EPA600/R-04/101
September 2004
On-line Tools for Assessing Petroleum Releases
by
James W. Weaver
Ecosystems Research Division
National Exposure Research Laboratory
Athens, Georgia 30605
National Exposure Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, NC 27711
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Notice
The U.S. Environmental Protection Agency through its Office of Research and Development
funded and managed the research described here. It has been subjected to the Agency's peer and
administrative review and has been approved for publication as an EPA document. Mention of
trade names or commercial products does not constitute endorsement or recommendation for use.
The author acknowledges contributions and interactions with colleagues in the New York State
Department of Environmental Conservation, Pennsylvania Land Recycling Program, North
Carolina Department of Natural Resources, Ohio Bureau of Underground Storage Tank
Regulations, Delaware Department of Natural Resources and Environmental Control, Wisconsin
Department of Natural Resources , EPA Regions 9 and 4.
11
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Foreword
The National Exposure Research Laboratory's Ecosystems Research Division (ERD) in Athens,
Georgia, conducts research on organic and inorganic chemicals, greenhouse gas biogeochemical
cycles, and land use perturbations that create direct and indirect, chemical and non-chemical
stresses, exposures, and potential risks to humans and ecosystems. ERD develops, tests, applies
and provides technical support for exposure and ecosystem response models used for assessing
and managing risks to humans and ecosystems, within a watershed / regional context.
The Regulatory Support Branch (RSB) conducts problem-driven and applied research, develops
technology tools, and provides technical support to customer Program and Regional Offices,
States, Municipalities, and Tribes. Models are distributed and supported via the EPA Center for
Exposure Assessment Modeling (CEAM).
The Internet tools described in this report provide methods and models for evaluation of
contaminated sites. Two problems are addressed by models. The first is the placement of wells
for correct delineation of contaminant plumes. Because aquifer recharge can displace plumes
downward, the vertical placement of well screens is critical to obtain proper characterization
data. The second is the use of models where data are limited. In this case some form of
uncertainty analysis is necessary to evaluate transport behavior. The remainder of the report
describes a series of tools for estimating various model input parameters.
Rosemarie C. Russo, Ph.D.
Director
Ecosystems Research Division
Athens, Georgia
in
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Table of Contents
Notice ii
Foreword iii
List of Figures vi
List of Tables viii
1. Introduction 1
The On-line Calculators 1
2. Plume Diving 3
Vertical Delineation and Transport of Contaminants 3
1) East Patchogue, New York Plume Diving 4
2) Average Borehole Concentration Calculator 6
The Embedded Data 8
Calculation Method 9
Example Results 9
3) Observation of Vertical Gradients 11
Well Cluster Example 14
Application to Plume Diving 15
Recharge-Driven Plume Diving Calculation 16
Theory 17
East Patchogue Example 18
Required Input 19
Flow System Hydraulics 19
Source and Observation Well Locations 20
Model Results 21
Drawing Options 21
3 Contaminant Transport 22
First Arrival Versus Advective Travel Times 22
Uncertainty Range Determination 26
Approach 27
Simulation 29
Results 30
Concentration Uncertainty Model Input and Output 34
4. Model Input Parameters 37
Retardation Factor Calculator 37
IV
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Example 38
Ground Water Velocity 39
Seepage Velocity Calculator 39
Hydraulic Gradient Calculator 40
Example 40
Longitudinal Dispersion Coefficient Calculator 43
Half-Lives to Rate Constant Conversion Calculator 46
Effective Solubility from Mixture Calculator 47
Shallow Ground Water Temperature in the United States 47
Temperature Dependence of MTBE, Benzene and Toluene Solubility 48
MTBE 49
Benzene 49
Toluene 52
Example Values 53
Temperature-Dependent Henry's Law Coefficient Calculator 55
Example Values 56
Diffusion Coefficient Calculator 58
5. Conclusions 60
References 61
Appendix 1 Calculator Reference 64
Appendix 2 Acronyms and Abbreviations 65
Appendix 3 Plume Diving Calculator Equations 66
Appendix 4 One-Dimensional Transport in a Homogeneous Aquifer 69
Appendix 5 Estimation of Temperature-Dependent Henry's Law Coefficients 71
Appendix 6 Diffusion Coefficients 73
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List of Figures
Figure 1 Vertical cross section through the MTBE, and benzene plumes. The gasoline source is
located at the right hand edge of the sections and flow is to the left. Each of the plumes
dives into the aquifer with transport in the aquifer 4
Figure 2 Consequences of sampling only the top ten feet of the aquifer at East Patchogue, New
York. The MTBE plume would disappear (top); the benzene plume would be shortened
by two thirds; and the total xylenes plume appears at the same length because its extent
did not reach the gravel pit where diving dominates the contaminant distribution 6
Figure 3 Average borehole concentration calculation 6
Figure 4 Illustration of components of borehole flow calculator graphics 8
Figure 5 Estimated borehole concentration
of 120 |J.g/L for a twenty-foot long well screen located 20 feet below the land surface.
10
Figure 6 Estimated borehole concentration of 5677 \ig/L for a five-foot well screen located 50
feet below the land surface 10
Figure 7 Estimated borehole concentration of 995 |J-g/L for a ten-foot long well screen placed 70
feet below the land surface 10
Figure 8 Definition of relationships for vertical gradient calculations: dw is the depth to water, d
the depth to the top of the well screen, and s is the screen length 11
Figure 9 Submerged, water table and dry conditions of a well screen 11
Figure 10 Assumed distances for vertical gradient calculation: H = high, M=medium, L=low.
12
Figure 11 Gradients in wells that are screened in heterogeneous materials 13
Figure 12 Input and output screen for the plume diving calculator 16
Figure 13 Example plume diving calculation for the East Patchogue site 18
Figure 14 Plume Diving calculator hydraulic and hydrologic property entry 19
Figure 15 Plume Diving calculator location entry 20
Figure 16 Plume Diving calculator graphical output 20
Figure 17 Plume Diving calculator mass balance output 21
Figure 18 Plume Diving calculator display options 21
Figure 19 Illustration of transport by advection only (hypothetical) and transport by advection
and dispersion 23
Figure 20 Illustration showing the relationship between the first arrival time, maximum
concentration and duration of contamination. The first arrival time and duration are
determined relative to a given threshold concentration, that is usually a maximum
contaminant level or other concentration of concern 24
Figure 21 Relationship of uncertainty to model data availability 26
Figure 22 Output from the Concentration Uncertainty applet showing the wide range of
breakthrough curves that are possible given specified ranges of input parameters 28
Figure 23 Fixed parameters required for the Concentration Uncertainty model 34
Figure 24 Potentially variable parameters required for the Concentration Uncertainty model.
34
VI
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Figure 25 Output from the Concentration Uncertainty model showing the extreme values for
each model output: first arrival, maximum concentration, duration above threshold and
risk factor 35
Figure 26 Generic values (low or high) of parameter sets that generate extreme values of each
output of the model: first arrival time, maximum concentration, duration and risk factor.
35
Figure 27 Graphical presentation of Concentration Uncertainty model output showing a
comparison of the breakthrough curves for the extremes of each model output: first
arrival time, maximum concentration, duration, and risk factor with the breakthrough
curve for the average value of all inputs 36
Figure 28 Magnitude and direction of gradient for example site 40
Figure 29 Data tabulation from Gelhar et al. (1992) showing published longitudinal dispersivity
as a function of scale 43
Figure 30 Gelhar et al. (1992) dispersivity tabulation with scale-related estimates 44
Figure 31 Shallow ground water temperatures throughout the United States 47
Figure 32 Graph of MTBE solubility versus temperature showing decline in solubility with
increasing temperature 48
Figure 33 Graph of benzene solubility versus temperature showing modest increase in solubility
with increasing temperature 49
Figure 34 IUPAC data on benzene solubility showing the lower 95% confidence limit,
recommended value, and upper 95% confidence limit 51
Figure 35 IUPAC data on toluene solubility showing the lower value, best value, and upper
value 52
Figure 36 Water table mounding due to unequal recharge and hydraulic conductivity 67
vn
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List of Tables
Table 1 Effects of well screen length on concentrations given in |J-g/L observed wells screened
from 20 feet to 90 feet below the land surface at East Patchogue, New York 10
Table 2 Example field data for vertical gradient determination 14
Table 3 Example gradient estimates indicating the smallest, midpoint to midpoint, highest and
the value for a piezometer 15
Table 4 Gradient magnitude for a difference in head of 0.01 ft and various distances between
well screens 15
Table 5 Parameter inputs, their treatment in the model as fixed or variable and the values used
in the model uncertainty example 28
Table 6 The problem definition, its treatment in the model as fixed or variable and the values
used in the uncertainty example 29
Table 7 Model results for four scenarios showing a comparison of best and worst cases for the
four outputs (first arrival time, maximum concentration, duration and risk) 31
Table 8 Comparison of data sets giving the worst cases for each of four model outputs 32
Table 9 Retardation factors for benzene under varying conditions 38
Table 10 Retardation factors for MTBE under varying conditions 39
Table 11 Data for gradient calculation example 41
Table 12 Relationships between plume length, data scatter on the Gelhar (1992) tabulation and
the Xu and Eckstein (1995) formula 45
Table 13 IUPAC benzene solubility data 50
Table 14 IUPAC toluene solubility data 52
Table 15 Example effective solubilities for benzene and MTBE at temperatures of 5°C, 15°C
and 25 °C 53
Table 16 Range of variation of effective solubilities for benzene and MTBE from 5°C to 25 °C.
54
Table 17 Unit sets for Henry's Constants 56
Table 18 Constants needed for Henry's law unit conversions 56
Table 19 Estimated Henry's law coefficients for benzene, MTBE, perchloroethene and
trichloroethene at temperatures of 5 °C, 15 °C, and 25 °C 57
Table 20 Diffusion coefficient calculation input parameters for benzene, MTBE,
perchloroethene, and trichlorethene 58
Table 21 Estimated air and water diffusivities for benzene, MTBE, perchloroethene and
trichloroethene at temperatures of 5 °C, 15 °C, and 25 °C 59
Table 22 Web (URLs) for models and associated calculations described in the text 64
Table 23 Web (URLs) for formulas/model inputs described in the text 64
Table 24 Exponent "n" used in calculation of enthalpy of vaporization 71
Vlll
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1. Introduction
Sites that are contaminated with fuels and dissolved components of petroleum products
are commonly assessed through a combination of field data collection and application of models.
The field data typically consist of two types: first, field-measured aquifer parameters and
second, contaminant and water level observations. The first type may consist of grain size
distributions, hydraulic conductivity, and aquifer geometry. The second, contaminant and water
level data, may consist of soil core data, aqueous concentrations, water and product levels in
wells. Generally the second type of data is most abundant. The distinction between the two
types of data is made because the first type of data correspond, even if roughly, to inputs to
models and the second type to outputs from models.
Rarely are field data alone used in assessing a site. Some means are needed to extract
information from the data. Usually this is done with models of various types. Models may take
the form of simple formulas or complex numerical calculations. The simpler formulas and
approaches may not be perceived as models, but they share similar characteristics. Each is based
upon a set of assumptions which in some cases are well-matched by site conditions. Generally
each also require ancillary data which are often obtained from the literature. In some cases these
ancillary data are not easily obtained. The purpose of the methods and models described in this
report provide a basis for evaluating field observations and extracting information from a variety
of field observations.
The On-line Calculators
Each of the methods and models described in this document are available as a set of on-
line calculators. They may be found on the Internet1 at:
http ://www. epa. gov/athens/onsite
The calculators were developed from interactions with state environmental agencies. The most
important collaborations have been with the Region 2 Office of the New York State Department
of Environmental Conservation and the Pennsylvania Land Recycling program2.
Four types of calculations are found on the web site:
JUse of the calculators require that Javascript be enabled and that the Java 2 plug-in be
available. See http://www.j ava.com/en/download/windows_automatic.j sp for more information.
2Other significant interactions have occurred with the North Carolina Department of
Natural Resources, Ohio Bureau of Underground Storage Tank Regulations, Delaware
Department of Natural Resources and Environmental Control, Wisconsin Department of Natural
Resources , EPA Regions 9 and 4.
1
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formulas,
• models,
scientific demos, and
• unit conversions.
The formulas generally represent inputs to transport models that are derived from field
observations, literature data, or a combination of the two. These address concepts of field data
assessment discussed in the Introduction. The models are intended to transfer knowledge gained
from working on various sites. Because of limitations inherent to use of the Internet, it is not
desirable to reproduce complex models as web applications. Primarily the ability to store data is
severely restricted. The models are thus simple, but intended to introduce important concepts,
rather than serve as competitors to PC models. The term "calculator" also represents the models,
because they are intermediate between a hand calculation and a complex modeling application.
Two of the models have been selected for highlighting in this report. These appear in
Chapters 2 and 3. The first model addresses the diving of contaminant plumes into aquifers.
This behavior may be caused by recharge of rainwater. In light of the uncertainties in model
parameters, this calculation is not expected to give some sort of exact answer, but is intended to
be used to place well screens in the best possible vertical positions.
The second of the models is called "ConcentrationUncertainty" because it addresses the
ranges of possible model behavior given uncertainly in model input parameters. As will be
described, each parameter of the model is subject to some amount of uncertainty and this
application highlights that uncertainty. To move beyond simply identifying this problem, the
model can also determine generic parameter sets that always produce the worst (or best) case
results. For some applications, single model runs with these parameters could be used for more
certain decision-making, than the typical average parameter values used by modelers.
The remainder of the document (Chapter 4) is devoted to describing specific inputs to
models, focusing on those that drive transport. Each of these calculations is described, along
with literature and other input data. For each method an examination of typical results is given.
For reference, Appendix 1 gives the Internet addresses for each calculator described in the text.
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2. Plume Diving
This chapter describes the first of two models and its approach to their site-specific usage.
The first model was designed to aid in the placement of monitoring wells by assessing the
contribution of aquifer recharge to the vertical position of contaminant plumes. This topic will
introduced through data from a site and three on-line tools. The on-line tools allow exploration
of the apparent concentrations in boreholes that result from screen length and well placement
choices3, the estimation of vertical gradients from nested wells4, and the approximation of
vertical displacement of plumes due to aquifer recharge.5 The first two of these calculators
introduce the issues of vertical transport and the third allows site-specific data to be used for
estimating plume diving and subsequent placement of well screens.
Vertical Delineation and Transport of Contaminants
Primarily contaminants are transported by flowing ground water. Thus the direction that
the contaminant takes is determined by the direction of the flowing ground water. When ground
water moves deeper into an aquifer, the contaminants are also transported deeper or "dive". This
has an implication for sampling and well construction: the sample intervals must be located
appropriately or diving plumes might be missed.
Data from the East Patchogue, New York gasoline release site (Weaver et al., 1996) are
used to illustrate the consequences of ill-placed well screens. At this site there was an intensive
characterization using vertically-discrete samplers. As a result the vertical distribution of
contaminants is well-known. The second example is derived from this data and allows
placement of well screens of differing lengths at differing depths in the aquifer. By
encapsulating this idea in an applet, the implications of screen length and position can be seen
directly from a data set.
These examples illustrate problems that can be created by inappropriate sampling
locations. A simple calculation may assist in estimating the required placement of monitoring
wells when recharge dominates vertical flow. This calculation is available in a calculator that is
described below.
3The Average Borehole Concentration calculator:
http://www.epa.gov/athens/learn2model/part-two/onsite/abc.htm.
4The Vertical Gradient calculator: http://www.epa.gov/athens/learn2model/part-
two/onsite/vgradient.htm.
5The Plume Diving calculator: http://www.epa.gov/athens/learn2model/part-
two/onsite/diving. htm.
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1) East Patchogue, New York Plume Diving
The gasoline release at East Patchogue, New York created large BTEX and MTBE
plumes (Weaver et al., 1996). The plumes were detected when a private water supply well was
constructed, used for a short time period, and then found to be contaminated. This well was
located 4000 feet downgradient from the source. The well screen was about 50 feet below the
water table, right where much of the MTBE mass was located. The site investigation started at
this point and went upgradient to identify the source. Because of the importance of the aquifer
for drinking water supply, the State of New York undertook an extensive investigation of the site
which included vertical characterization of the plumes. Multilevel samplers with 6 inch screens
on five foot intervals were used. The resulting vertical section through the plume showed that
BTEX and MTBE tended to dive into the aquifer with distance from the source (Figure 1).
Further, it was noted that a significant amount of diving occurred as the BTEX plumes passed
under a gravel pit. By studying the well logs and performing a detailed hydraulic
characterization of the aquifer with a borehole flowmeter, vertical migration controlled by
stratigraphy was ruled out because the hydraulic conductivities varied by less than a factor of two
over the aquifer. This left recharge as the most likely explanation for the plume diving. The
model described in the sidebar was used to simulate the site and provided additional evidence for
recharge as the cause of the diving.
The focus in this example is on how recharge pushes the plume downward, but water also
discharges from aquifers. Where water comes up at discharge points, so will the contaminants.
This can happen along streams and rivers, at lakes, or the ocean. The latter is the expected
destination of the MTBE plume at East Patchogue. The ground water flow system discharges
into Great South Bay adjacent to the southern shore of Long Island and ground water moves
upward as it approaches its discharge point in the bottom of the Bay.
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Gravel Pit
Source
0(ft)
benzene
-120 (ft)
MTBE
0(ftJ
-120 (ft)
7000 (ft)
Distance from Source (ft)
OfftJ
Figure 1 Vertical cross section through the MTBE, and benzene plumes. The gasoline source is
located at the right hand edge of the sections and flow is to the left. Each of the plumes dives
into the aquifer with transport in the aquifer.
Consequences
What about the consequences of plume diving, or more to the point, the consequences of
missing a diving plume? The East Patchogue data set can be averaged to show what the plume
would appear to be like if sampled only from long-screened wells. The data were averaged over
the top ten feet of the aquifer to simulate twenty foot well screens "10 feet in and 10 feet out" of
the aquifer. The data are plotted in Figure 2. This figure shows the maximum concentrations
of MTBE, benzene and total xylenes along the length of the plume. These concentrations were
the highest measured in the plume. Also plotted are the averages from the top ten feet of the
aquifer. In this case the MTBE concentrations all fall below the State's threshold of 10 ug/L.
With only these data we would have concluded that there was no MTBE plume at this site.
Interestingly, the effect on benzene and total xylenes are that their plumes become the same
length. The separation we expect to occur because of the differing tendency for sorption of
benzene and xylenes has been negated by the sampling strategy. The total xylenes plume itself
has not been shortened, because its length just reached the distance where diving would cause the
plume to drop below the bottom of the monitoring wells. Benzene, however, did drop out of the
monitoring network, and the simulated 10 foot long monitoring wells made the plume appear to
be about one-third of its actual length.
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LJJ
DQ
i8
"o
£ c
o
~ 6000 -
§ 4000 -
§ 'BI 2000 -
o D ~
c 3000 -i
o
2 2000 -
"c ^ .
§ § 1000-
o -
; 20000 -
\ 16000 -
E 12000 \
] IT 8000 \
5 §> 4000 -
} •=- o :
^^^ — Maximum
— — — — Average t V
^^^"^^"^^ Average of Top 10 Feet r
/-' — ^ \
^v. x.^^^-x \
^fc^» •*"^^. ^^^^ ^i^^^^^^
I^T~T^TV iV i i \r i prT^^l™™!
k-
1000 2000 3000 4000
Distance From Source (ft)
5000 6000
Figure 2 Consequences of sampling only the top ten feet of the aquifer at
East Patchogue, New York. The MTBE plume would disappear (top); the
benzene plume would be shortened by two thirds; and the total xylenes
plume appears at the same length because its extent did not reach the
gravel pit where diving dominates the contaminant distribution.
2) Average Borehole Concentration Calculator
The Average Borehole Concentration calculator demonstrates borehole dilution in
screened wells. Borehole dilution occurs for at least two reasons: the placement of the screen
relative to the contaminant and hydraulic conductivity distributions, and the length of the screen.
The distribution of hydraulic conductivity and the vertical distribution of the contaminant are
required for calculating the expected dilution. Since both of these are not commonly collected,
the calculator has an embedded example. With these the user can see the effect of borehole
dilution in hypothetical screened wells.
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Average
Calculate/Redraw
Reset
. O x
P" Drawing ?
ignore Conductive Variation ?
Select Parameters:
Site
Screen Length
i East Patchogue
Depth of Top of Screen i 50.0 ft
15.0ft
Formation Data
Water Table
max. rel. Ks = 2.740
Well
Concentrations...
Range of Measured Values
Estimated Borehole Concentration =
(12.0 ugfl to ugfl)
ugfl
Figure 3 Average borehole concentration calculation.
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The Embedded Data
The distribution of flow in the aquifer is represented by the relative hydraulic
conductivies over the thickness of the aquifer. These data were collected from a borehole
flowmeter on 1 foot to 3 foot intervals. Figure 4, part A, shows the distribution of hydraulic
conductivity for a well at East Patchogue. Concentration data can be collected from a variety of
vertical profiling methods. Each of these has a sampling interval that generally does not provide
continuous coverage of the aqufier. In this illustration the sampling interval was 6 inches (Figure
4, part B). Between the actual sampling points, the calculator assumes linear variation of
concentration. The user has the opportunity to select a screen depth and length (Figure 4, part C).
With these the average borehole concentration that would have been seen in this well is
calculated from the flow and concentration distributions. The users select the site, with its
embedded data set, the depth of the top of the screen and the screen length.
Figure 4 Illustration of components of borehole flow calculator graphics. "A" represents the
distribution of hydraulic conductivity as measured by the borehole flow meter. "B" represents
the distribution of concentration as measured at discrete sampling points. These correspond to
the horizontal bars. The measured sampling points are connected by the diagonal lines. "C"
represents the well with a screened interval indicated at some depth below the watertable.
The user may turn off the labeling of the drawing. Labels may fall on top of drawing
features of interest and this option removes the labels. The calculation may also be performed
with the hydraulic conductivity variation ignored. Running the calculation with both options
shows how much influence hydraulic conductivity variation has on the results.
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Calculation Method
The average borehole concentration is calculated by averaging the mass flux from each
layer in the profile. Each increment of mass flux is determined from
A4 = Artfvc(z)A (
where AJ; is the increment of mass flux for depth increment i, Kri is the relative hydraulic
conductivity for the increment, c(z) is the depth varying concentration over the increment, and z;_
and zi+ are the depths of the top and bottom of depth increment A, respectively. The integration
is completed by assuming that c(z) follows a linear function between the measurement intervals.
Example Results
The calculator provides a graphic which shows the superimposed distributions of
hydraulic conductivity, the concentration distribution and an assumed screened monitoring well
(Figures 3 and 4). The screened interval is indicated by a bar drawn over the data that originates
in the well (Figure 4, part C). The bottom of the screen gives the results from the calculation
along with the range of measured concentrations. Here (Figure 3) the concentrations measured
from six-inch screens range froml2.0 |o,g/l to 6400.0 |ig/l. The five-foot long screened interval
was placed 50 feet below the water table and had an expected concentration of 5677 |ig/l.
Because of the fortuitous placement of the well screen near the location of maximum
concentration, the expected concentration is not off by much.
The expected concentrations differ with varying depth and screen length. The
concentrations for a 20 foot screen located at 20 feet, 5 foot screen at 50 feet and and 10 foot
screen at 70 feet below the water table are 120 |ig/l, 5677 |ig/l, and 994.7 |ig/l, respectively.
Figures 5 to 7 illustrate the well configurations for these examples.
With the top of the screen at 50 feet below land surface, well screens with length of 5
feet, 10 feet and 20 feet would give concentrations of 5677 |ig/l, 4849 |ig/l, and 3028 |ig/l,
respectively (Table 1). Thus the effect of increasing the screen length at this depth is to reduce
the apparent concentration by one-half. When the screen is at 50 feet, the five-foot screen spans
the point with highest concentration. Increasing the screen length includes depths that only have
lower concentrations. The resulting ratio of concentration measured from a 20 foot screen to that
measured from a five-foot screen is about 0.5 (given in the last row of Table 1). At other depths,
say 40 feet, increasing the well screen length causes depths with higher concentrations to be
included. In this case, the apparent concentration increases with screen length. Thus, varying the
screen length may increase or decrease the apparent concentration depending upon its placement
relative to the underlying contaminant distribution. When the ratio in Table 1 is less than one,
increasing the screen length decreases concentration and vice versa. In these examples there are
as many cases with the screen length increases the apparent concentration as that decrease it.
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Screen
Length
(feet)
5
10
20
Depth of Well Screen Top
20
feet
0
0
120
30
feet
280
210
761
40
feet
1054
1805
3524
50
feet
5677
4849
3028
60
feet
877
1054
1130
70
feet
1155
995
743
80
feet
561
476
316
90
feet
80
62
37
Concentration ratio (20 ft screen: 5 ft screen)
—
2.7
3.3
0.53
1.3
0.64
0.56
0.46
Table 1 Effects of well screen length on concentrations given in |J-g/L observed wells
screened from 20 feet to 90 feet below the land surface at East Patchogue, New York.
Formation Data
Well
Water Table
max. rel. Ks = 2.740
--= 6400,0.
Figure 5 Estimated borehole concentration
of 120 |J.g/L for a twenty-foot long well
screen located 20 feet below the land
surface.
Formation Data
Well
Water Table
max. rel. Ks = 2.740
Figure 6 Estimated borehole concentration
of 5677 |J.g/L for a five-foot well screen
located 50 feet below the land surface.
10
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Formation Data
Well
WaterTable.
max. rel.Ks= 2.740
~~nraxrc=6480Jl.
Figure 7 Estimated borehole concentration
of 995 |J.g/L for a ten-foot long well screen
placed 70 feet below the land surface.
3) Observation of Vertical Gradients
Estimation of vertical gradient are one tool for assessing the possibility of vertical flows.
Water levels in nested well clusters (wells located closely together) indicate upward or
downward flowin aquifers or flow between adjacent geologic units. Flow is governed by Darcy's
Law:
„ Change of head
A.
Distance
(2)
where q is the Darcy flux (volume of water per unit area per unit time) [L/T] and K is the
hydraulic conductivity [L/T]. The change of head (roughly water level) divided by the distance
determines the magnitude and direction of flow. Figure 8 shows the relative relationships
between two wells. With respect to each other, one is shallow and the other deep. The figure
also indicates the important dimensions: depth to water (dw), depth to the top of the screen (d)
and the screen length(s).
11
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shallow
deep
Figure 8 Definition of relationships for vertical
gradient calculations: dw is the depth to water, d
the depth to the top of the well screen, and s is
the screen length. In relation to each other one
well is shallow and the other deep.
submerged
water-table
dry
depth to
water
(three
possibilities)
Figure 9 Submerged, water table and dry
conditions of a well screen.
One choice for calculating the gradient is to use the mid-points for determining the
distance in the denominator of equation 2 (as illustrated in Figure 8). In performing the
calculation there are three possibilities for how the water levels relate to the well screen (Figure
9):
• Submerged. The water level is above the top of the well screen.
• Water-table. The well screen intersects the water-table. The gradient is calculated from
the mid-point of the water level and the bottom of the well screen.
• Dry. The well is dry and no calculation is performed.
Theoretically, the gradients are determined from piezometers that are only open at the bottom and
thus have an effective screen length of zero. In practice, since wells with screens of various
lengths are used to calculate the gradients, the screen lengths may have an influence on the
calculated gradients.
12
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:H
M:M
H:L
Figure 10 Assumed distances for vertical
gradient calculation: H = high, M=medium,
L=low.
Five choices are made concerning screen lengths and are illustrated in Figure 10:
Distance is from top of screen to top of screen (H:H)
• Distance is from mid-point of screen to mid-point of screen (M:M)
Distance is from bottom of screen to bottom of screen (L:L)
• Distance is from top of screen to bottom of screen (H:L)
Distance is from bottom of screen to top of screen (L:H)
For screens of equal length the first three choices all give the same result, no matter the relative
depth of the screens. In addition they give the same value as a piezometer open at the midpoint
depth of the screen. By supplying results for all these possibilities, a range of values is provided
that would bracket the "true" value of the gradient.
The previous discussion presumes that the well is screened in a uniform or fairly uniform
material. To avoid complexities of heterogeneity, this placement is desirable. Figure 11
illustrates the difficulty when different materials are present. Here sand is assumed to overly a
tight clay. If there was an upward gradient, the head in the clay would be higher than the head in
the sand. The upward gradient might be best represented by assuming that the head in the well
represented a point at the bottom of the screen rather than the top where the sand exists.
13
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screen
sand
tight clay
Figure 11 Gradients in wells that are screened
in heterogeneous materials.
Well Cluster Example
Data from a field site in Michigan are shown in Table 2. The site had three closely-
spaced wells with screens located at different depths. The wells were intended to serve a cluster
and were placed as close together as practical. These were used to generate the gradient
estimates that are summarized in Table 3. For the first pair of wells (W-404S and W-405M) the
gradients are directed downward for all estimates and range in magnitude from 0.0057 to 0.016, a
difference of 65%. The midpoint to midpoint estimate of 0.0083 is close to the assumed
piezometer value of 0.0068. The difference between the two is due to the different screen
lengths in the two wells.
Flow is upward between the second pair of wells (W-405M and W-406D) and varies by
26%. The values are lower by a factor often, reflecting the smaller difference in depth to water
than for the first pair. Since the screen lengths are the same for these two wells, the piezometer
and midpoint estimates are the same.
Well
W-404S
W-405M
W-406D
Elevation of Top
of Casing
(ft)
626.44
626.73
626.77
Depth to Water
(ft)
9.10
9.51
9.54
Screened Interval
(ft)
7.6- 17.6
25.6-30.6
40.2-45.2
Table 2 Example field data for vertical gradient determination.
14
-------
Wells
W-404S &
W-405M
W-405M &
W-406D
Estimated Gradients
direction
down
up
Smallest
0.0057
0.00051
Midpoint to
Midpoint
0.0083
0.00069
Piezometer
0.0068
0.00069
High Value
0.016
0.0010
Table 3 Example gradient estimates indicating the smallest, midpoint to midpoint, highest
and the value for a piezometer.
Application to Plume Diving
At the East Patchogue field site, the average annual recharge rate was estimated to be 22
in/yr. This value, coupled with an estimated hydraulic conductivity of 400 ft/d, can be used to
estimate a bounding vertical gradient. If all the recharge water moved only vertically in the
aquifer, the maximum gradient would be 0.0000125 ft/ft.6 Could this gradient be observed in
monitoring wells?
Vertical Distance
between Screens (ft)
10
20
100
1000
Gradient Magnitude
(ft/ft)
0.001
0.0005
0.0001
0.00001
Table 4 Gradient magnitude for a difference in
head of 0.01 ft and various distances between
well screens.
The gradient is determined from the difference in water elevation and the distance
between the measuring points (Equation 2). Here the difference in water elevation could be
6The gradient was calculated from 22 in/yr (0.005 ft/d) divided by 400 ft/d.
15
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assumed to be the smallest measurable difference: 0.01 feet. By selecting various distances
between the screens, Table 4 shows that a difference of 1000 ft in screen position is necessary to
detect a vertical gradient of 0.00001. Thus it is unlikely that recharge-driven plume diving would
be detected by water level differences in monitoring wells.
Recharge-Driven Plume Diving Calculation
The prospects for plume diving should be considered when placing wells at all sites. The
first consideration should be indications of dipping strata from well logs. These can control the
distribution of contaminants down gradient from the source. Recharge can also cause plume
diving. The calculator shows the effect due to recharge only. The implication of the results are
that for some aquifers the recharge of clean water to the aquifer can push the plume deeper into
the aquifer.
The plume diving calculator was designed to be used as a tool for site assessment by
following these steps:
• Estimate the required parameters for the each segment of the flow system:
Up and down gradient heads,
• Aquifer hydraulic conductivity, and the
Recharge rate.
• Run the calculator for the proposed well location,
Check the plume depth at the location, and
• Locate the well screen in appropriate vertical intervals.
The calculator (see Figure 12) uses a simple aquifer model, in complex geological settings a
more complex model would be required to show the effects of recharge on plumes.
16
-------
Calculate/Redraw
Reset
P" DrawWatertable?
F[ Draw UpparBound ofPlumi ?
Upgradient Head (at A)
Downgradient Head (at B)
Source Location
Well Location
Plume Depth at Well
% Balance Error
Results...calculated for:
Source Location at 10.0
Well Location at 400.0
50.0
40.0
10.0
400.0
I ft
\ ft
I ft
I n
3.332233133053878
<1e-6
P* Vertical Exaggeration ?
P" Drawing ?
50.0 ft
40.0 ft
Hydraulic Conductivity
Segment Length
Recharge
Bottom Elevation
10.0
250.0
5.0
flM
ft
10.0
250.0
5.0
ffcJ
ft
in/yr
10.0
250.0
5.0
fl/d
ft
iniyr
I 0.0 ft
Figure 12 Input and output screen for the plume diving calculator.
Theory
The recharge calculator is based upon two solutions of the one-dimensional Dupuit
equation for flow in unconfmed aquifers. These solutions are combined to determine the
position of the phreatic surface (water table) and a streamline originating at the water table, the
gradient, ground water fluxes and travel times for sorbing contaminants. The methods used for
performing these calculations are given in Appendix 3.
17
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East Patchogue Example
This method was developed for application to the MTBE and benzene plumes at East
Patchogue, New York. At this site the plume appeared to drop when it moved below a sand and
gravel mining operation. Data collected from a borehole flow meter on the variation of hydraulic
conductivity with depth showed no direct connection between contaminant concentrations and
hydraulic conductivity. From this it was concluded that statigraphy did not cause the diving of
the plume. The plume diving calculation was developed to assess the contribution of recharge to
the dropping of the plume. The results of the calculations are shown in Figure 13. The line
labeled "A" represents the water table, which was calibrated to observed elevations. The line
labeled "B" was determined from the plume diving calculator, using the calibrated water levels,
hydraulic conductivity calibrated from a transport model (Weaver, 1996), and assumptions
concerning the recharge. Most importantly, the average recharge on Long Island is estimated to
be 22 in/year by the USGS. In the area of the sand and gravel pit (Segment 2 on Figure 13) the
entire annual rainfall of 44 in/year was assumed to recharge the aquifer. The combination of
these inputs was sufficient to produce line "B" which is the best estimate of the upper bound of
the contaminant plume.
Although the calculation reproduces the field behavior, the best use of the calculator is
where vertical data are not yet available. Using recharge estimates and land use characteristics an
estimate of plume diving can be made. Sampling can then be made at appropriate elevations.
Since there are many factors that influence the accuracy of the method, not the least being the
assumption of one-dimensional flow, the calculator predictions are not intended to give the "last
word" on plume diving. Rather the calculator results should guide placement of well screens.
18
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§
CD
annum
LU
40
0
-40
-80
-120
6000
4000
(ft)
I
0
Figure 13 Example plume diving calculation for the East Patchogue site. "A"
represents the watertable which was calibrated to the field data. "B" represents an
uncalibrated prediction of the top of the plume determined from the plume diving
calculator.
Required Input
The plume diving calculator requires input to 1) Define the hydraulics of the flow system
and 2) Contaminant source and observation well location. The single screen of the plume diving
interface is used for parameter entry, reporting of results and graphing the solution. The
following drawings show an exploded view of the interface.
Flow System Hydraulics
The aquifer is divided into segments and for each the hydraulic conductivity, recharge
rate, and length must be specified (Figure 14). A uniform aquifer can be simulated by using the
same parameter values for each segment.
Segment 1
Segment 2
10.0
250.0
5.0
10.0
Segment 3
10.0 ! Wtj
250.0
2500
5.0
5.0
Figure 14 Plume Diving calculator hydraulic and hydrologic property entry.
19
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Source and Observation Well Locations
Two heads are specified, one for the upgradient end of the flow domain at point (A) and
one at the downgradient end (B). Within this domain, the locations for the source of
contaminants and the well are then entered on the side panel (Figure 15).
Mpgradient Head (at A)
powngradientHead (at B)
Source Location
Wilt Location
Figure 15 Plume Diving calculator location entry.
50.0
40.0
10.0
400.0
I ft
I ft
I ft
! ft
For a source that covers a large area (free product zone, landfill, etc.) the source location should
be taken as the down gradient edge of the source.
The well location is used to show how much plume diving to expect at a point down
gradient from the source (Figure 16). The well is indicated by a red line on the output that shows
the difference between the water table and the top of the plume.
60.Oft
watertable
plume top
Figure 16 Plume Diving calculator
graphical output.
20
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Model Results
In addition to the graphical output, the depth below the water table is output, as is the
over all mass balance error. The source and well locations are echoed in this section (Figure 17).
Plume at We!! 3.332233133053879
'% Mass Balance Error < 1 e-6
Results...ealeulated for:
Source Location at 10.0
Well Location at 400.0
Figure 17 Plume Diving calculator mass
balance output.
Drawing Options
The features graphed on the drawing can be customized by (Figure 18)
Labeling
• Drawing the water table
Drawing the top (upper bound) of the plume
• Vertically exaggerating the drawing
P~ DrawWatertabte ? P" Vertical Exaggeration ?
W Draw UpperBound of Plume ? P Label Drawing ?
Figure 18 Plume Diving calculator display options.
Drawing the top (upper bound) of the plume automatically turns on drawing of the water table.
When vertical exaggeration is turned off, the horizontal and vertical scales of the drawing are the
same.
21
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3 Contaminant Transport
When confronted with the possibility of contaminated ground water, questions often
asked by the public include:
• Will my well become contaminated?
• When will contaminants reach my drinking water supply?
• How bad will it be? (i.e., How high will the contaminant level be?)
• What are the effects of drinking this water on my children or myself?
The ability to answer these questions presumes a predictive capability that cannot be achieved by
monitoring alone. Thus many agencies and individuals turn to models to provide answers to
these questions. Models are chosen for this task because
they have an evident ability to predict future concentrations,
• they have a scientific basis,
they have the ability to include the effects of many different factors, and
• they have become accepted as predictive tools.
A number of factors, however, influence and limit the ability of models to predict future
contamination. The proper usage of models depends on the details of their construction and how
they are used for a specific problem. As will be seen to be the major thrust in the following
section, an accounting for uncertainty should be included as a part of model usage. A calculator
called the ConcentrationUncertainty calculator7 will be used to show the impacts of parameter
variation on contaminant transport and the generation of generic worst-case parameter sets.
First Arrival Versus Advective Travel Times
Before beginning the discussion on model application, several concepts in subsurface
transport are introduced in this section.
Frequently the advective travel time is used as a rough guide for contaminant transport. If
contaminants were transported only by movement with the ground water, then the travel time
from a source to a receptor would be the advective travel time. Figure 19 shows a schematic
illustration of transport by advection only compared against transport by advection and
dispersion.
7The ConcentrationUncertainty calculator can be found at
http://www.epa.gov/athens/learn2model/part-two/onsite/uncertainty.htm
22
-------
c
o
"S
"E
o»
o
j-
o
O
Adveetion Only
Adveetion and Dispersion
Distance
Figure 19 Illustration of transport by advection only (hypothetical)
and transport by advection and dispersion.
The advective travel time, tadvective, is calculated from Darcy's Law and a retardation factor:
advecttve
receptor
v/R
S
(3)
where xreceptor is the distance to a receptor in the aquifer (i.e., a well) [L], R is the retardation
factor [dimensionless], and vs is the seepage velocity [L/T]. The retardation factor is the ratio of
transport velocities of a conservative tracer to that of a sorbing chemical. The seepage velocity
is defined from Darcy's Law:
0
- Ki
0
(4)
where vd is the Darcy flux [L/T], K is the hydraulic conductivity [L/T], i is the hydraulic gradient,
and 0 is given variously as the effective porosity or total porosity [L3/L3].
In contrast to this simple approach (Figure 19 and Equation 3), transport is also subject to
dispersion8 and biodegradation. Transport by advection alone does not occur in ground water
because there is always some spreading of a sharp front (illustrated by the abrupt drop of
concentration in the advection only case). At some time after a release, the contaminant arrives
at a given location. Because of spreading caused by dispersion this time will be sooner than if
only advection is included.
8Dispersion is properly interpreted as differential advection through materials of various
hydraulic conductivity. Were this properly accounted in a simple formula as Equation 3, there
would be less of a need for the approach that follows.
23
-------
Advective travel time is an appealing quantity because of its conceptual and calculational
simplicity. But, the first arrival time of a contaminant can be calculated easily and can
incorporate the threshold concentration of concern, biodegradation, dispersion, advection,
retardation, source concentration and source lifetime. Figure 20 illustrates the first arrival time
and other concepts to be applied in following calculation. Given a receptor located some
distance from a contaminant source, no contamination is observed at the receptor when the
release begins. Because of the distance some amount of time is required to transport the
contaminant from the source to the receptor. In the Figure, this time period is about 1400 days.
After that time, the contaminant concentration rises. The gradual rise occurs because subsurface
transport occurs through a heterogeneous medium with different rates in different parts of the
medium. The time when the contaminant first goes above the concentration of concern (CoC) is
called the first arrival time. As more contamination arrives at the receptor, the concentration
increases until a maximum is reached. This is called the maximum concentration. The
maximum concentration cannot exceed the source concentration and may be reduced much lower
from dispersion or biodegradation. In the case shown in Figure 20, the maximum concentration
does equal the source concentration. If the source of contaminant is finite, at some time the
concentration at the receptor will decline and drop below the concentration of concern. The time
period that the concentration is above the level of concern is called the duration.
It is possible for dispersion and biodegradation to cause the contaminant concentration to
be reduced to below a threshold level of concern before the chemical reaches a specified
receptor. In this extreme case, the concept of advective travel time has no relevance to the
receptor. In other, less extreme cases, contamination may rise above the level-of-concern sooner
than the advective travel time, and a protection strategy based upon advective travel time is non-
conservative.
The concentration of concern would normally be chosen as an maximum concentration
level (MCL) or other value. For example the federal MCL for benzene is 5 |J-g/L. The US EPA
Office of Water has given a drinking water advisory for MTBE at a level of 20 |J-g/L to 40 |J,g/L.
These levels could be chosen as CoCs. The first arrival time, duration and maximum
concentration all can vary from those shown in the Figure, depending upon the source and
transport properties.
24
-------
Breakthrough Curve
25
IT 20 -
O)
CO
•tj
OJ
O
I—
o
O
15
10 -
5 -
0
0 1000*2000 3000 4000 50
First
Arrival
Duration
DO 6000
Time (days)
Figure 20 Illustration showing the relationship between the first arrival
time, maximum concentration and duration of contamination. The first
arrival time and duration are determined relative to a given threshold
concentration, that is usually a maximum contaminant level or other
concentration of concern.
25
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Uncertainty Range Determination
Models are commonly viewed as useful tools for understanding contaminant transport
(Oreskes et al., 1994) and determining future risk (ASTM, 1995). The degree of predictive
capability of subsurface transport models has, in fact, not been established (e.g., Miller and
Gray, 2002, Eggleston and Rojstaczer, 2000). Given that the values of all the parameters
(hydraulic conductivity, dispersivity, retardation factor, biodegradation rate constant) and the
forcing function (source concentration and source duration) were known, and that the
assumptions behind the model were exactly met, the model equations (Equation 24) could be
solved for the required outputs. In the real world, however, the values are not exactly known and
no aquifer would meet the required assumption of homogeneity. At leaking underground
storage tank (LUST) sites, it would typically be expected that hydraulic conductivities would be
measured through slug tests and the other input parameters not measured. The source
concentration and duration would be unknown. Dispersivity, since it represents unaccounted
heterogeneity, is clearly not a fundamental parameter and is best viewed as a fitting parameter:
thus its appropriate value is also unknown.
Models are more likely to provide a framework for understanding transport than for
predicting future exposure and risk. Commonly, models are calibrated to field data to
demonstrate their ability to reproduce contaminant behavior at a site. This process implies a
degree of correctness in understanding and provides the first step toward demonstration of
predictive ability. For screening sites or where rapid response is required, sufficient data may not
be collected for calibrating a model. How then should models be used in situations where they
can not or will not be calibrated? What are the plausible ranges of output given uncertainty in
inputs? Can worst case parameter sets be selected that always provide a bound on plausible
outcomes?
Figure 21 shows a conceptual relationship between uncertainty and data availability.
With small amounts of either measured input data or calibration data, the resulting model
uncertainty is high. Models may still be useful in these cases, but their uncertainty should be
quantified so that their results are not taken falsely as inerrant.
i: ^vt. Estimates and a few site-specific parameter values
|!
Uncertainty |;
^ ^!=%* Model calibrated to heads/concentration data
||
I; Calibrated model tested
i: against prediction
Data: Input Parameters or Calibration Data
Figure 21 Relationship of uncertainty to model data
availability.
26
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Approach
Several approaches to uncertainty analysis have been developed. Generally these require
knowledge of parameter values and their statistical distributions including correlations between
individual parameters. Here site investigations are not sufficiently detailed to determine values
for some of the parameters, let alone their statistical distributions and correlations. A brief
accounting of the model inputs is given as follows: Porosity and dispersivity are essentially
never determined on a site-specific basis, despite their importance in determining model
outcomes. Biodegradation rate constants may be estimated from simple techniques (Buscheck
and Alacantar, 1995), but these require adherence to a suite of restrictive assumptions that limits
the results by the same considerations that we are attempting to address in this work. Parameters
measured in the field are subject to uncertainty because of spatial variability (hydraulic
conductivity and fraction organic carbon) or temporal fluctuations (gradients). The forcing
parameters of the model, initial concentration and duration, are rarely known, because
contamination is normally discovered years after a release occurs.
There is a similar lack of knowledge of statistical distributions of the inputs. A widely-
used alternative is to assume knowledge of the statistical properties by using scientific literature
values as substitutes. These approaches allow assignment of probabilities to the various
outcomes, but suffers from obvious lack of site-specificity. Where results depend strongly on
assumed distributions, it is not possible to determine how much error is introduced into the
results from the distributions. Alternatively, if it is assumed only that plausible ranges of input
parameters are known, similar outcomes can be determined, but probabilities cannot be assigned.
Because of lack of knowledge of the underlying probability distributions, a method based on
ranges of inputs was developed.
Nine parameters of the one-dimensional model (Equation 24) are assumed to be variable.
Tables 5 and 6 list parameters and their treatment in the model. All seven parameters of the
model were allowed to be variable, as were the concentration and duration of the source. The
chemical, distance to receptor, and minimum concentration of concern are taken as fixed for a
given analysis. With this selection of inputs there are two values each for nine parameters: the
minimum value and the maximum value. This leads to a total of 29 or 512 unique combinations
of parameters. This calculation highlights an assumption of this method: That each parameter
value is equally likely and can occur in combination with each other parameter value. In other
words that each parameter is uniformly distributed and uncorrelated. The large number of
parameter combinations is the reason to seek models that execute rapidly. Hence the interest in
the one-dimensional model.
Figure 22 shows an example output where breakthrough curves representing various
breakthrough curves have been plotted. These have been selected from the 512 outputs of the
model to represent the significant outputs of the model: earliest and latest first arrival times,
minimum and maximum peak concentration, and the longest and shortest duration. Along with
27
-------
these the breakthrough curve for the average values of the inputs was plotted. It's clear that a
wide variety of breakthrough curves is possible from this model, given the selected input range.
r Graphical Output-
C
24.0_
16.0
8.0.
0.0
0.0
Breakthrough Curves
4300.0
8000.0
12000.0
E a rl i est F i rst Am va I
Latest First Arrival T
Min. Cone.
Max. Cone.
Shortest Duration
Longest Duration
Mean value of ail ii
Figure 22 Output from the Concentration Uncertainty applet showing the wide range of
breakthrough curves that are possible given specified ranges of input parameters.
Concentrations in milligrams per liter are plotted against time in days.
28
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Quantity
Treatment
Example Problem Values
Low
High
Model Parameters
Hydraulic Conductivity
Porosity
Gradient
Fraction Organic Carbon
Organic Carbon Partition Coefficient
Dispersivity
Half Life
variable
variable
variable
variable
variable
variable
variable
Low Scenario
High Scenario
15ft/d
108 ft/d
0.20
0.001
0.0001
31L/kg
0.1 * estimate from Xu and Eckstein
(1995)
Low Scenario
High Scenario
100 days
4000 days
50 ft/d
328 ft/d
0.25
0.005
0.001
106 L/kg
10 * estimate from Xu and
Eckstein (1995)
730 days
6000 days
Table 5 Parameter inputs, their treatment in the model as fixed or variable and the values used in
the model uncertainty example.
Quantity
Treatment
Example Problem Values
Low
High
Problem Definition
Source Concentration
Source Duration
Chemical
Distance to receptor
Minimum Concentration of Concern
variable
variable
fixed
fixed
fixed
lOmg/L
1500 days
benzene
Low Scenario
High Scenario
0.005 mg/L
50ft
500ft
30 mg/L
3000 days
benzene
50ft
500ft
0.005 mg/L
Table 6 The problem definition, its treatment in the model as fixed or variable and the
values used in the uncertainty example.
Simulation
In order to compare various scenarios, four outputs are generated from the modeled
breakthrough curves: 1) first arrival time, 2) maximum concentration, 3) duration of
29
-------
contamination, and 4) risk factor. This first three of these are illustrated in Figure 22. Cancer
risk is normally determined from an expression of the form (US EPA, 1989)
D. , ,.,,-, CW ED EF IR CE,
Risk = ISF = SF (5\
BW AT l '
where I is the intake in mg/kg-day, SF is the cancer slope factor (kg-day/mg), CW is the
concentration in water (mg/L), ED is the exposure duration (days), EF is the exposure frequency
(days/year), IR is the injestion rate (liters/day), BW is the body weight (kg), and AT is the
averaging time (years). Since concentrations on the breakthrough curve change with time, the
effect of the transient in concentration is included in the risk equation by using the substitution:
CWED = *°CW(i)dt (6)
where CW(t) are the modeled concentrations, t0 is the contaminant first arrival time, ta is the last
time that the concentration is above the threshold. Thus the integral of concentration versus time
gives a measure of relative risk. The model accumulates results and determines the best and
worse cases for each of the four chosen breakthrough curve outputs.
In addition to variable parameters, four scenarios were created to simulate a variety of
conditions and determine if the model behavior was similar despite variation in parameter values.
Two ranges each of hydraulic conductivity and half life were selected (Table 6). These variables
were chosen to vary because they have a direct effect on model outputs: Hydraulic conductivity
affects advective transport rates and thus the arrival times and duration of contamination, and the
half life impacts maximum concentration. Risk is affected by both concentration and duration.
The scenarios are generally comparable with each other with the exception that the receptor is
closer to the source in the low conductivity scenario. This selection was made so that there
would be complete breakthrough curves for all combinations of parameters in each scenario.
Results
Table 7 shows the extreme cases for four problem scenarios (see Table 5): High and low
conductivity aquifers, and high and low biodegradation rates. These results show the magnitude
of possible outcomes given the range of inputs used. The high and low conductivity scenarios
have a different distance to the receptor (500 ft versus 50 ft), so the first arrival time results are
not directly comparable. In going 10 times further in the high conductivity scenario the arrival
time is approximately 2.5 times greater than the low conductivity scenario (20 days/7.9 days),
indicating proportionately earlier first arrival in the high conductivity scenario. The minimum
durations are in part determined by the source duration (which at a minimum is 1500 days).
With high biodegradation rates the minimum concentrations can be greatly reduced (compare
row a and b and row c and d in column 3) in either scenario.
30
-------
First Arrival
Time
(days)
Earliest
(1)
Latest
(2)
Maximum
Concentration
(mg/1)
Lowest
(3)
Highest
(4)
Duration
(days)
Shortest
(5)
Highest
(6)
Risk Factor
(day-mg/1)
Best
(7)
Worse
(8)
High Conductivity Scenario (receptor located 500 feet from the source)
(a) Low Biodegradation
(b) High Biodegradation
20
20
1140
1760
6.4
0.0081
30
28.3
1580
1340
8310
7210
2.01e5
562
1.95e6
1.54e6
Low Conductivity Scenario (receptor located 50 feet from the source)
(c) Low Biodegradation
(d) High Biodegradation
7.9
7.9
604
740
6.9
0.079
30
30
1580
1580
9210
7610
1.71e5
3330
1.76e6
1.55e6
Table 7 Model results for four scenarios showing a comparison of best and worst cases for the
four outputs (first arrival time, maximum concentration, duration and risk).
Table 8 shows a comparison of the parameters across the scenarios. This comparison was
made to determine if the extreme cases were generated by the same sets of parameters, despite
changes in the average values of the parameter. These results imply that it is possible to
determine a generic set of worst case parameters for three of the outputs (first arrival time,
maximum concentration and duration above the threshold). In some cases the results are
insensitive to a given parameter and either parameter could generate the worst case. For
example, the first arrival is independent of release duration in these cases, because the release
duration was much greater than the arrival times. Generally the results were consistent for the
first arrival, maximum concentration and duration. Definition of the worst case for risk,
however, was less clear as the parameter sets were not the same for each simulation. The
porosity, fraction organic carbon, dispersivity and half life were different for the scenarios and
suggested that a generic set of parameters did not exist for risk.
Note that each of these generic results rests upon the assumption that the model is a valid
representation of contaminant transport at a site. At fractured rock or karst sites, sites where
pumping wells dominate flow, or for transport in multi-layer aquifers, a more powerful model is
required. These models have differing sensitivities to parameters and even different required
parameters. Thus a complete assessment depends also on uncertainty associated with the degree
of correspondence between the simulation code and the conceptual model of the site.
Uncertainty in model inputs results from spatial variability and incomplete or imperfect
sampling methods. Running all combinations of input parameters gives bounds on the plausible
31
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outputs of the model. For each individual parameter set selected for analysis, the worst case
parameters varied with the output of interest (i.e., first arrival time, maximum concentration,
duration, and risk). Given the chosen output, however, the worst case parameter sets were the
same for each combination of hydraulic conductivity and biodegradation rate for the first three
outputs. In contrast, the results for the risk calculation, perhaps because it depends strongly
upon both the concentrations and the duration of the breakthrough curve, had no consistent set of
worst case parameters. For risk the uncertainty analysis must be performed individually for each
parameter set, while for the others this analysis showed a that a generic set of worst case
parameters existed. Because this problem is one of transient transport concentrations differ over
the duration of exposure. For a steady state model, however, risk could be calculated directly
from concentration and there does exist a generic worst case parameter set for this problem.
32
-------
Scenario
Hydraulic Conductivity
£•
I
Gradient
Fraction organic carbon
1
Dispersivity
Half Life
Source cone
SourceDuration
Earliest First Arrival
Low Conductivity, Low Biodegradation
Low Conductivity, High Biodegradation
High Conductivity, Low Biodegradation
High Conductivity, High Biodegradation
H
H
H
H
L
L
L
L
H
H
H
H
L
L
L
L
L
L
L
L
H
H
H
H
E
E
E
E
H
H
H
H
E
E
E
E
Highest Maximum Concentration
Low Conductivity, Low Biodegradation
Low Conductivity, High Biodegradation
High Conductivity, Low Biodegradation
High Conductivity, High Biodegradation
H
H
H
H
L
L
L
L
H
H
H
H
E
E
E
E
E
E
E
E
H
H
E
H
H
H
H
H
H
H
H
H
E
E
E
E
Longest Duration
Low Conductivity, Low Biodegradation
Low Conductivity, High Biodegradation
High Conductivity, Low Biodegradation
High Conductivity, High Biodegradation
L
L
L
L
H
H
H
H
L
L
L
L
H
H
H
H
H
H
H
H
H
H
H
H
H
H
E
H
H
H
H
H
H
H
H
H
Highest Risk
Low Conductivity, Low Biodegradation
Low Conductivity, High Biodegradation
High Conductivity, Low Biodegradation
High Conductivity, High Biodegradation
L
L
L
L
H
H
E
L
H
H
L
L
H
H
E
H
H
H
E
L
L
L
H
H
L
L
E
H
H
H
H
H
H
H
H
H
H = high value, L = low value, E = either value
Table 8 Comparison of data sets giving the worst cases for each of four model outputs.
33
-------
Concentration Uncertainty Model Input and Output
The inputs required for the concentration uncertainty model define the fixed and variable
parameters. It is presumed that the distance to the receptor, the chemical of concern and the
minimum concentration are all fixed (Figure 23). The remaining parameters of the model can be
variable (Figure 24). These include the formal parameters of the transport equation, and
parameters that describe the forcing function (source concentration and duration). Each of these
parameters can be individually fixed by entering the same value as both "low" and "high" on the
input screen. Each parameter that is fixed to a single value reduces the number of model runs by
a power of two.
WMMe inputs
QutprtB
iplir Until
Cpuparlsan »Htt Heflltn
Minimum Canetutstlon of Canes n
Figure 23 Fixed parameters required for the Concentration Uncertainty
model.
i Biekinunil
: M
i [liai Inputs Porosity
1 Varialils Inputs
: foe
•
H_________ ^
!
• r s DIs^irsMI?
: GnmtKliuii iMI Metfltn
i ~ I
! Si*
source DWMUI
I Run 1
108.0
0.2
0.0010
1.0E-4
31.0
I
10.0
1500.0
j
!i(jh Viillli!
324.0
0.25
0.0050
0.0010
106.0
•IttVHH
300
3000.0
Figure 24 Potentially variable parameters required for the Concentration
Uncertainty model.
34
-------
Model Outputs
The major outputs of the model are displayed on three output screens. The first of the
output screens shows the values of all the key outputs of the model: first arrival time, maximum
concentration, duration above threshold concentration and the risk factor (Figure 25). These are
each show for the best and worst case. In addition to displaying the value of the parameter of
interest, say the first arrival time, each line of output also shows the associated values of all other
outputs.
If it could be determined that a generic set of best and worst case parameter existed, then
it would not be necessary to run an uncertainty analysis every time the model was run. Figure 26
indicates which parameter (low or high value) resulted in each model outcome. From these it has
been shown that generic worst cases exist for some of the outputs. (See the example problem
outputs and discussion.)
Thirdly, the output breakthrough curves are plotted against the breakthrough curve for the
average of all input values. These plots show the wide range of possible model outcomes (Figure
27). The families of curves can be moved closer together by fixing values of variable input
parameters. Exploring which parameters could benefit from improved estimates can help focus
site assessment activities.
Background
Select a Model
Fixed Inputs
Variable Inputs
Outputs
Parameter Results
graphical Output
Cases Tested
Earliest First Arrival
Latest First Arrival
Lowest Max, Cone.
Highest Max. Cone,
Shortest Duration
First arrival
Max. Cone.
Duration
Risk Factor
Comparison with Median I Longest Duration
About I Lowes! Risk
Highest Risk
Pause
Stop
Figure 25 Output from the Concentration Uncertainty model showing the extreme values for
each model output: first arrival, maximum concentration, duration above threshold and risk
factor.
35
-------
Background
Low Value
High Value
Select a Model
Fixed Inputs
Variable Inputs
Outputs
Parameter Results
=i Earliest First Arrival
=| Latest First Arrival
"1 Lowest Max. Cone.
1 Highest Max. Cone.
Graphical Output
Comparison wtth Median j
About
! Shortest Duration
! Longest Duration
I Lowest Risk
Highest Risk
Run
Figure 26 Generic values (low or high) of parameter sets that generate extreme values of each
output of the model: first arrival time, maximum concentration, duration and risk factor.
1 1
i BacKgfGHiwI :
i Select a Model j
i Fixed MptAs :
i Variable Inputs I 24°-
\ a"*1*8 ; «0_
\ Parameter ResMts i
^::::::::::::::::::::::::::::::::::::::^^^^ 12^0__
i Graphical Output E
j C««|iarssBnii»«h Median i 00-
i »"* 1 OJ)
0
i nm l
Breakthrough Curves
- """ "" H
Mean ¥a!ua ot all inpute
: E a rH sst First Arrival Tim
Latest First ftrii«ai Time
! , "•»• c»««^
| Lowsst Risls Factor
1 Highest RisSc Factor
/ ^.
j
1 ! \ -
0 1000,0 2000.0 3000,0 4000.0 5000.0 8000.0
Time
Figure 27 Graphical presentation of Concentration Uncertainty model
output showing a comparison of the breakthrough curves for the extremes
of each model output: first arrival time, maximum concentration, duration,
and risk factor with the breakthrough curve for the average value of all
inputs.
36
-------
4. Model Input Parameters
The simulation models described above require input parameters that can be estimated in
various ways. The transport equation given above (equation 24) requires six input parameters:
the retardation factor, R,
• the seepage velocity, vx,
the dispersion coefficients in x, y and z (Dx, Dy, and Dz), and
• the loss rate constant, X.
The next sections describe available methods for estimating or manipulating these parameters.
Retardation Factor Calculator9
The retardation factor expresses the amount of sorption of an organic chemical on aquifer
solids. This concept is found in the majority of common transport models and is itself a model,
as it represents a simplified conceptualization of sorption. The retardation factor is calculated
from
where R is the dimensionless retardation factor, pb is the bulk density of the aquifer material
[M/L3], 0 is the porosity [L3/L3] and kd is the soil-water distribution coefficient [L3/M]. The bulk
density is related to the porosity
P* = P,0 - 6)
= 2.65^- (1 - 0) (8)
ml
where ps is the solids density [M/L3] which is commonly taken as 2.65 g/ml: the density of
quartz. The use of bulk density reflects the fact that aquifer materials are composed of both
solids and void space.
The soil-water distribution coefficient, kd, is commonly estimated as the product of the
fraction of organic carbon in soil [dimensionless] and the organic carbon partition coefficient, Koc
[L3/M]. Koc has been tabulated or estimated and values are available for most contaminants of
concern.
9http://www.epa.gov/athens/learn2model/part-two/onsite/retard.htm
37
-------
Example
Retardation factors for benzene and MTBE are given in Tables 10 and 10. These values
are presented for varying conditions: low organic carbon (foc = 0.0001), medium organic carbon
(foc = 0.001); and porosity of 0.25 versus 0.15. The range of Koc's is taken from data currently in
the calculator and is illustrative of the range of possible variation. For benzene (Table 9) the
Koc range is taken as 38 L/Kg to 100 L/Kg. This results in insignificant differences in the
retardation factor for low values of organic carbon. At the medium organic carbon level
(0.0001), the differences in R become more pronounced as the porosity decreases. For MTBE
the range of variation in Koc is low, so the differences in R are attributable to the porosity and
organic carbon content. Table 10 shows that the highest R for MTBE (1.2) occurs with the
lowest porosity and highest organic carbon content (0.15 and 0.001, respectively).
Porosity
Fraction
Organic
Carbon
Organic Carbon
Partition
Coefficient
(L/Kg)
Retardation
Factor
Benzene
0.15
0.25
0.0001
0.001
0.0001
0.001
38
65
83
100
38
65
83
100
38
65
83
100
38
65
83
100
1.1
1.1
1.1
1.2
1.6
2.0
2.2
2.5
1.0
1.1
1.1
1.1
1.3
1.5
1.7
1.8
Table 9 Retardation factors for benzene under varying conditions.
38
-------
Porosity
Fraction
Organic
Carbon
Organic Carbon
Partition
Coefficient
(L/Kg)
Retardation
Factor
MTBE
0.15
0.15
0.25
0.25
0.0001
0.001
0.0001
0.001
ll
14
11
14
11
14
11
14
1.0
1.0
1.2
1.2
1.0
1.0
1.1
1.1
Table 10 Retardation factors for MTBE under varying conditions.
Ground Water Velocity
Seepage Velocity Calculator10
On average, the velocity of a contaminant is governed by the seepage velocity11, vs, which
is the Darcy Flux divided by the porosity.
v* = ~ -fit (9)
where K is the hydraulic conductivity [L/T], 0 is the porosity [L3/L3], and i is the hydraulic
gradient [L/L]. Equation 9 gives the magnitude of the gradient as a scalar quantity. Velocity is,
obviously, directional and the gradient establishes the direction12.
10http://www.epa.gov/athens/learn2model/part-two/onsite/seepage.htm
nSeepage velocity is also know by the term average linear velocity and others.
12
In non-isotropic media, the direction is also determined by the hydraulic conductivity
tensor.
39
-------
Hydraulic Gradient Calculator13
The magnitude and direction of the hydraulic gradient can be estimated by fitting a plane
through the ground water surface. The equation of the plane is
ax + by + c = h (10)
where x and y are the coordinates of a well, h is the hydraulic head, and a,b, and c are constants.
The constants can be evaluated by fitting the plane to at least three wells. When more than three
points are used, the coefficients are calculated by least-squares fitting of the data to the plane.
The magnitude of the gradient is calculated from the square root of a2 + b2, the direction from
North is determined from the arctangent of a/b or b/a, depending on the quadrant.
Example
Water level data from four wells on a service station are shown in Table 11. These were used to
calculate the magnitude and direction of the gradient for six dates from 1991 to 1998. Note that
the locations (x and y) and elevations of the top of casing for each well are necessary for
calculating the gradient. In this case the x and y coordinates were scaled from a map and the
elevations were determined by a survey. Figure 28 shows the resulting directions and
magnitudes. Clearly, these results are not consistent over time. This result may be due to the
very small scale covered by the data, and the high proportion of pavement at the site. Note that
when the direction is indicated to be about 90 degrees (East), the magnitude of the gradient is
small. These times correspond to nearly identical water levels in the wells (Table 11, data from
3/20/1994 and 4/22/1998), and for this reasone these flat gradients may not be accurate. Also,
this site is located just to the east of the Mississippi River, and the expected direction of flow
would be to the West. To determine the gradient more definitively, wells are needed beyond the
boundaries of the service station property and the regional ground water flow direction needs to
be considered.
13http://www.epa.gov/athens/learn2model/part-two/onsite/gradi ent4plus-ns.htm
40
-------
.o
"o
CD
Q
_o
LJ_
Q)
"OB
^
C
O
CD
1
'c
D)
03
0
360^
270
-
180-
—
90-
o-
0.1 -i
0.01 -i
0.001 -i
.0001 -
5/7
^ ^
• •
^
^
•
* • * *•
iii
790 1/31/93 10/28/95 7/24/98 4/1 £
Date
Figure 28 Magnitude and direction of gradient for example site.
41
-------
Date
5/7/1991
3/20/1994
10/11/1995
10/8/1997
4/22/1998
10/29/1998
Well
MW-1
MW-2
MW-3
MW-4
MW-1
MW-2
MW-3
MW-4
MW-1
MW-2
MW-3
MW-4
MW-1
MW-2
MW-3
MW-4
MW-1
MW-2
MW-3
MW-4
MW-1
MW-2
MW-3
MW-4
Coordinates (ft)
East
133
123
100
62
133
123
100
62
133
123
100
62
133
123
100
62
133
123
100
62
133
123
100
62
North
99
38
101
89
99
38
101
89
99
38
101
89
99
38
101
89
99
38
101
89
99
38
101
89
Elevation
TOC1
(ft)
499.11
499.19
498.83
498.63
499.11
499.19
498.83
498.63
499.11
499.19
498.83
498.63
499.11
499.19
498.83
498.63
499.11
499.19
498.83
498.63
499.11
499.19
498.83
498.63
Depth to
Water (ft)
48.08
48.21
47.87
47.73
49.27
49.37
49.01
48.77
49.31
49.52
49.06
48.83
48.79
48.7
48.2
48.42
47.99
48.08
47.72
47.47
48.87
49.89
48.6
48.37
Elevation
Water
(ft)
451.03
450.98
450.96
450.9
449.84
449.82
449.82
449.86
449.8
449.67
449.77
449.8
450.32
450.49
450.63
450.21
451.12
451.11
451.11
451.16
450.24
449.3
450.23
450.26
1 TOC stands for too of casing.
Table 11 Data for gradient calculation example.
42
-------
Longitudinal Dispersion Coefficient Calculator14
As discussed previously, the dispersion coefficient appearing in the transport equation
(24) represents differential advection, is not measured at LUST sites, and is best treated as a
fitting parameter. Data have been collected on dispersion coefficients, however, that provide
insight into the dispersion parameter. Gelhar et al. (1992) published a review of dispersivities
(D/vs) which is replotted in Figure 29. Because the data do not plot on a horizontal line, These
data show scale-dependence of dispersivity. This fact implies that no single value represents a
contaminant plume. From this follows the conclusion that dispersivity should be treated as a
fitting parameter. Various methods have been devised to estimate dispersivity for a specific site
from the Gelhar et al. (1992) tabulation. Two of these are shown in Figure 30 (see Xu and
Eckstein, 1995). Without comparing values generated from these formulas to the underlying
data, a false sense of certainty may be perceived. At a given scale the tabulated dispersivities can
range over three orders of magnitude. The formulas, of course, give specific values.
In order to give a sense of the formula values and the range of variability, the dispersivity
calculator produces estimates from two formulas, and an evident range from the tabulation.
Table 12 gives values for various plume lengths (an indicator of the problem scale) and the
estimated range of values in the tabulation. From these results one can conclude that the range of
literature variability of this parameter is very high and that in the absence of fitting to observed
concentrations there is no way to prove that a chosen value is appropriate.15
14
http://www.epa.gov/athens/learn2model/part-two/onsite/longdisp.htm
15The converse is also true: it is not possible to prove values are inappropriate in the
absence of fitting.
43
-------
Gelhar, Welty and Rehfeldt (1992) Dispersivity Data
\5
>,
•>
'w
k_
0
Q.
(O
b
"oj
_c
J
'6)
c
o
104
103
102
101
1
1C'1
1C'2
10'3
1C
: Reliability:
• High [
r A Intermediate
: D Low
r D
: n
n n n
n cr u n
A n n
: A n, n n
r— 1 \\ n_l n-n— i
II II i-li
g ^^ f °
: Aj g °A n A/^ na n
[jA[Hw^ft
A^jpA A • «
: ^'^A^ A *
nA~^ A
n A
; n
: n n
- A A nA
: A
iii
r1 1 io1 io2 io3 io4 K
Scale (m)
Figure 29 Data tabulation from Gelhar et al. (1992) showing published longitudinal dispersivity
as a function of scale.
44
-------
(0
CD
Q.
CO
CO
C
)
c
o
Gelhar, Welty and Rehfeldt (1992) Dispersivity Data
105
Reliability:
one tenth"
•^-« Xu and Eckstein
• High
A Intermediate
n Low
Scale (m)
Figure 30 Gelhar et al. (1992) dispersivity tabulation with scale-related estimates.
45
-------
Plume Length
(ft)
100
500
1000
2500
5000
Approximate
Lower Bound
(ft)
0.084
0.95
1.7
3.9
6.3
Xuand
Eckstein
Formula
Result
(ft)
7.06
17.9
24.5
35.1
44.6
Approximate
Upper Bound
(ft)
670
2800
5200
12000
19000
Table 12 Relationships between plume length, data scatter on
the Gelhar (1992) tabulation and the Xu and Eckstein (1995)
formula.
Half-Lives to Rate Constant Conversion Calculator
16
Laboratory microcosm data are often reported in terms of representative first-order rate
constants, X. The rate constant is also required as input for most models that use this concept.
The rate constant can be represented as a half-life, through a simple conversion.
The equation for first-order decay is the starting point for determining the conversion
factor:
dC
dt
(11)
where C is the concentration, t is time, and X is the first-order rate constant. The solution to
equation 11 is given by
C8exp[-A.(f -
(12)
where C(t) is the concentration at time t, C0 is the intial concentration, and t0 is the initial time.
The half life, t1/2 is obtained by setting the concentration equal to one-half its initial value in
equation 12. After rearranging and taking the natural log (In) of both sides, the half life becomes
16http://www.epa.gov/athens/learn2model/part-two/onsite/halflife.htm
46
-------
-) = - M/2 (13)
When the log of 1A is evaluated, the conversion is seen to be
0.6931
Additional Parameter Values
There are a number of other parameters that are used in models of various types. These will be
discussed in the following sections: the effective solubility from a mixture, henry's constants and
diffusion coefficients.
Effective Solubility from Mixture Calculator17
The concentration of a chemical in equilibrium with a water-immiscible mixture differs from the
solubility of that chemical in equilibrium with water alone. The solubility resulting from the
mixture is called the effective solubility and depends on the pure component solubility (water
alone) and the amount of chemical in the mixture (mixture properties). Raoult's Law is used to
characterize the partitioning and is stated as:
Seff = XiSi (15)
where Seff is the effective solubility of a chemical in a mixture [M/L3], x; is the mole fraction of
the chemical in the mixture [M/M] and S; is the solubility of the chemical in water [M/L3]. In this
formulation the activity of the chemical in the mixture has been assumed equal to one, following
the work of Cline et al. (1991) for gasolines. Effective solubilities, thus depend on the
composition of the mixture (the mole fraction) and the solubility of the chemical of interest.
These solubilities may in turn vary with temperature as described in the next section.
Shallow Ground Water Temperature in the United States
Average shallow ground water temperatures range from 5 °C in the North to 25 °C in
southern Florida (Figure 31). These values give an indication of the range of temperature
dependence that might be encountered on a nationwide scale.
17http://www.epa.gov/athens/learn2model/part-two/onsite/es.htm
47
-------
Figure 31 Shallow ground water temperatures throughout the
United States.
Temperature Dependence of MTBE, Benzene and Toluene Solubility
Temperature dependence in effective solubility is introduced through data on methyl tert-
butyl ether (MBTE) and benzene solubilities over the range of 0 °C to 40 °C. These data are
drawn from Peters et al. (2002) and Montegomery (1996). Some comments are needed on these
data:
The temperature-dependent effective solubility calculator uses the assumptions concerning
Raoult's law described above:
Mixture properties are approximated by the average properties of the fuel
• Unitary activity coefficients
Inconsistent solubility data reported in the scientific literature is the rule rather than the
exception.
As new or improved data become available the calculator will be updated.
• The MTBE solubilities from Peters et al. (2002), do not match the commonly used value
of about 50 g/L at 25 °C. The Peters et al., data do, however, roughly match two data
points reported by Fischer et al. (2004) , and the value reported given in the Handbook of
Chemistry and Physics (Lide, 2000)
48
-------
80
MTBE Solubility
* Peters et al.. 2DD2
• Fischer et al. 2004
Lid e, 2 GOD
Temperature (C)
Figure 32 Graph of MTBE solubility versus temperature
showing decline in solubility with increasing temperature.
MTBE
The calculator uses the data points from Peters et al. (triangles on Figure 32) and linear
interpolation to estimate the MTBE solubility. Other data that would more fully establish the
temperature-dependent solubility of MTBE do not exist.
Benzene
The benzene data are taken from a larger list of contradictory data presented by
Montegomery, but these (Stephens and Stephens, 1963) data were selected on the basis of their
agreement with the reported solubility of benzene at 25 °C (Figure 33).
49
-------
2500 —1
2000
500 —
Benzene Solubility
(Stephens and Stephens, 1963)
1 I
3 10
I
20
I
30
I
40
Temperature (C)
Figure 33 Graph of benzene solubility versus temperature
showing modest increase in solubility with increasing
temperature.
The International Union of Pure and Applied Chemistry (IUPAC) has evaluated benzene
data from many sources and prepared a set of recommended values or best values for temperatures
from 0 °C to 80 °C. Over the range of 0 °C to 25 °C the IUPAC data and its ranges encompasses
the Stephens and Stephens (1963) data (Figure 33). IUPAC, however, shows that the solubility
of Benzene remains roughly constant over this range. The data point for 0 °C is a "best" value
rather a "recommeded" value, because of more uncertainty at 0 °C. Table 13 shows the values and
ranges plotted in Figure 34.
50
-------
Temperature
°C
0
5
10
15
20
25
Lower 95%
Confidence Limit
mg/L
1372
1753
1743
1723
1733
1753
IUPAC Best or
Recommended
Value
mg/L
1693
1803
1783
1763
1763
1773
Upper 95%
Confidence Limit
mg/L
2013
1853
1823
1803
1793
1793
Table 13 IUPAC benzene solubility data.
3200
2800
1200
IUPAC Benzene Solubility
•Lower95% Confidence Limit
Best (0 C) and Recommended
'Va!ues(5Cto 80 C)
•Upper95% Confidence Limit
I ' T
20 40
Temperature (C)
r
60
I
SO
Figure 34 IUPAC data on benzene solubility showing the
lower 95% confidence limit, recommended value, and
upper 95% confidence limit.
51
-------
Other sources of chemical data can be found on the ERD chemical properties page at
http://www.epa.gov/athens/research/regsupport/properties.html
Toluene
The IUPAC prepared data on toluene (Figure 52) and prepared a set of best values for
temperatures from 0 °C to 60 °C. The following table shows the ranges of values for toluene and
lower and upper ranges. These data were judged to be of generally lower quality and only "best"
values rather than "recommended" values were given.
1000 •
800
200
IUPAC Toluene Solubility
Lower Value
Best Value
UpperValue
\ ' r
20 40
Temperature (C)
\
60
Figure 35 IUPAC data on toluene solubility showing the
lower value, best value, and upper value.
52
-------
Temperature
°C
0
5
10
20
25
Lower 95%
Confidence Limit
mg/L
660
620
550
540
510
IUPAC Best or
Value
mg/L
690
630
590
570
530
Upper 95%
Confidence Limit
mg/L
720
640
630
600
550
Table 14 IUPAC toluene solubility data.
Example Values
Table 15 shows effective solubilities for benzene and MTBE for temperatures of 5 °C, 15
°C and 25 °C. Since the effective solubility depends on the mass fraction of the chemical in the
fuel, the table includes values for example fuels. These fuels contain varying amounts of benzene
and MTBE and the choices used in the table reflect a range of possibilities. The effective
solubilities also depend strongly on the presumed temperature-dependent solubilities for each
chemical. In every case the effective solubilities are far less than the solubilities. For a given
temperature, though, the effective solubility increases with mass fraction in the fuel. The
variation of effective solubility with temperature is show in Table 16. It is evident that most, if
not all the variation is due to the variation of solubility itself.
53
-------
Chemical
Solubility
mg/L
benzene mass fractions:
5°C
15 °C
25 °C
1480
1570
1680
MTBE mass fractions:
5°C
15 °C
25 °C
84500
59000
37500
Effective Solubilities (mg/1) for
indicated % mass fraction
0.398 (1)
7.9
8.4
8.9
1.52(4)
1530
1070
678
0.453 (2)
9.0
9.6
10.
10.4(5)
10300
7180
4560
0.845 (3)
16.8
17.8
19.
12.2(6)
12100
8420
5350
(1)93 Octane gasoline, MTBE
(2)93 Octane gasoline, non-MTBE
(3)87 Octane gasoline, Low MTBE
(4)93 Octane gasoline, Low MTBE
(5)87 Octane gasoline, MTBE
(6)93 Octane gasoline, Low MTBE
Table 15 Example effective solubilities for benzene and MTBE at
temperatures of 5°C, 15°C and 25 °C.
54
-------
Chemical
% Range in
Solubility
benzene mass fractions:
% variation 5 °C to 25 °C
11.9%
MTBE mass fractions:
% variation 5 °C to 25 °C
55.6%
% Variation in Effective
Solubilities for indicated % mass
fraction
0.398 (1)
11.2%
1.52(4)
55.7%
0.453 (2)
10%
10.4(5)
55.7%
0.845 (3)
11.6%
12.2(6)
55.8%
(1)93 Octane gasoline, MTBE
(2)93 Octane gasoline, non-MTBE
(3)87 Octane gasoline, Low MTBE
(4)93 Octane gasoline, Low MTBE
(5)87 Octane gasoline, MTBE
(6)93 Octane gasoline, Low MTBE
Table 16 Range of variation of effective solubilities for benzene and MTBE from
5°C to 25 °C.
Temperature-Dependent Henry's Law Coefficient Calculator18
Henry's constants represent equilibrium partitioning between a chemical in water and the
air. If the concentrations in water, Cw, and air, Ca are given in units of mg/L, then the nominally
dimensionless Henry's law coefficient, Hcc is given by
(16)
The concentrations may be expressed in differing unit sets, however, and the corresponding
Henry's Law coefficients differ from the dimensionless values. Staudinger and Roberts (1996,
page 292) give the relationships between the four Henry's constant unit sets listed in Table 17.
The relationships between the values of the four unit sets was given by Staudinger and Roberts
(1996) by the formula:
MW
PG
MW,
G
(17)
18http://www.epa.gov/athens/learn2model/part-two/onsite/esthenry.htm
55
-------
where MWG and MWL are the molecular weights [M] of the gas (G) and liquid (L) respectively,
pG and pL are the densities [M/L3] of the gas and liquid, T is the temperature in K, and R is the
universal gas constant. Table 18 gives values of the constants required for performing the unit
conversions. Equation 18 gives the conversion factors for ambient conditions (pressure of 1
atm).
0.2194
0.2194
H
12,186
pc
(18)
Concentration Representations: Air/Water
Concentration/Concentration
Mole Fraction Y / Mole Fraction X
Partial Pressure / Mole Fraction X
Partial Pressure / Solubility
Symbol
Hcc
Hyx
Hpx
Hpc
units
(dimensionless-volumetric
basis)
(dimensionless)
(atmospheres)
(atm mVmol)
Table 17 Unit sets for Henry's Constants.
Quantity
molecular weight of water
density of water
molecular weight of air
universal gas constant
Symbol
MWL
PL
MWG
R
Value
0.0 18 kg/mole
1000kg/m3
0.029 kg/mole
82.06* 10'6atm
m3/mol K
Table 18 Constants needed for Henry's law unit conversions.
Example Values
Values of Henry's contants are given in Table 19 for benzene, MTBE, perchloroethene
(PCE) and trichloroethene (TCE). Values are given for both the OSWER and Washington (1996)
methods. The two sets of results show generally close agreement. As illustrated by MTBE, data
may not be available for each chemical in each method. Over the range of temperatures of 5 °C to
25 °C, the Henry's constants vary by about 50 % to 70%.
56
-------
Chemical
Estimated Henry's Law Coefficient
OSWER
Method
(dimensionless)
% variation
5 °C to 25 °C
Washington
(1996)
Method
(dimensionless)
% variation
5 °C to 25 °C
Benzene
5
15
25
0.0887
0.145
0.227
60.9%
0.106
0.152
0.214
50.5%
MTBE
5
15
25
.. w
-»
-»
.»
0.0101
0.0165
0.0262
61.5%
Perchloroethene (PCE)
5
15
25
0.245
0.441
0.752
66.5%
0.232
0.415
0.711
67.4%
Trichloroethene (TCE)
5
15
25
0.156
0.263
0.421
62.9%
0.137
0.230
0.372
63.2%
(a)Data not available for calculation.
Table 19 Estimated Henry's law coefficients for benzene, MTBE, perchloroethene and
trichloroethene at temperatures of 5 °C, 15 °C, and 25 °C.
57
-------
Diffusion Coefficient Calculator19
The estimates of diffusion coefficients are developed from methods presented by Tucker
and Nelken (1990). The calculator uses three methods to make estimates of diffusion coefficients
in air: Fuller, Schettler and Giddings (FSG), the LaBas modification of FSG (FSG-LaBas) and
Wilke and Lee (WL), and one method for the diffusion coefficient in water: Hayduk and Laudie
(HL). Each of these are described briefly in Appendix 6, with complete details given in the
reference (Tucker and Nelken, 1990).
Example Input and Output Values
Table 20 shows the values of inputs to define four chemicals: benzene, MTBE,
perchloroethene (PCE) and trichloroethene (TCE). The calculator requires the input of the
numbers of atoms in the molecule, the number of aromatic rings, special conditions for oxygen
and nitrogen, pressure, temperature and boiling point. The calculation is applicable to chemicals
containing only the atoms displayed (hydrogen, carbon, nitrogen, oxygen, sulfur, fluorine,
chlorine, bromine and iodine) and, if rings are present, they are only aromatic (6-carbon) rings.
Adjustments are made internally for oxygen in the form of esters or ethers, in acids or joined to
sulphur or nitrogen. Adjustment are also made for amine nitrogen and double-bonded nitrogen.
The boiling point of the chemical is needed for the WL method. The values appearing in Table
58 were obtained from Aldrich Chemical Company (2003).
Table 21 shows estimated air and water phase diffusivities for the four chemicals. The
values were estimated by each of the methods described above (FSG, FSG-LaBas, WL for air, and
HL for water). Because the estimation methods depend upon the volume occupied by each
molecule either in air or water, the values vary little and tend to be on the order of 10"2 cm2/s for
the air phase and 10"6 cm2/s for water. The values given in Table 21, are generally of this
magnitude and also show little variability with temperature.
Chemical
benzene
MTBE
PCE
TCE
Number of atoms
hydrogen
6
12
0
1
carbon
6
5
2
2
oxygen
0
1
0
0
chlorine
0
0
4
3
Aromatic
Rings
1
0
0
0
Oxygen
conditions
n/a
higher
ethers
n/a
n/a
Nitrogen
conditions
n/a
n/a
n/a
n/a
Boiling
point
(°Q
80
55.5
121
86.7
Table 20 Diffusion coefficient calculation input parameters for benzene, MTBE,
perchloroethene, and trichlorethene.
19http://www.epa.gov/athens/learn2model/part-two/onsite/estdiffusion.htm
58
-------
Chemical
Diffusion Coefficients in Air
(cm2/s)
FSG
FSG-
LaBas
WL
Diffusion
Coefficient in
Water
(cm2/s)
HL
Benzene
5°C
15 °C
25 °C
0.0792
0.0842
0.0894
0.0817
0.0869
0.0923
0.0859
0.0919
0.0980
5.6 e-6
7.8 e-6
1.0 e-5
MTBE
5°C
15 °C
25 °C
0.0714
0.0759
0.0806
0.0710
0.0755
0.0802
0.0751
0.0803
0.0856
4.7 e-6
6.5 e-6
8.6 e-6
Perchloroethene (PCE)
5°C
15 °C
25 °C
0.0673
0.0716
0.0760
0.0671
0.0713
0.0757
0.0711
0.0760
0.0811
4.7 e-6
6.6 e-6
8.7 e-7
Trichloroethene (TCE)
5°C
15 °C
25 °C
0.0738
0.0785
0.0833
0.0737
0.0784
0.0832
0.0779
0.0832
0.0888
5.3 e-6
7.3 e-6
9.7 e-6
Table 21 Estimated air and water diffusivities for benzene, MTBE,
perchloroethene and trichloroethene at temperatures of 5 °C, 15 °C, and 25 °C.
59
-------
5. Conclusions
A series of on-line tools has been created for assisting in assessing contaminated sites.
These tools include simple methods for estimating model inputs. The direct model input tools
include the retardation factor, hydraulic gradient, seepage velocity, dispersion coefficients, rate
constant/half life conversions. Other tools provide methods and data for estimating the effective
solubility of contaminants from fuels, temperature-dependent Henry's constants, and air and water
phase diffusivities. These parameters are needed for certain models and the effective solubility
can be sued to determine the maximum possible concentration resulting from fuel contamination.
Because few sites are evaluated based on field data alone, models or other calculations are
used in site assessment. As stated previously, models are chosen for this task because
they have an evident ability to predict future concentrations,
• they have a scientific basis,
they have the ability to include the effects of many different factors, and
• they have become accepted as predictive tools.
A number of factors, however, influence and limit the ability of models to predict future
contamination. This report focused on uncertainty in parameter values. For the prediction of
plume diving, recharge estimates are critical, but may be difficult to obtain. Here the model is
used as a tool for the best placement of well screens. Subsequent data collection would confirm
the prediction, but, more importantly, would be of higher quality because of the inclusion of this
theory in well placement. Parameter uncertainty in contaminant transport showed that a wide
variety of breakthrough curves could be generated from a set of inputs. Where the inputs are not
completely know, this behavior of the model is a truer representation of the results than a model
run using averaged parameters. If, however, through uncertainty analysis it can be shown that
generic best and worst cases exist, then bounding results could be generated. For the one-
dimensional contaminant transport model studied here, generic worst/best cases can be established
for the first arrival time, maximum concentration and duration. The parameter sets that generate
these do not generate each of the worst/best cases. When risk is considered to be based on
transient concentrations, a generic worst case doesn't exist at all. For these two reasons the
existence of universal extreme parameter sets it limited, but may be useful in some cases.
60
-------
References
Aldrich Chemical Company, 2003 Aldrich Handbook of Fine Chemicals and Laboratory
Equipment.
ASTM, 1995, Standard Guide for Risk-Based Corrective Action Applied at Petroleum Release
Sites, American Society for Testing and Materials, ASTM Designation E 1739095.
Bear, J., 1972, Dynamics o f Fluids in Porous Media, American Elsevier, New York.
Buscheck, T.E. and C.M. Alacantar, 1995, Regression Techniques and Analytical Solutions to
Demonstrate Intrinsic Bioremediation, Intrinsic Bioremediation, Hinchee, R.E., J.T. Wilson, D.C.
Downey, eds., Battelle Press, 3(1).
Cline, P.V., JJ. Delfino, P.S.C. Rao, 1991, Partitioning of aromatic constituents into water from
gasoline and other complex solvent mixtures, Environmental Science and Technology, 23, 914-
920.
Eggleston, J.R. and S. A. Rojstaczer, 2000, Can We Predict Subsurface Mass Transport?,
Environmental Science and Technology, 34, 4010-4017.
Fischer, A., M. Muller, J. Klasmeier, 2004, Determination of Henry's law constant for methyl tert-
butyl ether (MTBE) at groundwater temperatures, Chemosphere, 54, 689-694.
Gelhar, L.W., K.R. Rehfeldt, and C.A. Welty, 1992, A critical review of data on field-scale
dispersion in aquifers, Water Resources Research, 28(7), 1955-1974.
Haitjema, H.M., 1995, Analytic Element Modeling of Groundwater Flow, Academic Press,
304pp.
Leij, FJ. and S.A. Bradford, 1994, 3DADE:A computer program for evaluating three-dimensional
equilibrium solute transport in porous media, Research Report No. 134, U.S. Salinity Laboratory,
Riverside, CA.
Lide, D.R., 2000, CRC Handbook of Chemistry and Physics, 81st ed., CRC Press, page 8-95.
Miller, C.T. and W.G. Gray, 2002, Hydrological Research: Just Getting Started, Ground Water,
40(3), 224-231.
Montegomery, J.H., 1996, Groundwater Chemicals Desk Reference, 2nd ed., CRC Press, Lewis
Publishers, Boca Raton, Florida, page 70-71.
Oreskes, N., K. Shrader-Frechette, K. Belitz, 1994, Verification, Validation, and Confirmation of
Numerical Models in the Earth Sciences, Science, 263, 641-646.
61
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Peters, U., F. Nierlich, M. Sakuth, and M. Laugier, 2002, Methyl Tert-Butyl Ether, Physical and
Chemical Properties, Ullmanns Encyclopedia of Industrial Chemistry, Release 2003, 6th ed.,
VCH Verlag GmbH & Co. KGaA, Wiley, DOT, 10.1002/14356007.al6_543.
Shaw, D.G., 1989, Solubility Data Series, Hydrocarbon with Water and Seawater, Part I:
Hydrocarbons C5 to C7, Volume 37, International Union of Pure and Applied Chemistry,
Pergamon Press, Oxford.
Strack, O.D.L., 1984, Three-dimensional streamlines in Dupuit-Forchheimer models, Water
Resources Research, 20, 812-822.
Staudinger and Roberts, 1996, A Critical Review of Henry's Law Constants for Environmental
Applications, in Critical Reviews in Environmental Science and Technology, 26(3):205-297
Stephen, H. and T. Stephen, 1963, Solubilities of Inorganic and Organic Compounds, Vol. 1.,
Macmillan, New York.
Tucker, W.A., and L.H. Nelken, 1990, Chapter 17: Diffusion Coefficients in Air and Water,
Handbook of Chemical Property Estimation Methods. Warren J. Lyman, William F. Reehl and
David H. Rosenblatt, eds., first edition, American Chemical Society.
United States Environmental Protection Agency, 1989, Risk Assessment Guidance for Superfund
Volume 1, Human Health Evaluation Manual (Part A), Interim Final, Office of Emergency and
Remedial Response, Washington, DC, EPA/540/1-89-002.
United States Environmental Protection Agency, 2001, Fact Sheet: Correcting the Henry's Law
Coefficient for Temperature, Office of Solid Waste and Emergency Response,
http://www.epa.gov/athens/learn2model/part-two/onsite/doc/factsheet.pdf
van Genuchten, M. T. and W. J. Alves, 1982, Analytical Solutions of the One-Dimensional
Convective-Dispersive Solute Transport Equation, United States Department of Agriculture,
Agricultural Research Service, Technical Bulletin Number 1661.
Washington, J., 1996, Gas partitioning of dissolved volatile organic compounds in the vadose
zone: Principles, temperature effects and literature review, Ground Water, 34(4), 709-718.
Weaver, J.W, J.T. Haas, and J.T. Wilson, 1996, Analysis of the gasoline spill at East Patchogue,
New York, Procedings of Non-Aqueous Phase Liquids (NAPLs) in Subsurface Environment:
Assessment and Remediation, ed. L. Reddi, American Society of Civil Engineers, Washington,
D.C., November 12-14, pp. 707-718.
Weaver, J.W, 1996, Application of the Hydrocarbon Spill Screening Model to Field Sites,
Procedings of Non-Aqueous Phase Liquids (NAPLs) in Subsurface Environment: Assessment and
62
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Remediation, ed. L. Reddi, American Society of Civil Engineers, Washington, D.C., November
12-14, pp. 788-799.
Xu, M., and Y. Eckstein, 1995, Use of weighted least-squares method in evaluation of the
relationship between dispersivity and scale, Groundwater, 33(6), 905-908.
63
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Appendix 1 Calculator Reference
The calculators described in the text are available on the EPA web site at
http://www.epa.gov/athens/onsite. The specific web links for each calculator are given in the
following tables: Table 22- models, Table - formulas/model input paramters, and Table - unit
conversions.
Method
Average
Borehole
Concentration
Vertical
Gradients
Plume Diving
Uncertainty in
Subsurface
Transport
Calculations
Name
Average BoreHole
Concentration
Vertical Gradient
Plume Diving
ConcentrationUncertainty
URL
http : //www. epa. go v/athens/learn2model/part-two/onsite/abc . htm
http://www.epa.gov/athens/learn2model/part-two/onsite/vgradient.htm
http : //www. epa. go v/athens/learn2model/part-two/onsite/di ving . htm
http://www.epa.gov/athens/leam2model/part-two/onsite/uncertainty.htm
Table 22 Web (URLs) for models and associated calculations described in the text.
Method
Retardation
Factor
Ground Water
Velocity
Dispersion
Coefficient
Half-Lives to
Rate Constants
Effective
Solubility
Henry's
Constant
Diffusivity
Name
Retardation Factor
Seepage Velocity
Hydraulic Gradient
Dispersion Coefficient
Half Lives and Rate
Constants
Effective Solubility
Henry's Constant
Diffusivity
URL
http://www.epa.gov/athens/learn2model/part-two/onsite/retard.htm
http://www.epa.gov/athens/learn2model/part-two/onsite/seepage.htm
http://www.epa.gov/athens/learn2model/part-two/onsite/gradient4plus.htm
http : //www. epa. go v/athens/leam2model/part-two/onsite/longdisp . thm
http://www.epa.gov/athens/learn2model/part-two/onsite/halflife.htm
http://www.epa.gov/athens/learn2model/part-two/onsite/es.htm
http://www.epa.gov/athens/leam2model/part-two/onsite/es-temperature.htm
http://www.epa.gov/athens/leam2model/part-two/onsite/esthenry.htm
http://www.epa.gov/athens/learn2model/part-two/onsite/estdiffusion.htm
Table 23 Web (URLs) for formulas/model inputs described in the text.
64
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Appendix 2 Acronyms and Abbreviations
BTEX Benzene, toluene, ethylbenzene and xylenes
CoC Concentration-of-concern
FSG Fuller, Schettler and Giddings Method for Air Diffusion
IUPAC International Union of Pure and Applied Chemistry
FIL Hayduk and Laudie Method for Water Diffusion
LUST Leaking underground storage tank
MCL Maximum contaminant level
MTBE Methyl tert-butyl ether
PCE Perchloroethene
TCE Trichloroethene
TOC Top of casing
URL Uniform Resource Locator
WL Wilke and Lee Method for Air Diffusion
65
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Appendix 3 Plume Diving Calculator Equations
Phreatic Surface
The first solution is used to determine the location of the phreatic surface Bear (1972, page
379) and is given by
h2 = - — + Cx + B (19)
A.
where h is the elevation of the phreatic surface above the datum (bottom of the aquifer) [L], N is
the recharge rate [L3/L2/T], K is the hydraulic conductivity of the aquifer [L/T], x is the distance
[L]. The constants C and B in equation 19 are determined from
,2 ,2 N
h ~ h ~ -
(20)
and
D , 2 N 2 „
B = h2 + -x2 - Cx2 (21)
where x1 and x2 [L] are up and down gradient locations in the aquifer, and hj and h2 [L] are the
corresponding phreatic surface elevations or heads. Bear's solution does not require that the water
table elevation decrease monotonically down gradient. Depending on the values input for
recharge, hydraulic conductivity and heads, there could be a ground water divide between the two
specified end points.
Aquifers Composed of Multiple Segments
If an aquifer can be conceptualized as being composed of a set of segments, then different
properties can be applied to each while still using the analytic solution given for the water table
Bear (1972). In concept, this is the same idea that underlies the analytic element method Haitjema
(1995). Of particular interest are situations where the recharge varies over the site. By supplying
differing recharge rates to segments of the model, the effects of localized recharge variation can
be incorporated intoan analytic solution. The coefficients, C and B, which are determined for
each segment, are chosen to satisfy continuity of head and flux as the segment boundaries are
crossed. Since there can be no jump in head across segment boundaries,
66
-------
(22)
where xsb and hsb are the position and the head of the segment boundary, respectively. The flux
condition is given by
, „ dh*. , „ ^
qsb = ~ ( 'J^' = "( z+1"^)/+1 (23)
where qsb is the flux across the segment boundary. A consequence of this formulation is that there
is a jump in gradient across the segment boundaries if the hydraulic conductivities differ. For
each pair of segments there are two conditions that are applied where the segments join, and two
conditions applied at exterior edges. Since the equations are linear, they can be solved using
gaussian elimination.
The recharge model is based on the assumption that the aquifer can be conceptualized as a
one-dimensional flow system. It is easy to generate parameter sets that result in mounding in the
middle of the domain— reducing the hydraulic conductivity and increasing the recharge rate for the
center section. The result is that there is flow toward both ends of the domain. This occurs even
though one head is higher than the other. It is the volume of water entering the aquifer from all
sources that determines the shape of the water table. In this case there can be enough water
entering through recharge swamp out the gradient that would be established by the constant head
boundaries.
By changing the hydraulic conductivity to 0.1 ft/d and increasing the recharge rate to 20
in/yr, a mound is generated in the example problem solution. Flow is toward the left hand
boundary (50 ft head) when the source location is chosen to be 275 ft (Figure 36).
67
-------
50.0ft
40.0ft
well
Figure 36 Water table mounding due to unequal recharge and hydraulic conductivity.
If the mounding is caused by a localized source, for example, a gravel pit, then the flow system
might really be two- or three-dimensional. In this case flow might go around the mound. The
one-dimensional mounding case is more realistic for large scale recharge zones as would occur
because of topography.
68
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Appendix 4 One-Dimensional Transport in a Homogeneous Aquifer
The first arrival time, maximum concentration and durations are calculated from the
transport equation. Transport in a one-dimensional homogeneous aquifer is governed by
D dc dc n n ~ ,
R — = - vv — + Z) - + Z) - + Z) - - Ac (24)
dt xdx xdx2 ydy2 zdz2 ( }
where R is the retardation factor [dimensionless], c is the concentration [M/L3]; t represents time
[T]; x, y, and z are the three cartesian coordinate directions [L]; vx, is the x-direction seepage
velocity [L/T], Dx, Dy and Dx are the three components of dispersion [L2/T], and A, is a first order
loss coefficient [T"1]. This form of the transport equation is based on the assumption that the
dispersion constants are independent of time and space that ground water flow is one-dimensional,
steady and uniform, that biodegradation is adequately represented by a first order process. With
boundary conditions specified as
c ; tt
, p (25)
dc f , n
— (°°) = 0
ax
a solution can be obtained for a one-dimensional case where the transverse and vertical
components of dispersion are assumed negligible (van Genuchten and Alves, 1982) or a similar
three dimensional case (Leij and Bradford, 1994). The one-dimensional case is useful because of
rapid computation of its results and is used in the following.
The solution for the one-dimensional case is
c(x,t) = co(B(x,t) - B(x,t- 0) (26)
where B(x,t) is defined by
nf ^\ / , f
B(x, 0 = - exp (— — - ) erfc
(27)
1 ,v + u >. ,. Rx + ut
- exp (— — - ) erfc
2
and u is defined as
69
-------
u
\
1 +
(28)
70
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Appendix 5 Estimation of Temperature-Dependent Henry's Law Coefficients
Two methods are used to calculate the temperature-dependence of Henry's law
coefficients. First is a method used to adjust the values using the Clausius-Clapeyron
relationship:
H..
TS
HRexp[
R
(29)
where HTS is the dimensional (atm-m3/mol) Henry's Law coefficient at the Kelvin temperature, Ts,
AHV TS is the enthalpy of vaporization at Ts in units of cal/mol, TR is the reference temperature for
Henry's Law (HR) in K, and Rc is the gas constant and is equal to 1.9872 cal/mol-K. The enthalpy
of vaporization, AHV TS is estimated from
A H
v,TS
- TJT
BC
(30)
where AHV b is the enthalpy of vaporization at the normal boiling point (cal/mol), TB is the normal
boiling point in K, and Tc is the critical temperature in K. The exponent n is selected from Table
24
Ratio TB/TC
<0.57
0.57 to 0.71
>0.71
Exponent n
0.30
0.74 (TB/TC)
-0.116
0.41
Table 24 Exponent "n" used in calculation
of enthalpy of vaporization.
All of the input values required for these equations were generated by the US EPA Office of Solid
Waste and Emergency Response (OSWER) and are available in a fact sheet (US EPA, 2001). The
fact sheet contains more information on these methods. Notably the normal boiling point
enthalpies of vaporization (AHV b) were estimated from the Antoine equation for vapor pressure.
The coefficients of the Antoine equation were themselves estimated from methods given in the
factsheet. These quantities are not recalculated in the Henry's law calculator because they were
tabulated in the attachment to the OSWER factsheet.
71
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The second method was developed by Washington (1996). For many compounds included
in this calculator, variation of Henry's Law Constant with temperature is quantified by the van't
Hoff Equation:
- Atf® AS.9
RT R
(31)
where In Kh is the natural logarithm of the Henry's constant, AH/ is the standard state enthalpy,
R is the universal gas constant, T is the temperature in Kelvin, and ASre standard state entropy.
This method is based upon the assumption that the head capacity is the same for the reactants and
products. Therefore the temperature range is not extrapolated beyond values used in generating
the coefficients used in making the estimates.
72
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Appendix 6 Diffusion Coefficients
The Fuller, Schettler and Giddings (FSG) method is based on the regression formula:
0.001 T1-75
<32>
where DBA is the diffusion coefficient of compound B in compound A in cm2/s, T is the
temperature in K, P is the pressure in atm, Mr a function of the molecular weights MA and MB of
compounds A and B, VA and VB are the molar volumes of air (A) and the gas (B) in question. Mr
is equal to (MA + MB)/MA MB. VB can be estimated from volume increments associated with each
element in the compound. These increments give the volume (cm3) per mole of atom present.
The values are given in the reference and have been programmed into the calculator.
The FSG-LaBas method uses the same formula as the FSG method but substitutes the
LaBas volume estimates for molar volume. The FSG-LaBas method allows for estimating the
diffusivities of more compounds of interest.
The Wilke and Lee (WL) method uses LaBas molar volumes and is based upon a series of
calculations of a "collision integral", Q, that represents collision between atoms. The estimate of
DBA is given by
(33)
where B' is a function of the molecular weights of A, s2BA is the average molal volume at the
boiling point of A and B, and Q is called the collision integral. The details of calculation of these
quantities are given (Tucker and Nelken, 1990).
Diffusion Coefficients in Water
The Hayduk and Laudie (FIL) method for estimating the diffusivity of an organic
compound in water in cm2/s is given by
n 13.26 jc 10" s
** 1.14 0.589 (34)
where DBW is the diffusion coefficient of compound B in compound A (water) in cm2/s, nw is the
viscosity of water in cp (corrected for temperature) and VB are the LaBas molar volume
increments.
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