dispersion,
sorption in the soil, and nuclide decay. The calculated output is the
aquifer discharge rate for each radionuclide.
EROSIO - The EROSIO module calculates the rate at which soil is eroded
by wind action. When the soil is contaminated, perhaps by a
combination of leaching with evapotranspiration, EROSIO provides the
quantity of radioisotopes released for atmospheric transport. EROSIO
is a modification of the U.S. Department of Agriculture WEROS (wind
erosion) code.[4]
ATMOS - The ATMOS module calculates the atmospheric transport of
contaminants by use of standard Gaussian plume techniques. For the
calculation of population dose, sector averaged air and ground
radionuclide concentrations are calculated. Annual average results can
be calculated based on an array of input weather conditions or results
can be calculated for a specified input weather condition. Both wet
and dry deposition and plume depletion are taken into account.
DOSE - This module calculates the maximum individual dose and
population dose for the following exposure pathways: (a) direct
exposure to an airborne cloud of contamination, (b) direct exposure
from ground contamination, (c) inhalation of airborne contamination and
(d) ingestion of contamination. The inhalation pathway includes
resuspended contamination as well as nuclides in the initial plume.
The ingestion pathway includes water via contaminated aquifer, leafy
vegetables contaminated by atmospheric deposition, produce contaminated
by root uptake, and beef and milk contaminated by animal ingestion of
contaminated pasture grass.
For the maximum individual dose, the food and milk ingestion
are based on quantities specified by NRC Regulatory Guide 1.109. For
calculation of the population dose, it is assumed that all foods
produced are eaten and thus contribute to population exposure. This
production based (rather than consumption based) ingestion model
eliminates the need for concern about foods exported from or imported
into the analysis area.
DIRECT - This subroutine uses formulas from Rockwell[5] and Foderaro[6]
to calculate the direct gamma radiation exposure from undispersed
wastes. Example problems are direct radiation from wastes in the open
pit or shielded or unshielded individual waste packages during
transportation or processing.
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330
DATA BASES
BURYIT extracts data from three data bases: a scenario data
base, a radionuclide inventory data base, and a data base of
radionuclide properties.
The scenario data base contains release pathway descriptors
for over fifty different release scenarios. These release scenarios
were derived for the following phases of operation and post-operation:
waste packaging, waste transportation, waste arrival phase (pre-entry
inspection), on-site handling and trench emplacement, storage in an
uncovered trench, waste burial (including backfill and compaction),
post-burial (early time period, <100 years) and post-burial (later time
period, >100 years). The 100-year breakpoint corresponds to an
anticipated change in post-burial institutional controls.
The radionuclide inventory data base contains source terms
for five sets of wastes: control rods and high activity components from
light water reactors (LWRs); miscellaneous wastes (resins, sludges,
trash, etc.) from LWRs; LWR decomissioning wastes; miscellaneous wastes
from hospitals and similar institutions; and typical waste
concentrations in a low-level waste shallow-land burial trench.
The radionuclide properties data file contains half lives,
dose factors and gamma emission data (by gamma energy group) for over
350 radionuclides.
INPUT AND OUTPUT
Input data falls into the following categories: scenario and
inventory selections, weather data, geologic data, demographic data and
option switches. Of these, the geologic data tends to be the most
bulky. We may, in the future, develop a geologic data file with
element and soil type dependent coefficients from which BURYIT can
extract the required soil data.
The output is the population or maximum individual dose given
by organ, pathway, isotope, population age group, and radial distance
from the source. Additional detail from the analysis process can be
output on option.
CODE EXECUTION EXPERIENCE
BURYIT has been used to generate some sample cases and to
perform some sensitivity studies, but has not yet been used to analyze
any existing or proposed low-level waste disposal facilities.
Execution times vary from a few CPU seconds on a DEC-10 computer for
simple problems involving the calculation of only direct gamma exposure
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331
to nearly an hour for complex problems involving many radionuclides,
many soil layers and hundreds of repeated wet/dry cycles for chronic
release problems involving groundwater transport and time periods of
thousands of years. Typical problems require less than five minutes
CPU.
In the sensitivity study, major variables affecting the
result for a given airborne release scenario were the population, the
agriculture and the weather pattern. Major water pathway sensitivity
parameters were aquifer travel time and turnover and nuclide
solubility.
Test cases for generic facilities indicated relatively high
population doses from incinerator accidents involving the airborne
release of respirable contamination. Chronic releases via soil water
pathways tended to be small because of nuclide binding in the soil.
REFERENCES
1. B. Amirijafari and B. Cheney, "Simulation of Radionuclide
Transport in the Unsaturated Zone Beneath the Radioactive Waste
Management Complex," report submitted to EG&G by Science
Applications, Inc. (June 1979).
2. L. G. King and R. J. Hanks, "Irrigation Management for Control
of Quality of Irrigation Return Flow," prepared for USEPA,
EPA-R273-265 (June 1973).
3. W. V. DeMier, M. 0. Cloninger, H. C. Burkholder, and P. J.
Liddell, "GETOUT - A Computer Program for Predicting Radionuclide
Decay Chain Transport Through Geologic Media," Pacific Northwest
Laboratory, PNL-2970 (August 1979).
4. N. P. Woodruff, and F. H. Siddoway, "A Wind Erosion Equation,"
Soil. Sci. Soc. Amer. Proc. 29 602-608 (1965).
5. Rockwell, Reactor Shielding Design Manual. TID-7004 (March 1956).
6. A. Foderaro, The Photon Shielding Manual, Pennsylvannia State
University (1975T.
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MODELS AND CRITERIA FOR LLW
DISPOSAL PERFORMANCE
Craig F. Smith
Jerry J. Cohen
Lawrence Livermore National Laboratory, University of California
Livermore, CA 94550
ABSTRACT
A primary objective of the Low Level Waste (LLW) Management
Program is to assure that public health is protected. Predictive
modeling, to some extent, will play a role in meeting this objective.
This paper considers the requirements and limitations of predictive
modeling in providing useful inputs to waste management decision
making. In addition, criteria development needs and the relation
between criteria and models are discussed.
PREFACE
This paper is a compilation and summary of informal remarks
delivered to the LLW Modeling Workshop at Denver, CO in December, 1980.
INTRODUCTION
Calculational models for the prediction of the consequences of
waste management activites should be consistent with applicable
criteria; however, definitive criteria for judging the effectiveness
of low-level waste management activities have not, as yet, been
established. Despite the absence of official criteria, there has been
a considerable amount of work on model development. Aside from the
development of definitive criteria, there are several other areas that
should be defined before beginning model development. Model
development requires consideration of what is to be predicted, how
accurate it must be, and how it can be validated.
Over three hundred existing models applicable to low-level
radioactive waste have been identified.! Yet, an extensive degree
of effort in model development continues. Given this situation, one
might logically ask the questions: Why isn't what we have good
enough?; What are the specific gaps that need to be filled?; and How
can we recognize an acceptable model should one evolve? These
questions should be resolved before more extensive model development
takes place. Failure to do so could result in an endless quest for
some unattainable state of perfection.
*This work was performed under the auspices of the U.S. Department of
Energy by the Lawrence Livermore National Laboratory under contract
No. W-7405-Eng-48.
333
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334
PATHWAYS FOR EXPOSURE
The determination of what processes are to be modeled requires the
identification of pathways for human exposure to low-level waste
materials. Figure 1 illustrates the major pathways of concern.
Water pathways are generally believed to be the most important
pathways for human exposure. The two forms of water pathways are
surface runoff and groundwater transport. Climatic conditions (arid
versus humid) strongly influence the importance of these pathways.
The airborne pathways can result from operational release during
waste disposal activities, release to the air through erosion, and
release in conjuction with the evolution of gases in the waste
material. The erosional release pathway is a consideration primarily
during the post-closure phase of the disposal operation.
Intrusion is another potential pathway for exposure to LLW
materials. Human intrustion can be either intentional or
inadvertent. In addition, plant and animal intrusion present
potential exposure pathways.
Other pathways include combinations of the previously discussed
modes. For example, surface runoff followed by subsequent wind
transport could be considered.
An important objective should be the identification of pathways so
trivial that attention can be more reasonably directed to those which
have a greater likelihood of transporting significant quantities of
hazardous material to the biosphere.
MODELING APPROACHES
Considering the many potential pathways for human exposure, a
review of previous modeling studies indicates that perhaps an
inordinate amount of effort has been devoted to the groundwater
pathways. This is particularly true in light of previous studies2»3
which indicate that in terms of maximum individual exposures, the
groundwater pathways constitute a relatively minor threat. Figure 2,
taken from reference 2, shows the results of such a study.
There are several major approaches to the modeling of
environmental transport. These include:
• Physical (Analytical)
• Empirical (Analog)
t Statistical (Probabilistic)
t Combinations
Physical models are determined by an analytical or theoretical
description of the process being modeled based on first principles.
Due to the infinite complexity of real world processes, physical
models can never be exact predictive tools.
Empirical models are based on experimental data without the
necessity for a detailed underlying theoretical basis. They represent
an acceptance of the inability to thoroughly understand the process
while making the best use of the data that we do have.
Statistical models use probabilistic inference to model the
process. In some cases this is because the underlying theory is
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Figure 1
Evnlronmental Pathways for LLW
1
[WATER |
T~~
i
SURFACE
RUNOFF
1 i
IGROUNDWATERI OPERATIOI
RELEASE
| PATHWAYS |
1
|
IAIR | IN-
i i
WL lEROSIONl GAS (INTENTION/
EVOLUTION
1 1
FRUSION | | OTHER
|
]
\L| INADVERTENT!
. 1 ! .
HUMAN I IPLANTIIANIMALI
CO
CO
en
-------
(0
TJ
T3
1
O
CL
8
X
CO
Q)
*-<
S
Qi
O
'ro
c
c
10-
10
T 10 102 103 104
Pu-239 interface concentration - juCi/cm3
10s
CO
CO
FIG. 2. Annual individual dose vs HLW/LLW interface concentrations
239
for Pu calculated for six exposure scenarios.
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337
probabilistic in nature, and in other cases it is a reflection of the
uncertainty about input parameters or the detailed underlying
physics. Models may also use a combination of these approaches.
MODEL LIMITATIONS
There are several well known limitations to predictive modeling.
These include data limitations (where the complexity of the model may
outrun the available data), computational limitations (where the
complexity of the model outruns the computational capabilities), and
developmental limitations (where time, economic, or other factors
limit the desirability or capability for model development). In
addition, the apparent search for perfection in predictive modeling is
not only unnecessary, but may even be counterproductive.
There appears to be a tendency in model development to equate the
level of complexity and detail with the credibility of the results.
Modelers should be cautioned to keep in mind the famous GIGO
admonition (garbage in, garbage out). Model complexity should never
be used to disguise either lack of understanding of phenomenology, or
deficiency in quality of data input. No degree of model complexity or
precision can compensate for a lack of either understanding or
information in the analytical process.
VALIDATION
In light of these limitations, It is important to consider the
question of model validiation. In the absence of experience
(particularly the required observations over prolonged time periods)
the process of validation is tenuous at best. How can we tell if a
model works or not? One possible approach is to use natural analogs
for the purpose.4
An application of this approach may be found in predictive
modeling for the effects of Iodine-129 in radioactive waste. This
radionuclide has been a source of concern because of its extremely
long ( -lO? yr) half-life.5 A recent study6 indicates that, as
a result of groundwater leaching of high level waste, a maximum
radiation dose of 3.3 rem/yr to the thyroid could result. Although we
are unfamiliar with the details of the model producing this result,
the prediction can be checked using the natural analog approach.
Assuming the high-level waste repository contained the waste from
GWe-yr of power production, it would have a total inventory of
about 10 gm of 1-129. Further, assuming the repository was 1000 m
deep and sited in a watershed of 104 km2, the total natural iodine
inventory in the top 1000 m layer of earth would be about 8 x
1Q12 gm. if the iodine in the waste leaches at a rate no more or
less than that in the soil, then, at equilibrium, the average ratio of
I-129/total iodine is 1.3 x 10'6 at the point of groundwater
discharge. Because of its low specific activity, it can be shown that
if all the iodine in the human thyroid were 1-129, the maximum dose to
that organ would be ~19 rem/yr. Therefore an individual receiving his
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338
total iodine intake from that watershed could receive a maximum
thyroid dose of only (1.3 x 10~6) x 19 = 2.4 x lO'5 rem/yr, or
about five orders of magnitude less than the predicted 3.3 rem/yr.
Although such predictive models are based on admittedly conservative
assumptions, it might be in order to question the degree of
conservatism. Excessive conservatism, sometimes bordering on the
absurd, can call the entire modeling procedure into question.
CRITERIA
As indicated previously, the development of models in the absence
of definitive criteria is problematic. Previously proposed guidelines
have tended to be vague. Such guidelines are not very useful in
planning the development of models.
While radiological protection should be the overriding concern,
the approaches to the required analysis is sometimes confusing.
Emphasis on the maximum individual dose results in "worst case
analysis". Two deficiencies of the "worst case" approach are: (1) it
is highly dependent on the imagination of the modeler in selecting his
assumptions, and (2) to the non-professional, the result may appear to
reflect reality instead of an extreme and unlikely situation. An
approach toward eliminating this problem would be to incorporate the
concept of probability into the modeling process. This approach
requires the determination of probabilistic criteria.7
In any case, definitive standards are vital. If you don't know
what you are looking for, the chances are that you won't find it. In
addtion, predictive modeling capabilities should not dictate
criteria - it should be the other way around.
SUMMARY COMMENTS
Definitive criteria are essential in the planning of waste
management activities. In particular, the strategy for model
development in support of the LLW program would logically depend on
the prior establishment of criteria in order to ensure reasonableness
and consistency.
Model development should take into account the question "how good
is good enough?" in addition to pathways (what is to be modeled) and
approach (how to do it). Recognizing the limitations of the modeling
process, it should also be noted that models are tools to assist in
the decision making process. They should not be considered to be an
end in themselves.
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339
REFERENCES
1. Mosier, J. E., et al., "Low Level Waste Management: A Compilation
of Models and Monitoring Techniques," ORNL/SUB-79/13617/2, April
1980.
2. Cohen, J. J. and W. C. King, "Determination of a Radioactive Waste
Classification System," UCRL-52535. March 1978.
3. Adam, J. A., and V. L. Rogers, "A Classification System for
Radioactive Waste Disposal - What Waste Goes Where?" NUREG-0456,
June 1978.
4. Cohen, B. L. "A Super Analog Computer for Evaluating the Safety of
Buried Radioactive Waste," Waste Managetnent-81, Tucson, 1981.
5. Kocher, D. C., "A Dynamic Model of the Global Iodine Cycle for the
Estimation of Dose to the World Population from Releases of
Iodine-129 to the Environment," ORNL/NUREG-59, 1979.
6. Croff, A. G., J. 0. Blomeke and B. C. Finney, "Actinide
Partitioning - Transmutation Program Final Report," ORNL-5566,
June 1980.
7. Cohen, J. J., "Suggested Nuclear Waste Management Radiological
Performance Objectives", NUREG/CR-0579. UCRL-52626. December 1978.
DISCLAIMER
This document was prepared as an account of work sponsored by an agency of
the United States Government. Neither the United States Government nor the
University of California nor any of their employees, makes any warranty, ex-
press or implied, or assumes any legal liability or responsibility for the ac-
curacy, completeness, or usefulness of any information, apparatus, product, or
process disclosed, or represents that its use would not infringe privately owned
rights. Reference herein to any specific commercial products, process, or service
by trade name, trademark, manufacturer, or otherwise, does not necessarily
constitute or imply its endorsement, recommendation, or favoring by the United
States Government or the University of California. The views and opinions of
authors expressed herein do not necessarily state or reflect those of the United
States Government thereof, and shali not be used for advertising or product en-
dorsement purposes.
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Workshop Summaries
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SUMMARY OF RELEASE MECHANISMS WORKSHOP
L. R. Dole, Chairman
D. E. Fields,* Secretary
4, Chemical Technology Division
Health and Safety Research Division
Oak Ridge National Laboratory
Oak Ridge, Tennessee 37830
T
The purpose of this workshop was to identify the parameters that
control the release rates of radionuclides from low-level waste (LLW)
shallow land burial sites in which radioactive waste has been and will
be placed. This panel (for participants, see Appendix A) searched for
deterministic relationships describing source terms for the low-level
waste trenches whose components are waste forms, containers, backfills,
and other "engineered" bariers.
The need for this release mechanisms workshop was established in
a previous group of workshops [1], during the special Interagency Meeting
on Low-Level Waste Environmental Modeling in Bethesda, Maryland,
July 30-31, 1980. All three July 1980 workshops, (1) Model Status and
Development, (2) Validation and Verification, and (3) Criteria for
Selecting and Applying, declared the inadequacy of the present descrip-
tion of the low-level waste disposal site source terms, and the lack of
models describing the leach mechanisms and the chemical forms of poten-
tial contaminant releases, resulting from shallow land burial sites
containing LLW.
Therefore, the goal of this panel was to discuss the past experience
and ongoing research concerning the chemical, biological, and mechanical
processes summarized in Table 1.
Due to the makeup of the panel, discussions focused most sharply on
Mechanical Processes. Based on several field studies at existing burial
sites and operating experiences, the discussions summarized below
(1) described the three common site geologies, (2) identified subsidence-
driven scenarios, (3) identified site surveillance requirements,
(4) determined the present "state-of-the-art" of in situ source term
characterization, and (5) outlined the course of action to effectively
improve our quantitative understanding of release from LLW burial
trenches.
The panel was skeptical that one "standard site model" could deal
with the range of geological and climatological settings in which LLW
343
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344
Table 1. Processes Related to Radionuclide Releases
from Low-Level Shallow Land Burial Sites
1. Chemical Processes
a. Leaching of waste forms
b. Canister corrosion and degradation
c. Nuclide speciation, complexation, etc.
d. Nuclide retardation in backfill systems
e. Others
2. Biological Processes
a. Barrier degradation
b. Mobilization of fission products and actinides
c. Gas generation
d. Others
3. Mechanical Processes
a. Compaction and subsidence
b. Failure of barriers
c. Water penetration of capillary barriers
d. Others
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345
burial sites are placed. A minimum of three typical settings were con-
sidered. First most sites west of longitude 100°W are located in porous
host rocks and in semiarid to arid climates. Since these sites gener-
ally lose more moisture through transpiration than is received from
precipitation, there is little or no hydraulic gradient under normal
conditions.
Conversely, the two classes of eastern sites receive up to forty
inches of annual precipitation. The permeable eastern sites are located
in a range of hosts from sandy soil to fractured shales. On the other
extreme, the third site classification includes eastern sites located in
clay sequences in which essentially no permeability can be measured.
Burial site operators from all of the above classes of burial sites
reported subsidence in their trenches. The impact of this subsidence on
the trench cover has ranged from minor fissures to gaps that a man could
fall into. In isolated cases, the trench contents were visible. These
cases have ranged from potential to actual short circuits of the pathway
to the biosphere.
The panel considered that the probabilities and consequences of sub-
sidence features generally represented a more significant risk than
ground water scenarios. Subsidence short-circuit scenarios include:
(1) funnel ing run off through the trench, accelerating ground water
contamination, (2) flooding and overflowing of the trenches, delivering
activity to the ground surface, and (3) exposing the waste to weathering.
Operating practices have been developed to mitigate some of the
causes of subsidence listed in Table 2. While vibratory compaction tech-
niques have proven effective in filling voids and providing dense back-
fills, there are problem wastes and some subsidence is inevitable.
Table 2. Causes of Waste Burial Trench Subsidence
1. Compaction of the waste forms
2. Failure of backfill to fill voids between
containers
3. Compaction of loose backfill material
4. Degradation of the waste by corrosion, biological
decay, and dissolution
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346
Therefore, a surveillance and trench cap repair program is required
for at least the first ten years after the trenches are closed. It is
conceivable that a trench subsidence surveillance program should be
carried on for upwards of 50 years.
Panel discussion explored the possibility that trench cover fissuring
may have more severe consequences in sites in impermeable formations. At
first glance, the "mother nature" permeable systems, which rely on long
travel times into and through the ground water to the biosphere, appeared
less vulnerable to the short-circuit delivery to the ground surface from
an overflowing trench —the so-called "bath tub effect." However, the
permeable sites are still subject to flash flooding and are not exempt
from this scenario. Also, disposal in an impermeable host is highly
desirable since travel times can be measured in geological eras.
The potential "bath tub" scenario does not preclude sites in imper-
meable hosts. It does demand that the trench covers be designed care-
fully, that waste forms that do not contribute to subsidence are used,
and that there is a rigorous surveillance program. Nevertheless, imper-
meable sites are good choices for the very long-lived nuclides.
In an attempt to coyer the processes in Table 1, the panel explored
the results of site studies, which included ground and trench water
sampling and trench exhumation. In spite of the diversity in waste forms
and sites, there were some consistent observations.
Shallow burial trenches generally are deep enough to insure anoxic
reducing conditions. With the exceptions of rare geochemistries and
aggressive reagents inside drums, corrosion of normal carbon steel can
be slow. There are cases of intact drums older than 10 years. Even the
biodegradation of clothing and cardboard at a South Carolina sites was
small after 10 years.
The inhomogeneity of the waste is an insurmountable problem. Most
of the trench contents are there because they were suspect and may have
less activity than the dirt that was moved to bury them. It has been
observed that 90% of the activity in a particular trench was in one small
group of boxes, which represented a negligible contribution to the volume
of the waste buried.
Regional ground water sampling, because of long travel times, is a
very long-range effort. It is difficult to obtain representative trench
water samples because of capillary holdup in the waste host, which
results in poor flow distribution through coarse sump materials. While
a "bath tub" trench might make leachate more readily accessible, the
inhomogeneity of the trench contents makes an overall quantitative
accounting questionable.
Nevertheless, attempts have been made to correlate the radionuclide
inventories from hopeless shipping records and the activity in trench
waters. Table 3 presents the guesstimates of yearly fractional release
(FR).
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347
Table 3. Estimates of Yearly Fractional Release (FR)
from LLW Shallow Land Burial Trenches
Site FR
Savannah River 10~8
Oak Ridge 10"6
West Valley 2.5 x
The uncertainty range in the FR of four orders of magnitude cannot
be reduced in the present situation. In the case of the existing trenches
as they are found, both the inhomogeneity of the contents and the uncer-
tain nuclide inventories preclude the possibility of refining our quan-
titative understanding of releases.
There are two solutions to this dilemma. The first requires two
assumptions. These are (1) that the nuclide retention in the trenches
is insignificant in respect to the travel times to the biosphere, and
(2) that the chemical buffer capacity of the host is sufficient to over-
come any adverse chemical effect on the nuclide retention from the trench
contents. These assumptions reduce the LLW trench source term problem
to estimating inventories.
The second course of action involves a ten-year research program.
This program entails selecting sites and waste categories, segregating
the waste, assaying the nuclide content, and preparing study trenches.
Because the processes controlling the source terms are slow, the in situ
study requires at least ten years. This field data will require an elab-
orate laboratory back-up study in order to interpret the results in terms
of quantitative deterministic algorithms which are based on specific
release mechanisms.
In summary, the decision to institute a lengthy, costly, and complex
study balances against the potential impact of a regulatory decision
requiring expensive waste conditioning and, possibly, extensive remedial
treatment of buried waste.
In the previous workshops, the premise was stated that a model must
first simulate before it can predict. This workshop indicated that a
data base to successfully test a simulation does not exist in the case
of shallow land burial. Furthermore, this workshop concluded that such
a data base is not likely to be generated from current burial sites.
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REFERENCES
1. R. S. Lowrie et al., Summary of the Meeting on Low-Level Waste
Environmental Modeling, July 30-31, 1980, Bethesda, MD,
ORNL/NFW-80/26.
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APPENDIX A
List of Attendees
RELEASE MECHANISMS WORKSHOP
Name
E. L. Albenesius
Jim Bennett
Clinton M. Case
Don A. Clark
Richard Codell
Leslie R. Dole
Ken Erickson
David E. Fields
Jesse Freeman
Gerry Grisak
Cheng Y. Hung
Donald Jacobs
Thomas Johnson
Craig A. Little
Steve Phillips
David E. Prudic
Andrew E. Reisenauer
George Saulnier
Jacob Sedlet
Robert W. Thomas
Edwin P. Weeks
Ken Whitaker
Charles R. Wilson
Location
Savannah River Laboratory
U.S. Geological Survey
Desert Research Institute
U.S. Environmental Protection Agency
U.S. Nuclear Regulatory Commission
Oak Ridge National Laboratory
Sandia Laboratories
Oak Ridge National Laboratory
U.S. Nuclear Regulatory Commission,
WM Uranium Recovery
Environmental Canada,
National Hydrology Research Institute
U.S. Environmental Protection Agency
Evaluation Research Corporation
Illinois State Geological Survey
Oak Ridge National Laboratory
Rockwell Hanford,
Research and Engineering
U.S. Geological Survey
Pacific Northwest Laboratory
TRW, Inc.
Argonne National Laboratory
Los Alamos Scientific Laboratory
U.S. Geological Survey
Chem Nuclear Systems, Inc.
Lawrence Berkeley Laboratory
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SUMMARY OF WORKSHOP ON OVERALL SYSTEMS MODELING
James 6. Steger, Workshop Chairman
Los Alamos National Laboratory
Los Alamos, NM 87545
Leroy E. Stratton, Workshop Secretary
Oak Ridge National Laboratory
Oak Ridge, TN 37830
This workshop was tasked to look into the need for and status of
overall systems modeling for low-level waste. In attendance were 15
modelers, 6 managers/users, and 5 people who claimed to be neither of
the above.
The approach consisted of providing the attendees with a written
list of 10 questions that were to be discussed in sequence. A consensus
position was to be obtained on all of the questions. The following is a
brief summary of the results.
Ql. What needs to be modeled? What submodels are needed?
This question led to a discussion that lasted in excess of one hour.
It was concluded that there need to be three different levels of
models.
I. Screening Model. A need exists for a single overall screening
model to make a first-order assessment. It should be able to
be expanded and improved upon as experience is gained in its
use. Probabilistic models should be considered as well as
deterministic models.
II. Site Evaluation. A more detailed level of models to evaluate
sites that pass the screening test is necessary. These models
should have submodels for source term, airborne pathway, sur-
face pathway, subsurface pathway, biological pathway, and human
intrusion.
III. Process/Mechanisms. Researchers trying to understand the many
processes/mechanisms involved in low-level waste disposal will
have needs for many models in their work to test theories,
explain experimental results and communicate with other
researchers. These models will be quite specific to the prob-
lem being investigated.
When applying models as a predictive tool for evaluation,
the three kinds of models listed above should be used in a
hierarchal sequence. If the simple screening model with con-
servative parameters shows that there are no problems, then
the modeling effort should stop. However, if there is an
351
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352
indication that limits are being approached, then a more
detailed analysis should be performed.
The group did not feel that the status of modeling was
adequate and did not agree with the DOE comment (see Large,
this volume) that no more models are needed. The group also
cautioned that there seems to be an over-emphasis placed on
groundwater pathway modeling.
Q2. Should the coupling of models proceed, and if so, how?
Coupling of models requires that they are capable of interact-
ing. In the case of Screening Models, they can probably be coupled,
but this level of detail is not necessary. Simple input/output rela-
tionships are adequate. For Site Evaluation Models, the pathway
models are not sufficiently developed to permit complete coupling.
For Process/Mechanism Models, coupling is problem dependent and
desirable in some cases.
Q3. Are different systems models needed for arid vs. humid sites? For
every site?
A single model can probably be developed for screening purposes
for both arid and humid sites. However, there are enough differences
between the arid and humid sites that a single site evaluation model
does not seem likely. It is desirable though that effort should be
made to keep the form and scope of these models as similar as possible
to facilitate comparison of the results. For process/mechanism model-
ing, specific models will be required.
Q4. What determines when models for a specific pathway are adequate?
This question generated considerable discussion, with the final
conclusion that the user must determine when a model was adequate.
A model is a tooT to help managers make decisions. The adequacy of
the tool will usually be proportional to the amount of time that the
modeler and user spend communicating. Once the user specifies what
he needs and the modeler understands these needs, it is up to the
modeler to assure model internal consistency and the proper balance
between components.
Q5. What level of accuracy should we target for?
As in the previous question, this question can only be answered
by the user. In any event, the status of modeling today will only
permit making relative comparisons. Obtaining absolute values are
beyond present capabilities.
Q6. Can enough data be obtained on an economical basis?
For Screening Models, enough data are already available or
relatively easy to obtain. The answer is not so clear for the Site
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Evaluation Model. Data will usually be obtainable on an economical
basis, but in some cases the data will be very difficult to acquire.
Process/Mechanism Models are usually designed to achieve complete
understanding and consequently usually require more data than is
easily obtainable. One caution was discussed at this time, and that
is that care must be used in taking data from other sources to use
in a model. Unless it is clearly understood how and why the data
were obtained, it is very easy to misuse others' data.
Q7. What is the major purpose of an overall systems model(site evalua-
tion, evaluation of operations, licensing, design of monitoring
systems)?
This question is probably not appropriate for the group's con-
sideration. Models contribute to all aspects of the problem, and
it is not clear how one judges what the major purpose is to be.
Q8. Can post-operational performance be modeled and predicted using
preoperational data?
The group felt that the answer to this question was yes with
the present level of experience and data, but recommended that moni-
toring data be used to adjust the model as necessary to keep it as
accurate as possible. Man-made modifications can be modeled better
than making predictions of nature's actions in the long-term
(300 years).
Q9. Do the models have the capability to handle different fuel cycles?
It was concluded that a different fuel cycle would not make
the problem any more difficult and therefore the answer to this ques-
tion is yes. The main problem at this time is obtaining adequate
information on waste properties and release mechanisms.
Q10. Are there any models that propose to include such processes as trans-
portation, regional siting, etc., for optimization of sites? Are
such models desirable?
The group did not know of any total systems models (generation
to dose-to-man), and felt that it would be unrealistic to try to
include all of these factors in a single model. For additional
information, several independent efforts are underway, which con-
sider parts of the overall system. EG&G is developing a waste
generator system model. Sandia is developing a transportation
model. USNRC is preparing an environmental impact statement to
assess the impact of the requirements of the proposed 10 CFR 61.
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SUMMARY
The group was very enthusiastic about the exchange which took
place. It is very rare when a working modeler can have a direct conver-
sation with the policy setters in the DOE, NRC, EPA, and USGS, as well
as his peers at the same time. There were many comments concerning how
valuable the workshop participants felt that the exchange this meeting
facilitated was.
There was surprisingly little difference of opinion or views remain-
ing after discussion and resolution of terminology. Possibly a glossary
of terms would be useful. Communication was a constant theme on almost
every question. Everyone agreed that it needs to be improved. The data
gatherers, model developers, and manager-users must fully understand
each other's needs and capabilities.
The issues of model adequacy and accuracy cannot be determined by
either the modelers or the users alone. There is a limit to what a
modeler can do, and the user must identify the minimum needs. Close
interaction is the only sure way to obtain satisfactory results.
The most serious limitation to overall system modeling is the lack
of information about waste properties and release mechanisms. There are
many different waste types and many different disposal environments.
This situation makes for a very complex series of possible actions and
reactions, which are impossible to model with existing data. Possibly
the only solution is to pretreat the waste and to tighten the specifica-
tion on disposal practices.
The final comment and the unanimous opinion of the group is that
there is a critical need for standards to be used as targets. Until one
knows what needs to be accomplished, it is difficult to determine how
best to get there or whether or not success has been achieved.
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APPENDIX A
Overall Systems Modeling Workshop Participants
Name
Affiliation
Richard Codell
Paul Dickman
David Grove
Ed Hawkins
Richard Healy
Richard Heystee
Preston Hunter
Tim Jones
Dan Kapsch
Ken Kipp
Eric Lappala
D. H. McKenzie
Jim Mercer
Lew Meyer
Bruce Napier
Yasuo Onishi
Fred Pail let
Dave Pollock
Jack Robertson
John C. Rodgers
Bob Root
Dale Smith
J. C. Sonnichsen
Jim Steger
Jim Stoddard
Leroy Stratton
Burnell Vincent
G. T. Yeh
U.S. Nuclear Regulatory Commission
EG&G Idaho
U.S. Geological Survey
U.S. Nuclear Regulatory Commission
U.S. Geological Survey
Ontario Hydro
Ford, Bacon & Davis
Battelle Pacific Northwest Laboratory
Monsanto Research Corporation
U.S. Geological Survey
U.S. Geological Survey
Battelle Pacific Northwest Laboratory
GeoTrans, Inc.
U.S. Environmental Protection Agency,
Office of Radiation Programs
Battelle Pacific Northwest Laboratory
Battelle Pacific Northwest Laboratory
U.S. Geological Survey
U.S. Geological Survey
U.S. Geological Survey
Los Alamos National Laboratory
Savannah River Laboratory
U.S. Nuclear Regulatory Commission
Westinghouse Hanford
Los Alamos National Laboratory
Science Applications, Inc.
Oak Ridge National Laboratory
U.S. Environmental Protection Agency,
Solid Waste Office
Oak Ridge National Laboratory
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SUMMARY OF WORKSHOP ON
ENVIRONMENTAL TRANSPORT PARAMETERS AND PROCESSES
E. L. Albenesius, Workshop Chairman
Savannah River Laboratory
Aiken, South Carolina 29801
C. A. Little, Workshop Secretary
Oak Ridge National Laboratory
Oak Ridge, Tennessee 37830
ABSTRACT
This workshop was held on the afternoon of
December 3, 1980 in Denver, Colorado. Nearly thirty
participants discussed aspects of modeling the
environmental transport of radionuclides from low-level
radioactive waste shallow land burial grounds to the
receiving human populations or individuals. Pathways of
transport considered included groundwater, surface water,
the atmosphere, and food chains". The ability to model
transport in these pathways was discussed. Several
modeling and data needs were brought to light and several
recommendations were made to regulatory and funding
agencies.
INTRODUCTION
The Workshop on Environmental Transport Parameters and Processes
was held on the afternoon of December 3, 1980 at the Denver Marina
Hotel, Denver, Colorado. Approximately thirty individuals participated
in the discussion; their names and affiliations are listed in
Table 1. The topics of discussion were limited to those that
considered only transport from the trench boundary to some human
receptor. Processes occuring within the trench were discussed in the
Workshop on Release Mechanisms. Questions of accuracy and precision
were largely discussed in the Workshop on Model Verification and
Validation.
PURPOSE OF THE WORKSHOP
The ultimate purpose of this workshop, as well as the complete
meeting, was to foster communication and cooperation between the
various regulatory and funding agencies, model users, and model
builders. In more specific terms, the purpose of the workshop was to
357
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358
Table 1. List of Attendees of Environmental Transport
Processes Workshop
NAME
AFF RATION
Ed L. Albenesius
Clinton M. Case
Don Clark
Ken Erickson
David E. Fields
Jesse Freeman
Richard Heystee
Cheng Y. Hung
Craig A. Little
James Mercer
Bruce Napier
Yasuo Onishi
David Pollock
Andy Reisenauer
Bob Root
George Saulnier
Jacob Sedlet
James Stoddard
R. 6. Thomas
Edwin P. Weeks
Charles Wilson
Ken Whitaker
G. T. Yeh
Savannah River Laboratory
Desert Research Institute
USEPA-Kerr Environmental Res. Lab.
Sandia National Laboratory
Oak Ridge National Laboratory
USNRC
Ontario Hydro
USEPA
Oak Ridge National Laboratory
Geo Trans. Inc.
Battelle Pacific Northwest Lab.
Battelle Pacific Northwest Lab.
USGS
Battelle Pacific Northwest Lab.
Savannah River Laboratory
TRW, Energy Systems
Argonne National Laboratory
Science Applications, Inc.
Los Alamos National Laboratory
USGS
Lawrence Berkeley Laboratory
Chem-Nuclear Systems, Inc.
Oak Ridge National Laboratory
discuss the role in modeling of groundwater, surface water, and
atmospheric transport processes and assess the relative importance of
each of these processes, the adequacy of existing models and data
bases, and the needs for further development. Table 2 lists the
processes and pathways considered by the workshop participants. As
mentioned earlier, processes occuring within the boundary of the
trench were specifically excluded from discussion by this workshop,
but were dealt with in the Release Mechanisms Workshop.
SURVEY OF MODELS
In an attempt to ascertain the state-of-the-art of environmental
transport modeling for low-level waste management, a pre-workshop
survey form was sent to each potential participant several weeks
prior to the Denver meeting. The survey form is appended to this
summary (Appendix A). For each pathway or process, respondents were
asked to submit names and characteristics of environmental transport
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Table 2. Pathways From Burial Ground Site to Man
and Processes That Were Considered by the Workshop
Groundwater Transport
Surface Water Transport
Resuspension
Atmospheric Transport
Irrigation
Food Chain Transfer
Direct Exposure
models with which they were familiar or had experience. This
exercise resulted in only a dozen model replies. Models were
identified for only three categories: groundwater, atmospheric
transport, and dose-to-man or "systems" models. These replies are
tabulated in Appendix B, Tables B.I - B.3. The groundwater model
capabilities ranged from one-dimensional to multidimensional
simulations of either water flow or transport of radionuclide decay
chains. The atmospheric codes listed included both chronic release
annual average models (AIRDOS EPA) and accidental release models
(CRAC). The dose-to-man models all included some representation of
the transport simulated by the groundwater and atmospheric models.
None of the models in Table B.3 are presently documented in public
literature. This is in contrast to the groundwater and atmospheric
codes listed in Tables B.I and B.2 which for the most part are
documented in the available literature. This brief exercise suggests
that, for low-level waste, dose-to-man modeling is much less well
developed than either of the other types of modeling.
WORKSHOP DISCUSSION
Several questions were discussed during the course of the
workshop. These included:
1. What is the relative ability to model each of the above
pathways or processes?
2. What needs to be done to improve existing or future models
of these processes?
3. If you were sponsoring the low-level waste modeling program
for the entire nation, on what research would you spend
your program budget?
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These questions were posed to generate discussion among the attendees
and for the most part were not answered directly or specifically.
Nevertheless, answers were discerned for each model type during the
ensuing discussion.
With regard to groundwater (GW) models, the point was made that
GW flow and transport codes were often so detailed and numerically
complicated that they have little utility for many regulatory uses.
The SWIFT code, for example, may cost as much as $200 per run. Uses
such as those by EPA where numerous sites must be screened would seem
to preclude using a code that is expensive to run. It was agreed
that, in terms of code development, the computer time was a small
expense compared to either development manpower or data collection.
The participants also agreed that, in general, the quality of
groundwater models greatly exceeds that of the data used for input
parameters (e.g., kj, storage coefficients).
Surface water flow and transport models were also discussed.
Onishi and his coworkers at PNL have recently reviewed 28 surface
water models and their characteristics (Onishi et al., 1981) The
group generally agreed that, with the exception of sediment
transport, surface water models are collectively adequate or better.
Sediment transport however, is not well developed. It was stated
that models, especially assessment models, usually strive for
description of average conditions and discount the effects of low
probability events. For example, the James River model developed for
the Kepone pollution case (Onishi and Wise 1978) has a deficiency in
the inadequate treatment of pulsed input such as wash from severe
flooding. In the case of a sediment transported pollutant, such as
Kepone, the greater human hazard may be neglected if sediment
transport resulting from low-probability events is not well modeled.
The relatively long time frames of low-level waste modeling may make
detailed modeling of either dissolved or sediment-attached pollutants
unnecessary; dilution over total surface water flow may be acceptable
with much less effort.
Consensus regarding atmospheric transport modeling was that the
Gaussian Plume model (Gifford 1968) is the best known and most widely
used model. The Gaussian Plume is reasonably accurate to distances
out to perhaps 80 km from the source except for situations of complex
meteorology. For modeling potential airborne impacts on humans from
low-level waste burial sites, resuspension will be the most difficult
process to model. Resuspension likely provides a large portion of
the material input to the transported plume. However, resuspension
models are at this point conceptually crude and for the most part
only empirical models (see Healy, this volume). Data to parameterize
resuspension models are also likely to be highly site-specific.
Food chain transfer models were seen to be a weak link in the
modeling of transport from burial grounds to humans. This is the
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case for two reasons. First, the models used to "transport"
radionuclides through food chains are generally simplistic
multiplicative chain or equilibrium models. For certain applications
such as integrated doses to a static population over a long time
period (several generations) these models may be adequate. However,
for any application resulting from short term releases the models are
crude. The second difficulty with food chain models is the lack of
quality element-specific data for most of the transfer factors in the
food chain (e.g., forage-to-milk transfer within cattle, gut-to-organ
transfer within humans).
A consensus among the modelers was that credible scenarios of
exposure need to be developed. So-called worst-case scenarios may
need to be discarded for the simple reason that they are not always
conservative; virtually any "worst-case" scenario may be made more
damaging by employing less realistic assumptions. Scenarios for
impact estimation from trench intrusion, site reclamation, on-site
spillage, etc. are needed for modelers to construct reasonable
modeling approaches. Many of the participants felt that some
important regulatory decisions were being made by modelers by default
when dose calculation scenarios were designed by modelers.
The environmental transport of radionuclides from a shallow land
burial ground is very complex. Most of the workshop attendees agreed
that the best use of modeling of low-level waste burial is as a
screening device for the selection of potential new disposal sites.
In this context, dose or risk estimates from models would be used in
a relative sense and would not be considered estimates of the
absolute radiological exposure or risk. Site selection and screening
processes should utilize not only models but surveys of geology,
hydrology, intuition, etc.
CONSENSUS OF THE WORKSHOP
The participants reached consensus on a number of questions and
issues. Five statements of agreement are summarized in this section.
Modeling expertise is generally adequate for the perceived needs
of modeling low-level waste environmental transport. However, some
of the submodels need improvement. In particular, submodels of
resuspension, sediment transport by surface water, and intra-trench
processes that lead to radionuclides leaching out of the trench need
conceptual improvement.
Even though modeling expertise is adequate, the predictive
capability of the models perse is not satisfactory. The lack of
faith in predictability arises from an awareness that the models do a
poor job of estimating absolute values. The models are of great
value in generating relative numbers. One such model application
would be ranking the suitability of several alternative sites.
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The participants agreed that to improve the ability to simulate
a given site the greatest need is for a more complete data base. An
example of needed types of data not readily accessible is given in
Table 3. Operators interested in predicting the performance of a
chosen new site would be advised to make an effort to acquire the
listed data. Many of these parameters require site specific
measurements and none of these data are readily available for
presently unidentified sites. Some data, such as radionuclide
inventory and physical form may be acquired through accurate record
keeping. Others, such as dispersivity values are not even measurable
quantities in the field, but are usually back-calculated from the
results of a hydrologic flow model.
The workshop agrees that of the numerous models listed in
several recent model compendia (e.g., Hosier et al., 1979), only a
few codes, probably less than fifty, were adequately documented.
Adequate documentation includes not only internal code comments, but
also a comprehensive user's manual describing theory, operation, and
a sample problem to check model results. It may also be a useful
task for some group to specify the use of given models in given
situations. In some ways this could be similar to the USNRC
specifying its Regulatory Guide models. What group would be
responsible for pursuing such an examination of models for low-level
waste is unclear.
A final consensus of the workshop is that credible scenarios of
human exposure to low-level waste buried in shallow trenches need to
be developed. Definition of allowed human activities under such
headings as intrusion, reclamation, etc. are, at present, not
standardized. This lack of definition makes difficult the
intercomparison of results from several models. Again, the location
of the responsibility for developing scenarios is not clear.
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Table 3. Necessary, But Not Readily Available, Data for
Modeling Low-Level Waste Impacts at Hypothetical
Burial Grounds
Trench Contents
Radionuclide inventory by activity
Chemical form of elements
Physical form of elements
Presence of extraneous material (pesticides, herbicides, etc.)
Distribution coefficients (k^)
Waste form interaction with biosphere
Geologic Characteristics
Geochemistry
Stratigraphy
Fracture distribution
Hydrologic Characteristics
Head distribution
Sorption coefficients (in presence of expected groundwater
chemistry)
Conductivity (for both saturated and unsaturated conditions and
in presence of expected groundwater chemistry).
Porosity
Dispersivity values
Boundary conditions
Water storage coefficients
Biological interactions (such as burrowing animals)
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REFERENCES
1. DeMier, W. V., M. 0. Cloninger, H. C. Burkholder, and
P. J. Liddell. 1979. GETOUT - A computer program for
predicting radionuclide decay chain transport through geologic
media. PNL-2970. Battelle Pacific Northwest Laboratories,
Richland, Washington.
2. Dillon, R. T., R. B. Lantz, and S. B. Pahwa. 1978. Risk
methodology for geologic disposal of radioactive waste: The
Sandia Waste Isolation Flow and Transport (SWIFT) Model.
SAND 78-1267, NUREG/CR-0424. Sandia Laboratories, Albuquerque,
New Mexico.
3. Gifford, F. A. 1968. An outline of theories of diffusion in
the lower layers of the atmosphere Chapter 3. In D. H. Slade
(ed.), Meteorology and Atomic Energy 1968 TID-24190. U. S.
Atomic Energy Commission/Division of Technical Information,
Washington, D.C.
4. Moore, R. E., C. F. Baes III, L. M. McDowell-Boyer,
A. P. Watson, F. 0. Hoffman, J. C. Pleasant, and C. W. Miller.
1979. AIRDOS-EPA: A computerized methodology for estimating
environmental concentrations and dose to man from airborne
releases of radionuclides. ORNL-5532. Oak Ridge National
Laboratory, Oak Ridge, Tennessee.
5. Mosier, J. E., J. R. Fowler, C. J. Barton, W. W. Tolbert,
S. C. Meyers, J. E. Vancil, H. A. Price, M. J. R. Vasko,
E. E. Rutz, T. X. Wendeln, and L. D. Rikerstsen. 1980.
Low-level waste management: A compilation of models and
monitoring techniques. ORNL/SUB-79/13617/2. Prepared by
Science Applications, Inc., Oak Ridge Tennessee for Oak Ridge
National Laboratory, Oak Ridge, Tennessee.
6. Nuclear Regulatory Commission. 1975. Reactor safety study: An
assessment of accident risks in U. S. commercial nuclear power
plants. WASH-1400 NUREG-75/014. U. S. Nuclear Regulatory
Commission, Washington, D.C.
7. Onishi, Y., R. J. Seme, E. M. Arnold, C. E. Cowan, and
F. L. Thompson. 1980. Critical review: Radionuclide
transport, sediment transport and water quality management
modeling; and radionuclide sorption/desorption mechanisms.
NUREG/CR-1322, PNL-2901. Pacific Northwest Laboratory,
Richland, Washington.
8. Onishi, Y. and S. E. Wise. 1978. Mathematical simulation of
transport of sediment and kepone in the James River Estuary."
PNL-2731. Pacific Northwest Laboratory, Richland.
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9. Sagendorf, J. R. and J. T. Gall. 1977. XOQDOQ program for the
meteorological evaluation of routing effluent releases at
nuclear power stations. NUREG-0324. U.S. Nuclear Regulatory
Commission, Washington, D.C.
10. Trescott, P. C., G. F. Pinder, and S. P. Larson. 1976.
Finite-difference model for aquifer simulation in two dimensions
with results of numerical experiments. U. S. Geological Survey
Techniques of Water Resources Investigations, Book 7, Chap. 1
USGS, Reston, Virginia. 116 pp.
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Model Output
What is the model output? For a groundwater flow model it might
well head distribution; for atmospheric transport, Chi/Q. Be as specific
as possible.
Data
What input data are necessary for model/code operation? Are these
data generic or site-specific? Are they presently available, included in
the model/code documentation? Are the included data credible? Again,
be as specific as possible.
Complexity
Is the model simple or complex? For example, solute transport models
for groundwater may be 1-, 2-, or 3-dimensional and may include constant
or changing chemical conditions. How about ease of operation?
Documentation
Is the model well documented? Are the assumptions clearly and com-
pletely stated? If the model has been coded for computer use, is the
necessary machine-intrinsic information such as JCL explained? Is there
an adequate user's manual? Is a sample problem included?
Reference
Give a complete reference to the model or code. If the work is
ongoing, please list the site and principal investigator or contact
persons.
Thank you for your cooperation.
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APPENDIX A
PREWORKSHOP SURVEY FORM
Due to the shortness of time and the potentially large number of
participants, it is imperative that this workshop be structured and
focused. The ultimate goal of this and the other workshops is to foster
cooperation and communication between the represented agencies and their
contractors. A more specific goal is the determination of the state-of-
the-art of and needs for environmental transport models useful for low-
level waste management. As preparation for the workshop, would you please
participate in the following exercise before arriving in Denver? The
tables we produce will become an important source of information.
Please complete one of the following forms for as many applicable
models as you wish (make additional copies if you need them). List those
models which you have used or know of; they may be in development, well-
known, for a single pathway, or "comprehensive."
You need not include every known model on your lists; we are interested
in specific examples of the existing or developing environmental transport
models so we can gauge where work is needed.
Please leave copies of your forms at the meeting registration desk
in the hotel. The data you provide will be compiled before the workshop
begins.
The following guidelines may be helpful:
Pathway or Process
Choose one of these (or add others): groundwater transport, surface
water transport, resuspension, atmospheric transport, irrigation, food
chain transport to humans, direct exposure, erosion, intrusion.
Model
Name a specific model or code of a model.
Model Type
Describe the type of model you have named. Possible entries might
include Gaussian Plume model, linear compartment model, equilibrium model,
time-dependent model, finite-element or finite difference solution
technique. Many other entries are possible.
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Pathway or Process:
Model:
Model Type:
Model Output:
Data:
Complexity:
Documentation:
Reference:
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APPENDIX B
RESULTS OF PREWORKSHOP SURVEY
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Table B. 1. Results of Pre-workshop Survey: Groundwater Models.
Name
GETOUT
Model lype
One-dimensional
transport;
axial dispersion
Output
C1/yr discharged 1n
time and space
Data Input
Sorption equilibrium
constants; groundwater
velocities; dispersion
coefficient; height and
width of Input release
band
Complexity
Complex in dealing
with numerical
accuracy and decay
chains; simple with
regard to transport
Documentation
Fair; well commented
code; W. V. DeMler et al.
(1979) PNL-2970
HYDRO
SWIFT
TRESCOTT
One-dimensional flow;
finite element
solution
Multi-dimensional
transport; finite
difference solution.
Groundwater flow;
3-dimenslonal finite
difference solution
Radionucllde concentration
in soil 1n time and space;
amount released to aquifer
in time;
Pressure, mass concent.
or temperature in space
and time;
Hyraullc head and well
draw down in time;
Site specific kd;
sorption coefficients;
multi-layer geological
structure; water
conductivities
Permeability, porosity
etc., boundary and
initial conditions
Initial head; storage
coefficient;
hydraulic conductivity;
transm1ssiv1ty;
recharge; grid
spacing.
Crank-Nicholson Finite
Element solution;
constant chemical
conditions; constant
solubilities
1-3 dimensions;
radioactive decay
chains; fairly
difficult to use
Constant input;
changing input parms
for transient runs
requires sequence of
runs.
Mo formal document
available;
Bahrain Amirijafari
SAI
Well documented;
Dillon, Lantz &
Pahwa (1978)
Well documented
Including theory,
input description;
Trescott, Pinder &
Larson (1976)
CO
-J
o
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Table B.2 Results of Pre-workshop Survey: Atmospheric Transport Models
Name
AIRDOS - II
AIRDOS - EPA
CRAC
XOQDOQ
Model Type
Gaussian Plume;
analytical;
chronic releases
Analytical;
reactor accident
releases
Gaussian Plume;
analytical
Output
Annual population and
maximum individual
dose
Population doses, latent
and chronic; population
fatalities and injuries;
accident costs
X/Q or D/Q for either
annual average or
Data Input
Extensive; includes
population dist.,
annual average
meteoro logy; agricul.
data
Population and agricul.
data; reactor releases
inventory and release
fraction; health effects
parameters
Site specific weather
data, stack height
Complexity
Fairly simple to
operate
Fairly large, but
straight forward
Fairly simple
model
Documentation
Good; sample problem
included; user's manual;
Moore et al. (1979)
Well -commented;
documented in WASH-1400
Appendix VI.
NRC (1975)
Rough draft document;
Sagendorf and Gall
CO
probabilistic
(1977)
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Table B.3. Results of Pre-workshop Survey: Dose-to-Man Models
Name
BURYIT
DITTY1
DOSTOMAN
Model Type
Analytical;
finite-element
Multiplicative
chain, equilibrium
model
Finite difference
and linear
compartment
Output
Population doses
Integrated population
dose
Compartment
Inventories vs
time
Data Input
Site-specific geology,
meteorology, pop.
distribution
Scenarios; releases;
population dlst.
Intercompartmental
transfer coefficients
Complexity
One-dimensional
fairly simple
Many pathways;
easy to use
Easy to run
Documentation
Early 1981; Science
Applications, Inc.;
David Lester
Early 1981; Battelle-
Pac1f1c Northwest Lab.;
Bruce Napier
In-house documentation
only; Savannah River
Lab.; Bob Root
10
>j
ro
MAX I
PRESTO
Multiplicative
chain; equilibrium
model
Analytical and
multiplicative
chain combination
Organ doses
Integrated pop.
dose and Individual
dose; population
health effects
Pathways; food
consumption rates,etc.
Site specific geology,
hydrology, meteorology,
source Inventory, pop.
distribution; generic
agricultural parameters
Easy after first run
Easy to run; I.e. One
scenario per run
Preliminary user's
mannual; Battelle
Pacific Northwest Lab.;
Bruce Napier
1981; Oak Ridge
National Laboratory;
Craig Little
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SUMMARY OF MODEL VERIFICATION/VALIDATION WORKSHOP
T. L. Jones, Workshop Chairman
Pacific Northwest Laboratory
Richland, Washington 99352
and
D. G. Jacobs, Workshop Secretary
Evaluation Research Corporation
Oak Ridge, Tennessee 37830
INTRODUCTION
The goal of the Model Validation/Verification Workshop was to deter-
mine how the various agencies could best cooperate in establishing the
credibility of models used in support of low-level waste management.
This workshop seemed particularly susceptible to problems of semantics.
Definitions for such widely used terms such as verification, validation
and calibration are not available. The group had the option of establish-
ing a set of definitions or of finding a way of avoiding them altogether.
The latter approach seemed to be the only practical option for such a
short workshop. The discussion shifted from rigorous definitions to
discussions of what each of us felt constituted "good modeling." The dis-
cussion focused on detailed accounts of actual modeling projects conducted
by various participants. There was remarkable agreement among the group
on methods and techniques used. It was soon realized that although each
modeler may have a different name for each step in the modeling process,
there was an underlying methodology common to all efforts discussed. The
common concern of all modelers seemed to be to have their models and model
results accepted. The basic ingredients needed to give a model credibility
were identified as: (1) documentation, (2) reasonable estimation of
important parameters, (3) mathematical checks, and (4) applications to
independent data sets.
Participants (see Appendix A) felt that proper documentation of models
was not currently available. Descriptions of models should include the
usual information such as the equations solved together with the basic
computer techniques used; however, more information is necessary. Explana-
tions of the basic physical or empirical assumptions used to formulate
the theory, examples of intended applications, and possibly subjective
estimates of the overall value of the model are very valuable to model
users. Documentation should include minimum computer requirements necessary
to run the code, estimates of run time, and a detailed list of input data
requirements. The discussion on making reasonable estimates of parameters
brought up the subject of interaction with experimental and field personnel.
The need for such interaction is so obvious it is difficult to discuss.
There were no new ideas on how to foster more communication, but it was
agreed that no modeling effort is complete without strong input from people
with the proper physical insight and experimental data,
373
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The last two items of the methodology list were handled together and
resulted in the major recommendation of the workshop. It was recommended
that the mathematical checks (number 3 above) and applications to
independent'data sets (number 4) should be accomplished by using standard
test cases to cross-check most, if not all, waste management models. A
good example cited for this type of activity was a recent Gordon confer-
ence sponsored by the petroleum industry where several models were applied
to the same test cases so that model results could be compared. Members
of the workshop who had attended this conference fel.t the technique was
successful and could be applied to models used for low-level waste manage-
ment. It was felt that test cases could be developed within major disci-
plines such as saturated and unsaturated flow, leach models, atmospheric
transport, etc. A conference similar to the present interagency meeting
could provide a forum for presenting the results of applying many models
to the same test data. The expertise of most workshop participants was in
the area of saturated moisture and solute transport. A detailed list of
specifications that test cases should meet was developed and is presented
below.
TEST CASES FOR MODEL INTERCOMPARISON
The description of a test case was divided into four areas: (1) data
sources; (2) data specifications; (3) simulation specifications; and
(4) points of comparison.
Data Sources
A primary source of data for making mathematical checks are the
analytical solutions to the transport equations. While these data sets do
not usually include complicated boundary conditions, they do provide
valuable data on the ability of the computer code to solve the fundamental
equations of interest. For more realistic data sets, it was felt that there
was nothing wrong with hypothetical test data. If people experienced in
the area of transport were given a chance, very realistic test cases could
be produced which would closely resemble real world applications. This
method seemed very useful for making relative comparisons between models
while not being too useful for any absolute evaluation of a model's
accuracy. The primary source of data for test cases is laboratory and field
experiments. These data not only provide the most useful information for
making absolute evaluations of models, but the constant use of field
information in model development will also help prevent the development of
models whose data requirements are unrealistic in terms of what is possible
to do in the field.
Data Specifications
For the area of saturated moisture transport it was decided that the
test cases should be three-dimensional (3-D) wherever possible. There
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are sufficient 3-D models available to warrant this, and also 3-D data
could be used in 1- and 2-D models by extracting the appropriate-subset
of data. The general consensus was that there is no need to worry about
the non-isothermal cases for low-level waste. The test cases could be
isothermal experiments. There were two fundamental data types which
should be included. In hydrology models,'it makes a difference if the
data is cross sectional in nature or planometric. Different models are
required for each type of data and test cases involving each type should
be developed.
Simulation Specifications
In discussing test cases, it was most difficult for the participants
to agree on the details of what scenarios and the specifics of what "pro-
cesses to include.
It was recommended that the specific radionuclides tritium,
strontium-90, carbon-14, and uranium-238 should be included. It was felt
that this list included sufficiently broad chemical and transport
characteristics to be appropriate. An important feature of the simula-
tion specifications is the form of output required. At a minimum, the
output time should be fixed. If models are going to be compared to one
another, they must produce equivalent output for the same time period.
One cannot compare heads with flux or water level at two years with water
level at two years and six months. The time scale of the test cases was
discussed briefly and 300 years seemed reasonable. Obviously, no experi-
mental data will be available for that time period, but test cases involving
experimental data should be extended beyond the range of available data.
Points of Comparison
It is necessary to define exactly how the model outputs are going to
be compared. This is not a trivial matter and can be the key to the
success of the effort. One basic prediction of flow models is the direction
of flow. Models that don't agree on anything else should still predict
the same direction of flow. Mass balance is also a fundamental parameter
of interest. While preserving mass balance is not a sufficient condition
to accept a model result, -it is necessary. If dispersion is being pre-
dicted, there are at least three measures to'examine. The breakthrough time
for the'solute, the peak concentration over some, time period, and an
estimate of the overall fit to the concentration-time graph or concentration-
space graph. A test procedure such as least-squares analysis may be
appropriate. It should be clear from the outset what basis of comparisons
will be. Other areas of comparison involve the computer characteristics
such as language, CPU time, memory allocation, and the size of the time
of a space step used in the simulation. There was some discussion about
making a standard time and space step part of the simulation specification
but no agreement was reached.
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376
The important concept was that test cases could and should be developed
to aid in the evaluation of models used in waste management, and it was
recommended that there should be interagency cooperation in this effort.
The discussion presented here is just an example of what the test cases
for saturated water and solute flow might contain. A serious attempt
at designing test cases would involve much more time and experts from
many fields.
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377
APPENDIX A
LIST OF ATTENDEES
VERIFICATION/VALIDATION WORKSHOP
Name
Affiliation
John A. Adam
Bahram Amirijafari
Thomas S. Baer
Jim Bennett
Richard Codell
Jerry Cohen
Paul Dickman
Gerry Grisak
David Grove
Joanna Hamilton
Ed Hawkins
Preston Hunter
Don Jacobs
Thomas Johnson
Tim Jones
Dan Kapsch
Kenneth Kipp
Eric Lappala
Linda Lehman
Dan McKenzie
G. Lewis Meyer
E. S. Murphy
Fred Paillet
Steve Phillips
John Pickens
Matthew Pope
David Prudic
Jack Robertson
John C. Rodgers
William Sayre
Dale Smith
Jack Sonnichsen
James Steger
Ford, Bacon and Davis, Utah
Science Applications, Inc.
Nuclear Experience Co., Inc.
U.S.G.S.
USNRC
Lawrence Livermore National Lab
EG&G - Idaho
Environment Canada
U.S.G.S.
Dames and Moore
USNRC
Ford Bacon and Davis, Utah
Evaluation Research Corp.
Illinois State Geological Survey
Battelle-Pacific Northwest Laboratory
Monsanto Research Corp.
U.S.G.S.
U.S.G.S.
USNRC
Battelle-Pacific Northwest Laboratory
USEPA
Battelle-Pacific Northwest Laboratory
U.S.G.S.
Rockwell Hanford
Environment Canada
EG&G Idaho
U.S.G.S.
U.S.G.S.
LASL
U.S.G.S.
USNRC
Westinghouse Hanford
LASL
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MODELING AND LOW-LEVEL WASTE MANAGEMENT:
AN INTERAGENCY WORKSHOP
Denver, Colorado
LIST OF ATTENDEES
John A. Adam
Senior Engineer
Ford, Bacon & Davis, Utah
2110 Craig Drive
Colorado Springs, CO 80908
(303) 488-2499
E. L. Albenesius
Research Manager
E. I. DuPont de Nemours
Savannah River Laboratory
Aiken, SC 29808
FTS: 239-2482
(803) 450-6211, Ext. 2482
Bahrain Amirijafari
Director of Petroleum Engineering
Science Applications, Inc.
1726 Cole Blvd.
Suite 350
Golden, CO 80401
(303) 279-0701
Thomas S. Baer
Vice President
Nuclear Engineering Co., Inc.
P.O. Box 7246
Louisville, KY 40222
(502) 426-7160
Jim Bennett
Hydro!ogist
U.S. Geological Survey
MS 430
USGS National Headquarters
Reston, VA 22092
(FTS) 928-6947
(703) 860-6947
Clinton M. Case
Research Professor
Desert Research Institute
Water Resources Center
P.O. Box 60220
Reno, Nevada 89506
(702) 673-7375
Hans Claassen
Research Chemist
USGS
Denver Federal Center
Denver, CO 80225
(FTS) 234-2115
(303) 234-2115
Don A. Clark
Research Chemist
EPA
RSKERL
P.O. Box 1198
Ada, OK 74820
(FTS) 743-2300
(405) 332-8800
Richard Codell
Senior Hydraulic Engineer
U.S. NRC
Washington, DC 20555
(FTS) 492-8473
(301) 492-8473
Jerry Cohen
Lawrence Livermore National
L-262
P.O. Box 808
Livermore, CA 94550
(FTS) 532-6449
(415) 422-6449
Lab.
379
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380
Paul Dickman
Senior Engineer
Low-Level Waste Program
EG&G - Idaho
P.O. Box 1625
Idaho Falls, ID 83401
(FTS) 583-0610
(208) 526-0610
Leslie R. Dole
Physical Chemist
Oak Ridge National Laboratory
P.O. Box X/Y-12
Building 9204-3
Oak Ridge, TN 37830
(FTS) 626-7421
(615) 576-7421
Ken Erickson
Sandia Laboratories
P.O. Box 5800
Albuquerque, NM 87185
(FTS) 844-4133
(505) 844-4133
David E. Fields
Oak Ridge National Laboratory
Health and Safety Research Division
P.O. Box X
Building 7509
Oak Ridge, TN 37830
(FTS) 624-5435
(615) 574-5435
Jesse Freeman
Radio!ogic Transport Analyst
USNRC, WM Uranium Recovery
MS SS-483
Bethesda, MD 20555
(FTS) 427-4543
(301) 427-4543
Gerry Grisak
Hydrogeologist
Environment Canada
National Hydrology Research Institute
562 Booth Street
Ottawa, Ontario, Canada
K1A OE7
(613) 733-6476
David B. Grove
Chemical Engineer
USGS. MS 413
Denver Federal Center
Denver, CO 80225
(FTS) 234-2404
(303) 234-2404
Joanna Hamilton
Engineer
Dames & Moore
1626 Cole Blvd.
Golden, CO 80401
(303) 232-6262
Ed Hawkins
Section Chief
U.S. NRC
Washington, DC 20555
(FTS) 427-4533
Richard W. Healy
Hydrologist
U.S. Geological Survey
P.O. Box 1026
Champaign, IL 61820
(FTS) 958-5365
(217) 398-5365
Richard Heystee
Geotechnical Engineer
Ontario Hydro
700 University Avenue
HI 5 D26
Toronto, Ontario
Canada, M5G 1X6
(416) 469-3786
Cheng Y. Hung
Hydrologist
U.S. EPA
401 M Street SW
ANR 460
Washington, DC 20460
(FTS) 557-8978
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381
Preston H. Hunter
Program Manager
Ford Bacon & Davis, Utah
P.O. Box 8009
375 Chipeta Way
Salt Lake City, Utah 84108
(801) 583-3773, Ext 287
Donald Jacobs
Manager, Oak Ridge Operations
Evaluation Research Corporation
800 Oak Ridge Turnpike
Oak Ridge, TN 37830
(615) 482-7973
Thomas Johnson
Geologist
Illinois State Geological Survey
514 E. Peabody
Natural Resources Bldg.
Champaign, IL 61820
(217) 344-1481
Tim Jones
Research Scientist
Pacific Northwest Laboratory
Water and Land Resources Dept.
P.O. Box 999
Rich!and, WA 99352
(FTS) 375-7511
(509) 375-2611
Dan Kapsch
Senior Research Chemist
Monsanto Research Corp.
Mound Facility
Miamisburg, OH 45342
(FTS) 774-4207
(513) 865-4207
Kenneth Kipp
Hydro!ogist
USGS
MS 413
Denver Federal Center
Box 25046
Denver, CO 80225
(FTS) 234-2404
(303) 234-2404
Dewey E. Large
Manager, ORO Low-Level Waste
Management Program
P.O. Box E
Oak Ridge, TN 37830
(FTS) 626-0715
(615) 576-0715
Eric G. Lappala
Hydrologist
USGS
MS 413, Bldg. 53
Denver Federal Center
Lakewood, CO 80228
(FTS) 234-2404
(303) 234-2404
Linda Lehman
Hydrogeologist
USNRC
Division of Waste Management
MS 905 SS
Washington, DC 20555
(FTS) 427-4177
(301) 427-4177
Craig Little
Research Associate
Oak Ridge National Laboratory
Health & Safety Research Division
P.O. Box X
Oak Ridge, TN 37830
(FTS) 626-2106
(615) 576-2106
Robert S. Lowrie
Program Manager
National Low-Level Waste Management
Program
Oak Ridge National Laboratory
P.O. Box X
Oak Ridge, TN 37830
(FTS) 624-7259
(615) 574-7259
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382
Dan McKenzie
Battelle Northwest
Robert Young Bldg.
Rich!and, WA 99352
(FTS) 444-6573
(509) 376-6573
James W. Mercer
Hydrologist
GeoTrans, Inc.
P.O. Box 2550
Reston, VA 22090
(703) 435-4400
G. Lewis Meyer
LLW Standard Program
USEPA
Criteria & Standards Division
Office of Radiation Programs
Washington, DC 20460
(FTS) 557-8977
(702) 557-8977
E. S. Murphy
Senior Research Scientist
Battelle Northwest
P.O. Box 999
Rich!and, WA 99352
(FTS) 555-4321
(509) 376-4321
Bruce Napier
Environmental Scientist
Battelle, Pacific Northwest
P.O. Box 999
Rich!and, WA 99352
(FTS) 444-4217
(509) 376-4217
Yasuo Onishi
Staff Engineer
Pacific Northwest Laboratories
P.O. Box 999
Rich!and, WA 99352
(509) 375-2425
Fred Pail let
Geophysicist
Borehole Geophysics, USGS
MS 403
Denver Federal Center
Lakewood, CO 80225
(FTS) 234-2617
Steve Phillips
Senior Engineer
Rockwell Hanford
Research & Engineering
Richland, WA 99352
(FTS) 373-3468
(509) 373-3468
John Pickens
Hydrogeologist
Environment Canada
National Hydrology Research Inst.
Inland Waters Directorate
562 Booth Street
Ottawa, Ontario
Canada K1A OE7
(613) 995-4045
David W. Pollock
Hydrologist
USGS
MS 431
National Center
Reston, VA 22090
(FTS) 928-6892
(703) 860-6892
Matthew Pope
Scientist
EG&G
P.O. Box 1625
Idaho Falls, ID 83401
(208) 526-0688
David E. Prudic
Hydrologist
USGS
Rm. 225, Federal Bldg,
705 N. Plaza St.
Carson City, NV 89701
(702) 882-1388
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383
Andrew E. Reisenauer
Research Scientist
Pacific Northwest Laboratory
Richland, WA 99352
(FTS) 444-7511, Ext. 375-2513
(509) 375-2513
John (Jack) Robertson
Hydrologist
USGS
National Center, MS 410
Reston, VA 22092
(FTS) 928-6976
(203) 860-6976
John C. Rodgers
Los Alamos Scientific Laboratory
P.O. Box 1663, MS 495
Los Alamos, NM 87545
(FTS) 843-3167
(505) 662-3167
Robert Root
Geologist
du Pont - Savanna River Laboratory
Bldg. 773-A
Aiken, SC 29801
(FTS) 239-2612
(803) 725-2612
George Saulnier
TRW, Inc.
405 Urban S C
Suite 212
Lakewood, CO 80206
(303) 986-5533
William W. Sayre
Research Hydrologist
USGS, WRD
Box 25046, MS 413, DFC
Lakewood, CO 80225
(FTS) 234-2404
(303) 234-2404
Jacob Sedlet
Section Head
Argonne National Laboratory
9700 S. Cass Ave.
Argonne, IL 60439
(FTS) 972-5644
(312) 972-5646
Dale Smith
Chief, LLW Licensing Branch
USNRC
Washington, DC 20555
(FTS) 427-4433
(301) 427-4433
Jack Sonnichsen
Principal Engineer
Westinghouse Hanford
Federal Bldg. 420
P.O. Box 1970
Richand, WA 99352
(FTS) 444-6802
(509) 376-6802
James G. Steger
Alternate Group Leader
Los Alamos Scientific Laboratory
LS-6, MS-495
P.O. Box 1663
Los Alamos, NM 87545
(FTS) 843-3331
(505) 667-3331
James A. Stoddard
Science Applications, Inc.
P.O. Box 2351
1200 Prospect St.
La Jolla, CA 92038
(714) 454-3811, Ext. 2294
Leroy E. Stratton
National Low-Level Waste Manage-
ment Program
Oak Ridge National Laboratory
P.O. Box X
Bldg. 1505
Oak Ridge, TN 37830
(FTS) 626-0504
(615) 576-0504
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384
Robert G. Thomas
Los Alamos Scientific Laboratory
P.O. Box 1663, MS-400
Los Alamos, NM 87545
(FTS) 843-3318
(505) 667-3318
Burnell W. Vincent
Environmental Engineer
EPA
Office of Solid Waste
Washington, DC 20460
(FTS) 755-9116
(202) 755-9116
Edwin P. Weeks
Hydrologist
USGS, MS 413
Denver Federal Center
Denver, CO 80225
(FTS) 234-2404
(303) 234-2404
Ken Whitaker
Project Engineer
Chem Nuclear Systems, Inc.
P.O. Box 726
Barnwell, SC 29812
(803) 259-1781
Charles R. Wilson
Staff Scientist
Lawrence Berkeley Laboratory
c/o University of California
Berkeley, CA 94720
(FTS) 451-6433
(415) 486-6433
George Yeh
Research Staff
Oak Ridge National Laboratory
P.O. Box X
Oak Ridge, TN 37830
(FTS) 624-7285
(615) 574-7285
*U.S. GOVERNMENT PRINTING OFFICE: 1981-740-062/102
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