&EPA Ground Water Issue
United States
Environmental Protection
Agency
Calculation and Use of First-Order Rate
Constants for Monitored Natural Attenuation
Studies
Charles J. Newell1, Hanadi S. Rifai2, John T.Wilson3, John A. Connor1,
Julia A. Aziz1, and Monica P. Suarez2
Introduction
This issue paper explains when and how to apply first-order
attenuation rate constant calculations in monitored natural
attenuation (MNA) studies. First-order attenuation rate constant
calculations can be an important tool for evaluating natural
attenuation processes at ground-water contamination sites.
Specific applications identified in U.S. EPA guidelines (U.S. EPA,
1999) include use in characterization of plume trends (shrinking,
expanding, or showing relatively little change), as well as
estimation of the time required for achieving remediation goals.
However, the use of the attenuation rate data for these purposes
is complicated as different types of first-order rate constants
represent very different attenuation processes:
Concentration vs. time rate constants (/fpoint) are used for
estimating how quickly remediation goals will be met at a site.
Concentration vs. distance bulk attenuation rate constants
( k) are used for estimating if a plume is expanding, showing
relatively little change, or shrinking due to the combined
effects of dispersion, biodegradation, and other attenuation
processes.
Biodegradation rate constants ( X ) are used in solute
transport models to characterize the effect of biodegradation
on contaminant migration.
Correct use of attenuation rate constants requires an
understanding of the different attenuation processes that different
first-order rate constants represent.
For further information contact John T, Wilson (580) 436-8534 at
the Subsurface Protection and Remediation Division of the National
Risk Management Research Laboratory, Office of Research and
Development, U.S. Environmental Protection Agency, Ada,
Oklahoma.
Why Are Attenuation Rate Constants Important?
Monitored natural attenuation (MNA) refers to the reliance on
natural attenuation processes to achieve site-specific remediation
objectives within a reasonable time frame. Natural attenuation
processes include a variety of physical, chemical, and/or biological
processes that act without human intervention to reduce the mass
1Groundwater Services, Inc., Houston, Texas
2University of Houston, Texas
3U.S. Environmental Protection Agency, Office of Research and
Development, National Risk Management Research Laboratory,
Subsurface Protection and Remediation Division, Ada, Oklahoma
or concentration of contaminants in soil and ground water. These
in-situ processes include biodegradation, dispersion, dilution,
sorption, volatilization; radioactive decay; and chemical or
biological stabilization, transformation, or destruction of
contaminants (U.S. EPA, 1999).
The overall impact of natural attenuation processes at a given
site can be assessed by evaluating the rate at which contaminant
concentrations are decreasing either spatially or temporally.
Recent guidelines issued by the U.S. EPA (U.S. EPA, 1999) and
the American Society for Testing and Materials (ASTM, 1998) have
endorsed the use of site-specific attenuation rate constants for
evaluating natural attenuation processes in ground water. The
U.S. EPA directive on the use of Monitored Natural Attenuation
(MNA) at Superfund, RCRA, and UST sites (U.S. EPA, 1999)
includes several references to the application of attenuation rates:
Once site characterization data have been collected and
a conceptual model developed, the next step is to evaluate
the potential efficacy of MNA as a remedial alternative.
This involves collection of site-specific data sufficient to
estimate with an acceptable level of confidence both the
rate of attenuation processes and the anticipated time
required to achieve remediation objectives.
At a minimum, the monitoring program should be sufficient
to enable a determination of the rate(s) of attenuation and
how that rate is changing with time.
Site characterization (and monitoring) data are typically
used for estimating attenuation rates.
The ASTM Standard Guide for Remediation of Groundwater by
Natural Attenuation at Petroleum Release Sites (ASTM, 1998)
also identifies site-specific attenuation rates as a secondary line
of evidence of the occurrence and rate of natural attenuation. In
addition, technical guidelines issued by various state
environmental regulatory agencies recommend estimation of rate
constants to evaluate contaminant plume trends and duration (New
Jersey DEP, 1998; Wisconsin DNR, 1999). For example, the
New Jersey Department of Environmental Protection (DEP) now
requires such calculations for establishing "Classification
Exception Areas (CEAs)" at sites where ground-water quality
standards are or will be exceeded for an extended time period.
The technical literature contains numerous guidelines regarding
methods for derivation of site-specific attenuation rate constants
based upon observed plume concentration trends (e.g., ASTM,
1998; U.S. EPA, 1998a; 1998b; Wiedemeier et al. 1995; 1999;
Wilson and Kolhatkar, 2002). Other resources, such as the
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BIOSCREEN and BIOCHLOR natural attenuation models (Newell
et al., 1996; Aziz et al., 2000), include use of first-order rate
constants for simulating the attenuation of dissolved contaminants
once they leave the source and the attenuation of the source
itself. However, many of these references do not clearly distinguish
between the different types of rate constants and their appropriate
application in evaluation of natural attenuation processes. The
objective of this paper is to address this gap by briefly describing
the derivation, significance, and appropriate use of three key types
of attenuation rate constants commonly employed in natural
attenuation studies.
Key Point:
Rate calculations can help those performing MNA studies evaluate
the contribution of attenuation processes and the anticipated time
required to achieve remediation objectives. There are different
types of rate calculations, however, and it is important to use the
right kind of rate constant for the right application.
Types of First-Order Attenuation Rate Constants
In general, there are three different types of first-order attenuation
rate constants that are in common use:
Concentration vs. Time Attenuation Rate Constant, where
a rate constant, in units of inverse time (e.g., per day), is
derived as the slope of the natural log concentration vs. time
curve measured at a selected monitoring location (Figure 1).
contaminant transport vs. transport of a tracer, or more
commonly, calibration of solute transport model to field data
(Figure 3).
Figure 1. Determining concentration vs. time rate constant
CfpoJ-
Concentration vs. Distance Attenuation Rate Constant,
where a rate constant, in units of inverse time (e.g., per day),
is derived by plotting the natural log of the concentration vs.
distance and (if determined to match a first-order pattern)
calculating the rate as the product of the slope of the
transformed data plot and the ground-water seepage velocity
(Figure 2).
,
II
. c
11
Distance from Source
Figure 2. Determining concentration vs. distance rate
constant (fc).
Biodegradation Rate Constant. The "biodegradation rate
constant" (X ) in units of inverse time (e.g., per day) can be
derived by a variety of methods, such as comparison of
Contam
racer
Figure 3. Determining biodegradation rate constant ( X).
Distinctions Between Rate Constants
To interpret the past behavior of plumes, and to forecast their
future behavior, it is necessary to describe the behavior of the
plume in both space and time. It is necessary to collect long-term
monitoring data from wells that are distributed throughout the
plume. Concentration vs. Time Rate Constants describe the
behavior of the plume at one point in space; while Concentration
vs. Distance Rate Constants describe the behavior of the entire
plume at one point in time. The Biodegradation Rate Constant is
usually applied over both time and space, but only applies to one
attenuation mechanism. Standard practice for the environmental
industry finds applications for each of these rate constants. Under
appropriate conditions, each of the three constants can be
employed to assist in site-specific evaluation and quantification
of natural attenuation processes. Each of these terms is identified
as an "attenuation rate." Because they differ in their significance
and appropriate application, it is important to understand the
potential for misapplication of each type of rate as summarized
below:
Concentration vs. Time Rate Constants: A rate constant
derived from a concentration vs. time (C vs. T) plot at a single
monitoring location provides information regarding the
potential plume lifetime at that location, but cannot be used to
evaluate the distribution of contaminant mass within the
ground-water system. The C vs. T rate constant at a location
within the source zone represents the persistence in source
strength over time and can be used to estimate the time
requiredto reach a remediation goal atthat particular location.
To adequately assess an entire plume, monitoring wells must
be available that adequately delineate the entire plume, and
an adequate record of monitoring data must be available to
calculate a C vs. T plot for each well. At most sites, the rate
of attenuation in the source area (due to weathering of
residual source materials such as NAPLs) is slower than the
rate of attenuation of materials in ground water, and
concentration profiles in plumes tend to retreat back toward
the source overtime. In this circumstance, the lifecycle of the
plume is controlled by the rate of attenuation of the source,
and can be predicted by the C vs. T plots in the most
contaminated wells. At some sites, the rate of attenuation of
the source is rapid compared to the rate of attenuation in
ground water. This pattern is most common when
contaminants are readily soluble in ground water and when
contaminants are not biodegraded in ground water. In this
case, the rate of attenuation of the source as predicted by a
C vs. T plot will underestimate the lifetime of the plume.
Concentration vs. Distance Rate Constants: Attenuation rate
constants derived from concentration vs. distance (C vs. D)
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plots serve to characterize the distribution of contaminant
mass within space at a given point in time. A single C vs. D
plot provides no information with regard to the variation of
dissolved contaminant mass over timeand, therefore, cannot
be employed to estimate the time required for the dissolved
plume concentrations to be reduced to aspecified remediation
goal. This rate constant incorporates all attenuation
parameters (sorption, dispersion, biodegradation) for
dissolved constituents after they leave the source. Use of the
rate constant derived from a C vs. D plot (i.e., characterization
of contaminant mass over space) for this purpose (i.e., to
characterize contaminant mass over time) will provide
erroneous results. The C vs. D-based rate constant indicates
how quickly dissolved contaminants are attenuated once
they leave the source but provides no information on how
quickly a residual source zone is being attenuated. Note that
most sites with organic contamination will have some type of
continuing residualsourcezone, even afteractive remediation
(Wiedemeier et al., 1999), making the C vs. D rate constant
inappropriate for estimating plume lifetimes for most sites.
Biodegradation Rate Constant: Another type of error occurs
if a C vs. D rate constant is used as the biodegradation rate
term ( X ) in a solute transport model. The attenuation rate
constant derived from the C vs. D plot already reflects the
combined effects of contaminant sorption, dispersion, and
biodegradation. Consequently, use of a C vs. D rate constant
as the biodegradation rate within a model that separately
accounts for sorption and dispersion effects will significantly
overestimate attenuation effects during ground-water flow.
These examples serve to illustrate the need to ensure an
appropriate match between the significance and use of each rate
constant. Further guidelines regarding derivation and use of
attenuation rate constants are provided below.
Key Point:
There are three general types of first-order rate constants that
are commonly used for MNA studies: (1) Concentration vs. Time,
(2) Concentration vs. Distance, and (3) Biodegradation.
Rate Constants vs. Half-Lives
Both first-order rate constants and attenuation half-lives represent
the same process, first-order decay. Some environmental
professionals prefer to use rate constants (in units of per time) to
describe the first-order decay process, while others prefer
half-lives. These two terms are linearly related by:
Rate constant = 0.693 / [ half-life ] and
Half-life = 0.693 / [ rate constant ]
For example, a 2 year half-life is equivalent to a first-order rate
constant of 0.35 per year. This document describes the first-
order decay process in terms of rate constants instead of half-
lives.
Key Point:
Rate constants and half-lives represent the same first-order decay
process, and are inversely related.
Appropriate Use of Attenuation Rate Constants in
Natural Attenuation Studies
Attenuation rate constants may be used for the following three
purposes in natural attenuation studies:
Plume Attenuation: Demonstrate that contaminants are
being attenuated within the ground-water flow system;
PlumeTrends: Determine if the affected ground-water plume
is expanding, showing relatively little change, or shrinking;
and
Plume Duration: Estimate the time requiredto reach ground-
water remediation goals by natural attenuation alone.
Appropriate use of the various attenuation rate constants for
evaluation of plume attenuation, trends, and duration is shown in
Table 1.
As described in the U.S. EPA MNA Directive (U.S. EPA, 1999):
Site characterization (and monitoring) data are typically
used for estimating attenuation rates. These calculated
rates may be expressed with respect to either time or
distance from the source. Time-based estimates are used
to predict the time required for MNA to achieve remediation
objectives and distance-based estimates provide an
evaluation of whether a plume will expand, remain stable,
or shrink.
To clarify the applicability of the various first-order decay rate
constants, appropriate nomenclature is useful to indicate the
significance of each term. For example, point decay rates (defined
Table 1. Summary of First-Order Rate Constants for Natural Attenuation Studies
Use of Rate Constant
'lume Plume
Attenuation Trends? Duration?
Point Attenuation
Rate (Fig. 1)
(kpoint, time per year)
C vs. T Plot
Reduction in contaminant
concentration over time at a
single point
NO*
NO*
YES
Bulk Attenuation Rate
(Fig. 2)
(k; time per year)
C vs. D Plot
Reduction in dissolved
contaminant concentration with
distance from source
YES
NO*
NO
Biodegradation Rate
(Fig. 3)
(X, time per year)
Model Calibration,
Tracer Studies,
Calculations
Biodegradation rate for
dissolved contaminants after
leaving source, exclusive of
advection, dispersion, etc.
YES
NO
NO
* Note: Although assessment of an attenuation rate constant at a single location does not yield plume attenuation information, or plume
trend information, an assessment of general trends of multiple wells over the entire plume is useful to assess overall plume attenuation
and plume trends.
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as /fpoint), derived from single well concentration vs. time plot, may
be used to determine how long a plume will persist (Plume
Duration). While concentration vs. time data at a single point in
the plume are useful for determining trends at that location (i.e.,
are concentrations increasing, showing relatively little change, or
declining), a rate constant calculated from concentration vs. time
data at a single location cannot be used to estimate the trend of
an entire plume.
Bulk attenuation rates (defined as k), derived from concentration
vs. distance plots, can be used to indicate if a plume is expanding,
showing relatively little change, or shrinking (Plume Trends).
Biodegradation rates ( X ), modeling parameters which are
specific to biodegradation effects and exclusive of dispersion, etc.,
can be used in appropriate solute transport models to indicate if
a plume is expanding, showing relatively little change, or shrinking
(Plume Trends).
For each of these first-order decay rate parameters, Table 2
summarizes information on the derivation and appropriate use
as well as providing representative values. In summary, different
types of first-order attenuation rate calculations are available to
help evaluate natural attenuation processes at contaminated
ground-water sites. These different types of rate constants
represent different types of attenuation processes, therefore, the
right type of rate constant should be used for the right purpose.
Examples 1-3 illustrate how the three types of rate constants are
calculated and applied.
Key Point:
In general, all three types of rate constants are useful indicators
that attenuation is occurring. Concentration vs. time rate constants
( k ) can be used to estimate the duration of contamination at a
particular location. Concentration vs. time rate constants for wells
encompassing the entire plume can be used to identify overall
trends and predict the duration of the plume. Concentration vs.
distance rate constants ( k) and biodegradation rate constants
( X) can be used to project the rate of attenuation of contaminants
along the flow path in ground water, and predict the spatial extent
of the plume.
Tables 1 and 2 provide more detail on use, calculations, and
analysis of the three types of rate constants. Examples 1-3
illustrate the use and application of the three types of rate
constants.
Other Types of Rate Constants
Mass-Based Rate Constants. The previous discussion focused
on concentration-based rates. It is also possible to calculate mass
vs. time rate constants and mass vs. distance rate constants. In
practice, these rates would be very similar to the concentration-
based rates.
Mass vs.Time Rate Constant. This constant compares changes
in the total mass of contaminants in the plume over time. A
Thlessen polygon networkcan be used to weight the concentration
data from all the available wells at a site to derive a comprehensive
estimate of the mass of contaminants in the plume at any particular
round of sampling. Mass vs. time decay rates (in units of inverse
time) are estimated by plotting the natural log of total dissolved
mass as a function of time and estimating the slope of the line.
This rate is similar to the concentration vs. time rate and since it
accounts for the entire plume, it is a good indicator of how long a
plume will persist. Many plumes change flow direction over time,
making it difficult to identify a stable centerline. Estimates based
on the entire plume are less subject to errors caused by changes
in flow direction. See Hyman and DuPont, 2001 and DuPont et
al.,1998 for discussion and details of the methods.
Mass Flux vs. Distance Rate Constant. A mass vs. distance
decay rate (in units of inverse time) can be calculated by plotting
the natural log of mass flux through different transects
perpendicular to the flow as a function of distance from the source
and multiplying the slope of the best-fit line by the seepage velocity.
Comparable to the bulk attenuation rate, this type of rate can be
used to indicate if a plume is expanding, showing relatively little
change, or shrinking. See Einarson and Mackay, 2001 for
examples of mass flux calculations. Another method for calculating
mass loss rates is described by the Remediation Technologies
Development Forum (RTDF, 1997).
Mass Flux-Based Biodegradation Rate Constant. Mass fluxes
across plume transects can be further analyzed to determine
whether the observed mass loss spatially and temporally can be
attributed to biodegradation and/or source decay. For this purpose,
the mass flux across the source area is compared to the mass
flux through the next downgradient section. Theoretically, mass
fluxes at the downgradient transect should mimic the trends
observed in the source transect if source decay, sorption, and
dispersion were the only mass reduction attenuation mechanisms.
If there is additional mass loss, it can only be attributed to
biodegradation since the other processes are already accounted
for in the mass flux calculation. Once the actual mass loss
attributable to biodegradation has been determined, it is plotted
as a function of time and a biodegradation rate is estimated using
linear regression or a first-order decay model fit to the data. See
Borden et al. (1997) and Semprini et al. (1995) for examples of
biodegradation rates calculated from mass flux across transects.
Mass-based rate constants are not often used in practice due to
the data needs for mass estimates including a dense well network
as well as localized gradients, conductivity measurements, and
aquifer thickness at monitoring points.
Average-Plume Concentration Rate Constants. Some researchers
and practitioners have calculated rate constants for the change
in average plume concentration. This rate constant reflects
primarily the change in source strength over time.
Effect of Residual NAPL on Point Decay Rate
Constant
When a monitoring well is screened across an interval that
contains residual NAPL, and when the rate of weathering of the
NAPL is slow, the well water may sustain high concentrations of
contaminants over long periods of time.
Effect of NA Processes on Rate Constants
Natural attenuation processes include a variety of physical,
chemical, or biological processes that act without human
intervention to reduce the mass or concentration of contaminants
in soil and ground water. These in-situ processes include
biodegradation, dispersion, dilution, sorption, volatilization,
radioactive decay, and chemical or biological stabilization,
transformation, or destruction of contaminants (U.S. EPA, 1999).
Each of these processes influences contaminant concentrations
in soil and ground water both spatially and temporally at a site.
Contaminant concentrations in ground water are reduced as they
travel downgradient from the source. Subject to source
degradation, contaminant concentrations will also be reduced with
time at any given distance downgradient from the source. These
concepts are illustrated in Appendices II and III. The data in
Appendix II illustrate the change in contaminant concentrations
downgradient from the source at a hypothetical site in response
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to the different attenuation processes. It can be clearly seen from
Appendix II that contaminant concentrations downgradient from
source areas are attenuated due to dispersion, sorption,
biodegradation and source decay.The data in Appendix III illustrate
the change in contaminant concentrations with time at two points
downgradient from the source at the hypothetical site (one point
near the source and the other point at the leading edge of the
plume). As can be seen from Appendix III, contaminant
concentrations near the source will attenuate with time only if
source decay is occurring. While source decay is also important
for the leading edge of the plume, maximum contaminant
concentrations in that zone are significantly attenuated from their
source concentration counterparts due to biodegradation,
sorption, and dispersion.
Uncertainty in Rate Calculations
Rate calculations can be affected by uncertainty from a number
of sources, such as the design of the monitoring network, seasonal
variations, uncertainty in sampling methods and lab analyses,
and the heterogeneity in most ground-water plumes. Appendix I
discusses uncertainty in rate calculations and provides methods
for managing this uncertainty.
ORD has developed software (RaCES) to extract rate constants
from field data. This software is intended to facilitate an evaluation
of the uncertainty associated with the projections made by
computer models of the future behavior of plumes of contamination
in ground water. The software is available from The Ecosystem
Research Division of the National Exposure Research Laboratory
in Athens, Georgia (Budge et al., 2003).
Notice
The U.S. Environmental Protection Agency through its Office of
Research and Development funded and managed the research
described here under Contract No. 68-C-99-256 to Dynamac
Corporation. 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.
Quality Assurance Statement
All research projects making conclusions or recommendations
based on environmental data and funded by the U.S.
Environmental Protection Agency are required to participate in
the Agency Quality Assurance Program. This project did not
involve the collection or use of environmental data and, as such,
did not require a Quality Assurance Project Plan.
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Table 2. Quick Reference Summary of Three Types of Attenuation Rate Constants
USED FOR:
ecay Rate Constant (
Plume Duration Estimate. Used to
estimate time required to meet a
remediation goal at a particular point
within the plume. If wells in the source
zone are used to derive /cpoint, then this
rate can be used to estimate the time
required to meet remediation goals for
the entire site. /cpoint should nof be
used for representing biodegradation of
dissolved constituents in ground-water
models (use X as described in the right
hand column).
Bulk Attenuation Rate Constant (k) Biodegradation Rate Constant (\ )
Plume Trend Evaluation. Can be used
to project how far along a flow path a
plume will expand. This information can
be used to select the sites for monitoring
wells and plan long-term monitoring
strategies. Note that k should not be
used to estimate how long the plume will
persist except in the unusual case where
the source has been completely
removed, as the source will keep
replenishing dissolved contaminants in
the plume.
Plume Trend Evaluation. Can be
used to indicate if a plume is still
expanding, or if the plume has reached
a dynamic steady state. First calculate X,
then enter X into a fate and transport
model and run the model to match
existing data. Then increase the
simulation time in the model and see if
the plume grows larger than the plume
simulated in the previous step. Note
that X should not be used to estimate
how long the plume will persist except in
the unusual case where the source has
been completely removed.
REPRESENTS:
Mostly the change in source strength
over time with contributions from other
attenuation processes such as
dispersion and biodegradation. /cpoint is
not a biodegradation rate as it
represents how quickly the source is
depleting. In the rare case where the
source has been completely removed
(for a discussion of source zones, see
Wiedemeieret al., 1999), /cpointwill
approximate k.
Attenuation of dissolved constituents due
to all attenuation processes (primarily
sorption, dispersion, and biodegradation).
The biodegradation rate of dissolved
constituents once they have left the
source. It does not account for
attenuation due to dispersion or
sorption.
HOW TO
CALCULATE:
Plot natural log of concentration vs.
time for a single monitoring point and
calculate /cpoint = slope of the best-fit
line (ASTM, 1998). This calculation
can be repeated for multiple sampling
points and for average plume
concentration to indicate spatial trends
Plot natural log of cone. vs. distance. If
the data appear to be first-order,
determine the slope of the natural log-
transformed data by:
1. Transforming the data by taking
natural logs and performing a linear
regression on the transformed data, or
2. Plotting the data on a semi-log plot,
taking the natural log of the y intercept
minus the natural log of the x intercept
and dividing by the distance between the
two points.
Multiply this slope by the contaminant
velocity (seepage velocity divided by the
retardation factor R) to get k.
Adjust contaminant concentration by
comparison to existing tracer (e.g.,
chloride, tri-methyl benzenes) and then
use method for bulk attenuation rate
(see Wiedemeier et al., 1999); or
Calibrate a ground-water solute
transport computer model that includes
dispersion and retardation (e.g.,
BIOSCREEN, BIOCHLOR, BIOPLUME
III, MT3D) by adjusting X; or
Use the method of Buscheck and
Alcantar (1995) (plume must be at
steady-state to apply this method). Note
this method is a hybrid between k and X
as the Buscheck and Alcantar method
removes the effects of longitudinal
dispersion, but does not remove the
effects of transverse dispersion from
their X.
Note this calculation does not account
for any changes in attenuation
processes, particularly Dual-Equilibrium
Desorption (availability) which can
reduce the apparent attenuation rate at
lower concentrations (e.g., see Kan et
al., 1998).
Distance from Source
Contam
racer
FincU
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Table 2. Continued...
HOW TO USE:
ate Constant (kp
To estimate plume lifetime:
The time (t) to reach the remediation
goal at the point where Kpoint was
calculated is:
-Ln
t = -
goal
start
point
To estimate if a plume is showing
relatively little change:
Pick a point in the plume but
downgradient of any source zones.
Estimate the time needed to decay these
dissolved contaminants to meet a
remediation goal as these contaminants
move downgradient:
-Ln
t--
^start
Calculate the distance L that the
dissolved constituents will travel as they
are decaying using Vs as the seepage
velocity and R is the retardation factor for
the contaminant:
If the plume currently has not traveled
this distance L then this rate analysis
suggests the plume may expand to that
point. If the plume has extended beyond
point L, then this rate analysis suggests
the plume may shrink in the future. Note
that an alternative (and probably easier
method) is to merely extrapolate the
regression line to determine the distance
where the regression line reaches the
remediation goal.
ving
To estimate if a plume is showing
relatively little change:
Enter X in a solute transport model that
is calibrated to existing plume
conditions. Increase the simulation time
(e.g. by 100 years, or perhaps to the
year 2525), and determine if the model
shows that the plume is expanding,
showing relatively little change, or
shrinking.
TYPICAL
VALUES:
Reid and Reisinger (1999) indicated that
the mean point decay rate constant for
benzene from 49 gas station sites was
0.46 per year (half-life of 1.5 years). For
MTBE they reported point decay rate
constants of 0.44 per year (half-life of 1.6
years). In contrast, Peargin (2002)
calculated rates from wells that were
screened in areas with residual NAPL;
the mean decay rate for MTBE was 0.04
per year (half life of 17 years) the rate for
benzene was 0.14 per year (half life of 5
years).
Newell (personal communication)
calculated the following median point
decay rate constants: 0.33 per year (2.1
year half-life) for 159 benzene plumes at
service station sites in Texas; and 0.15
per year (4.7 year half-life) for 37 TCE
plumes around the U.S.
For many BTEX plumes, k will be similar
to biodegradation rates X (on the order of
0.001 to 0.01 per day; see Figure 4) as
the effects of dispersion and sorption will
be small compared to biodegradation.
For BTEX compounds, 0.1 -1 %/day
(half-lives of 700 to 70 days)(Suarez and
Rifai, 1999). Chlorinated solvent
biodegradation rates may be lower than
BTEX biodegradation rates at some
sites (Figures 4 and 5).
For more information about
biodegradation rates for a variety of
compounds, see Wiedemeier et al.,
1999 and Suarez and Rifai, 1999.
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1rO.07
Q
o
Q.
O
o
UJ
0.01
0.001
0.0001
<
Q
700
X
o
cc
Q.
Q.
7,000
Figure 4. Biodegradation Rate Constants ( X ) and Bulk Attenuation Rate Constants (k) for BTEX compounds from the literature.
Source: Rifai and Newell, 2001.
LEGEND
i
Maximum
75 Percentile
Median
25 Percentile
Minimum
Constituent
Number of Sites
7 days
70 days 3
(-0.2 yrs) =
1
0)
ra
'x
o
700 days g;
(~1.9 yrs) <
7000 days
(~19yrs)
Figure 5. Biodegradation Rate Constants ( X ) for Trichloroethene (TCE), cis-Dichloroethene (cDCE), and Vinyl Chloride (VC)
compounds from BIOCHLOR modeling studies. Source: Aziz et al., 2000.
-------
EXAMPLE 1. Use of Concentration vs. Time Rate Constants (k oint)
INTRODUCTION: A leaking underground storage tank site in Elbert, Anystate, has a maximum source concentration of
1.800 mg/L of benzene at well MW-3. A remediation goal of 0.005 mg/L of benzene has been established. How long will it take
for this site to reach the remediation goal using MNA with no active remediation? (Data source: Mace etal. 1997)
DATA: 10
The following are data from well MW-3
for the period 1986 to 1991.
DATE
8/19/86
7/17/87
9/29/87
12/19/87
6/25/88
9/30/88
12/21/88
4/25/89
10/23/89
7/4/91
11/20/91
Years
Since
1/1/86
0.63
1.54
1.74
1.96
2.48
2.75
2.97
3.31
3.81
5.50
5.88
MW-3
Benzene
(mg/L)
1.800
0.440
0.370
0.320
0.270
0.260
0.260
0.220
0.110
0.030
0.018
1 -
0.1 -
o
O
0.01 -
0.001
KEY POINT: The kpoint degradation rate constant is +0.77 per year.
QUESTION: Why is the sign positive?
ANSWER: The rate constant is defined as a rate of degradation. The
slope of the line is the rate of change. If the slope is negative, then
concentrations are attenuating, and the rate of degradation is positive.
y =
1
.9568e-°'7676x
Benzene MCL (0.005 mg/L)
3456
Time (years since 1/1/86)
CALCULATION: Construct a plot of concentration vs. time. Although the plot can be developed in many ways, the clearest
way is to convert the time data to years using an arbitrary starting point (for this example we chose 1/1/86). By transforming the
concentrations to natural log concentration, and using a spreadsheet or calculator to get the slope (-0.77) and intercept (0.67),
the following equation of the line was generated:
Ln ( Cone. Benzene) = exp (°67-°77x> which resulted in the following rate equation:
Benzene concentration (mg/L) = 1.96 mg/L* exp <-°77yrssince1/1/86) where kpojnt = +0.77 per year.
Rearranging the equation:
Time (years since 1/1/86) = - Ln [ Cone. Benzene (mg/L) /1.96 ] / 0.77
For the case where the remediation goal is 0.005 mg/L benzene,
Time (years since 1/1/86) = - Ln [ 0.005 /1.96 ] / 0.77 = 7.7 years = late 1993
A statistical analysis of the uncertainty involved in the calculation can be performed by determining the "one tailed" 90%
confidence interval using the methods outlined in Appendix I. The "one tailed" 90% confidence limit on the time to remediation
is a time that is no longer than 8.6 years from 1/1/86, or late 1994.
Plume Attenuation?
The concentration vs. time rate
constant is positive, indicating that
attenuation at this location (the source
zone in this example) is occurring. The
attenuation is probably due to
weathering of the source caused by
dissolution of benzene from a residual
NAPL into flowing ground water.
Raoult's Law predicts that weathering
from dissolution will be a first-order
process.
Plume Trends?
The concentration vs. time rate
constant is positive, indicating that
concentrations in this portion of the
plume are going down and that at
least a portion of the plume may be
shrinking. However, from the
information obtained at a single
location, no conclusion can be drawn
regarding the overall plume trend.
Plume Duration?
The concentration vs. time rate
constant was used to show that if
current trends hold then the plume will
reach the clean-up goal in 1994. Note
this assessment does not consider any
other processes which could reduce
the observed attenuation rate (i.e.,
changes in water levels, availability
effects at low concentration as
described by Kan et al., 1998, etc.).
Key Point:
A concentration vs. time rate constant is one of the best ways to estimate how long MNA (or any type of remediation system)
might take to reach a clean-up goal. A second method is to perform a mass-based approach (i.e., see DuPont et al., 1998;
Hyman and DuPont, 2001; Newell et al., 1996 or Chapter 2 of Wiedemeier et al., 1999).
-------
EXAMPLE 2. Use of Concentration vs. Distance Rate Constants (k)
INTRODUCTION: This constant is estimated between wells along the inferred centerline of the plume. An MTBE plume at a
former fuel farm located at a U.S. Coast Guard Base has a maximum source zone concentration of 1.740 mg/L of MTBE. The
average calculated seepage velocity at the site was calculated to be 82 meters per year and the retardation factor, R, is
assumed to be equal to one. For the purpose of this example, a clean-up goal of 0.030 mg/L was assumed. Most importantly,
the site is strongly anaerobic, indicating that relatively high rates of MTBE biodegradation are possible. Is the MTBE plume
attenuating? How far should it extend? 10
(source: Wilson et al., 2000).
DATA: 3
The following is data from wells |"
along the plume centerline: ^
Well
CPT-1
CPT-3
CPT-5
ESM-14
ESM-3
ESM-9
ESM-10
GP-1
Distance from
Source(m)
0
40
70
104
134
180
195
250
MTBE
Cone.(mg/L)
1.74
0.823
0.672
0.383
0.319
0.001
0.0097
0.001
Key Point: The degradation rate
constant k is + 0.0033 per year.
0.01
0.001
100
150
200
250
300
Distance from Source (meters)
CALCULATION: First, plot the natural log of concentration vs. distance at a point in time and calculate the slope of the best-fit
line using linear regression analysis, as shown above. The slope of the C vs. D plot is -0.033 per meter of travel.
Next, calculate the bulk attenuation rate constant, k, by multiplying the negative of the slope of the regression by the contaminant
velocity. The contaminant velocity equals the seepage velocity divided by the retardation factor. In this case the retardation
factor is 1, and the contaminant velocity is 82 meters per year. The bulk attenuation rate is (+0.033 per meter) * (82 meter per
year) = 2.7 per yr. This corresponds to a dissolved-phase half-life of 0.26 yrs (0.26 yrs = 0.69 / 2.7 per yr) after the MTBE
leaves the source zone.
To estimate the travel time required for the concentration of MTBE to attenuate to the cleanup goal, use the equation in Table 2.
The travel time to reach the remediation goal at the down gradient margin of the plume is 1.5 years (1.5 yr = - Ln [0.030 mg/L/
1.74 mg/L] / 2.7 per y). Based on the calculated attenuation rate, an MTBE source concentration of 1.74 mg/L, and a cleanup
goal of 0.030 mg/L, the MTBE plume should extend 123 meters from the source (123 meters = 82 meters per yr * 1.5 yr travel
time).
A sensitivity analysis can be performed on the rate estimates. See Appendix I for a discussion of confidence intervals. The
one-tailed 95% confidence interval on the slope is -0.021 per foot. At a seepage velocity of 82 meters per year, this is
equivalent to a concentration vs. distance rate constant (k) of 1.7 per year. The plume would require 2.4 years of travel in the
aquifer to attenuate to the cleanup goal. At 95% confidence, the plume boundary would be no more than 200 meters from the
source. The estimate of seepage velocity is also subject to uncertainty. A reasonable upper boundary on the seepage velocity
at this site is 150 meters per year (Wilson et al., 2000). At the upper bound on seepage velocity, and at the 95% confidence
interval on the slope, the MTBE plume would extend no more than 360 meters.
Plume Attenuation?
The calculated concentration vs. distance
rate constant is positive, indicating that
attenuation of dissolved MTBE is
occurring after the MTBE leaves the
source zone. The rate constant of 2.7 per
year indicates that dissolved MTBE
concentrations will be reduced by 50%
every 0.25 yrs after the MTBE leaves the
source zone. It does not indicate the
entire plume will be reduced in
concentration by 50% in 0.25 yrs.
Plume Trends?
In theory, the concentration vs.
distance rate constant can provide
supporting evidence that the plume
may be showing relatively little change
or shrinking in the future. However, an
analysis of concentration vs. time data
for all locations within an adequately
delineated plume is a much more direct
and robust method for estimating
plume trends.
Plume Duration?
A concentration vs. distance rate
constant is not useful for
estimating plume duration (i.e.,
the time to reach a clean-up goal).
A mass-based analysis by Wilson
et al., 2000 indicated that
60 years might be required to
reach the clean-up goal.
Key Point:
Concentration vs. distance rate constants cannot be used for estimating remediation time frames, and are only marginally
useful for estimating plume trends. This type of rate constant is most useful to predict the boundaries of a plume. It can be used
to plan the location of monitoring wells or sentinel wells. This rate constant is also used with other information to calculate the
rate of biodegradation.
10
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Example 3. Use of Biodegradation Rate Constants
INTRODUCTION: A chlorinated solvent plume at the Cape Canaveral Air Force Base, Florida, has maximum source
concentrations of 0.056 mg/LTetrachloroethene (PCE), 15.8 mg/LTrichloroethene (TCE), 98.5 mg/Lcis-Dichloroethene (DCE),
and 3.08 mg/L Vinyl Chloride (VC), 33 years after the spill originally occurred. The calculated seepage velocity at the site is
111.7 ft per year. Based on the existing distribution of chlorinated solvents and degradation products, how far down the flow
path will the plume extend when it eventually comes to a steady state? This example is based on the example in Appendix A.6
of the User's Manual for the BIOCHLOR natural attenuation decision support system (Aziz et al., 2000). This model and the
user's guide can be downloaded at no cost from the EPA Center for Subsurface Modeling Support (CSMoS) at http://www.epa.gov/
ada/csmos/models.html.
Well Distance from Source (feet)
CCFTA2-9S 0
MP-3 560
CPT-4 650
MP-6 930
MP-4s 1085
PCE (mg/L)
0.056
<0.001
ND
<0.001
<0.001
TCE (mg/L)
15.8
0.220
0.0165
0.0243
<0.001
cis-DCE (mg/L)
98.5
3.48
0.776
1.2
0.556
VC (mg/L)
3.08
3.08
0.797
2.52
5.02
100
hData I
Available r
Projections of
Model into Future
H
o
O
0.01
- - -TCE Prediction
DCE Prediction
VC Prediction
TCE Field Data
* DCE Field Data
n VC Field Data
CALCULATION: The following
approach was used to determine
biodegradation rate constants for
each of the chlorinated solvents
using a solute transport model:
Step 1: Perform parameter estima-
tion and enter data into model.
Step 2: Bytrial-and-error, adjust the
first-order biodegradation rate
constants ( X) to match the
observed site data. The resulting
first-order biodegradation rate
constant for PCE was 2.0 per year
(half-life of 0.34 years), forTCE was
1.0 peryear (half-life was 0.7 years),
for cis-DCE was 0.7 per year (half-
life 1.0 years) and for VC was 0.4
per year (half- life of 1.7 years).
Step 3: Run the simulation forward
in time until it comes to an apparent
steady state.
Step 4: Compare the simulated distribution of contaminants to the existing data used to calibrate the model. As discussed
in Example 1, attenuation rates for declining concentration are positive values. When compared to values in the literature
(see Figures 4 and 5), the values appear to be reasonable. All plume lengths were projected to the boundary defined by the
MCL for Vinyl Chloride. Available data to calibrate the model extended 1085 ft from the source. The model was calibrated
to the first 33 years of the plume. When the simulation was extended to 100 years the projections reached a steady state.
At steady-state, there was no significant increase in the length of the TCE plume, but the cis-DCE plume was approximately
twice as long at the time data available for calibration were collected, and the VC plume was approximately three times as
long.
0.001
0
1000
2000
3000
4000
Distance From Source (ft)
Plume Attenuation?
The calculated biodegradation
rate constant is positive,
indicating that biodegradation of
dissolved chlorinated solvents
is occurring after the solvents
leave the source zone. PCE
and TCE had the highest rates,
while VC had the lowest rate at
this site.
Plume Trends?
The screening model used biodegradation rate
constants to project the future distribution of PCE,
TCE, cis-DCE, and VC. The model projects relatively
little change in the PCE, and TCE plumes, but the
model predicts that the cis-DCE and VC plumes are
expanding. To confirm the true behavior of the
cis-DCE and VC plume, it may be necessary to install
more monitoring wells to adequately delineate the
plume, and collect data on concentration vs. time in
all the wells in the plume.
Plume Duration?
A biodegradation rate
constant is not useful for
estimating the duration of
the plume (i.e., the time to
reach a clean-up goal).
Key Point:
Biodegradation rate constants cannot be used for estimating remediation time frames, but are useful for identifying possible
trends in the behavior of plumes using mathematical models.
11
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Appendix I. Uncertainty in Rate Calculations
Using Statistics to Estimate the Time Frame to
Achieve Remediation Objectives
As with any remediation method, one of the fundamental questions
that arises is "How much time will be required before remediation
objectives are achieved?" At the current state of practice, the
only practical approach available uses a statistical analysis of
long-term monitoring data from wells in the source area of the
contaminant plume. Many practitioners will calculate the Pearson
product moment correlation coefficient (R2) for the regression used
to extract the Point Decay Rate constant (kpaint)- If the coefficient
is near one (e.g., greater than 0.9 or 0.95), the regression is
accepted as being useful in a qualitative way. There are two
problems with this approach; it does not allow the user to select a
level of confidence for the comparison, and it does not give more
validity to regressions with many points compared to regressions
with only a few points.
The slope of the regression is the rate constant. A better approach
is to calculate a confidence interval on the slope of the regression.
The following data from Kolhatkar et al., 2000 will be used to
illustrate this approach. They collected long-term ground-water
monitoring data from three wells at a gasoline release site in New
Jersey. Their original data displayed extreme oscillations with
concentrations bouncing from a high value down to the analytical
detection limit of 1|o.g/L, and then back to a high value over
sequential sampling intervals. Although the scatter in the data
set is typical of the variation seen at many other sites, the influence
of these outliers on the statistical estimate of the rate of attenuation
was removed by editing the data set to remove those points where
the concentration of MTBE was less than the detection limit.
Table 1-1. Sources of Uncertainty in Calculated Rate Constants
Point Decay Rate (k int)
I Bulk Attenuation Rate
(constant (k)
JBiodegradation Rate
[constant (X)
Point Decay Rate (k int)
I Bulk Attenuation Rate
Iconstant (k)
iBiodegradation Rate
[constant (X)
I Bulk Attenuation Rate
(constant (k)
lAII rate constant
(calculations
Wells not in strongest source area may
not give representative indication of how
long entire plume will persist.
Wells not on centerline of plume can give
misleading indications about
concentration profile in plume.
A poorly designed monitoring well
network may give misleading information
about source strength, source size, and
centerline plume concentrations used for
calibration.
Can introduce additional scatter in data
used to develop k intrate constant.
Typically not a problem as all data are
collected at the same time.
Can be a problem if seasonal effects are
significant and the data used for
calibration are not collected
(concentration vs. distance) at the same
time.
Increases overall uncertainty in
calculation.
Increases apparent uncertainty.
Characterize source with several wells.
Estimate and report uncertainty in final result
(estimated time to reach clean-up standards).
Use a well-designed monitoring well network
with transects of wells in rows across the
plume rather than one set of wells down the
inferred centerline. Estimate and report
uncertainty in final result (estimated plume
length).
The source and plume need to be well
characterized to ensure representative
modeling results. Perform sensitivity analysis
on model.
Address as part of an uncertainty calculation
(see below). For strong seasonal effects, use
of data from the same season can be
considered.
Not applicable.
For strong seasonal effects, use data from
same season to help ensure representative
modeling results. Perform sensitivity analysis
on model.
Average results from multiple seepage
estimates along plume centerline. Improve
seepage velocity estimate. Estimate and
report uncertainty in final result (estimated
plume length).
Use worst-case data. Use transects to
capture plume heterogeneity. For regression-
based rate constants (k and kpoint), estimate
and report uncertainty in final result. For
modeling studies designed to determine X,
perform sensitivity analysis on model by
changing key variables to their upper and
lower expected range and evaluate how
modeling results change.
12
-------
Because there is natural scatter in the long-term monitoring data,
there is uncertainty in the estimate of the Point Decay Rate (kpom),
and in the projected time frame to achieve cleanup in that
monitoring well. To account for this uncertainty, a confidence
interval was calculated for each estimate of the Point Decay Rate
(fcp0int) at a pre-determined level of confidence of 90% and 95%.
The level of confidence is simply the probability that the true rate
is contained within the calculated confidence interval. A
confidence level of 90% is reasonable for many sites. At other
sites, a more stringent confidence level (e.g., 95%) may be more
appropriate, depending upon the level of risk that is acceptable.
In most applications of regression, the user wishes to calculate
both an upper boundary and lower boundary on the confidence
interval that will contain the true rate at the pre-determined level
of confidence. This is termed a "two tailed" confidence interval
because the possibility of error (the tail of the probability frequency
distribution) is distributed between rates above the upper boundary
and below the lower boundary of the confidence interval. As a
consequence, tables of critical values in statistical reference books
and computer applications provide a "two-tailed" confidence
interval. At a level of confidence of 80%, the estimate will be in
error 20% of the time. The true rate will be contained within the
calculated confidence interval 80% of the time, 10% of the time
the true rate will be faster than the upper boundary of the
confidence interval, and 10% of the time the true rate will be slower
than the lower boundary of the confidence interval. Using the
data provided above from MW-5, the slope of a regression of the
natural logarithm of concentration of MTBE on time is -0.188 per
year. The Point Decay Rate (/(point) is +0.188 per year. The
boundaries of the "two tailed" confidence interval on the rate at
80% confidence are 0.248 per year and 0.127 per year. This
means that 80% of the time the true rate will be between 0.248
and 0.127 per year, that 10% of the time the true rate is greater
than 0.248 per year, and 10% of the time the true rate is less than
0.127 per year. The true rate will be greater than 0.127 per year
90% of the time.
There is little value in estimating the shortest possible time that
would be required to reach the goals for cleanup; remedial options
are compared and evaluated based on the greatest time required
to reach goals. At the selected level of confidence, all the
possibility of error should be assigned to rates that are slower
than the lower boundary of the confidence interval. This is a "one-
tailed" confidence level; it includes all true rates that are faster
than the lower boundary of the confidence interval. A"one tailed"
Table I-2. MTBE Concentrations in the Three Most Contaminated Monitoring Wells at a Gasoline Spill Site
Date
9/17/93
9/23/94
5/17/96
8/10/96
11/7/96
12/8/97
3/27/98
7/23/98
9/18/98
12/16/98
3/1/99
6/21/99
9/7/99
9/7/99
12/30/99
3/20/00
6/22/00
MW-5
Concentration
dig/liter)
1,900
1,800
1,300
980
620
500
635
470
1,210
379
700
574
792
1,050
525
501
420
MW-6
Concentration
(ng/liter)
270
200
120
120
66
71.2
44
42.2
43.2
36
51.2
MW-11
Concentration
(ng/liter)
2200
880
660
339
426
419
144
123
464
195
155
220
173
146
13
-------
confidence interval can be calculated as the slower of the two
confidence intervals from a "two-tailed" test that has twice the
uncertainty. In the example above, where "two tailed" confidence
intervals were calculated for a confidence level of 80%, the true
rate will be greater than a rate of 0.127 per year 90% of the time.
The "one tailed" confidence intervals reported in the table below
were calculated in this fashion. Monitoring well MW-5 has the
highest concentration of MTBE and the lowest Point Decay Rate,
and can reasonably be expected to be the last monitoring well to
reach the goal. The other monitoring wells should reach the goal
much sooner; the best estimate of the lifetime of the plume is the
expected lifetime of MTBE in MW-5.
Note that for a given number of observations, as the level of
confidence is increased, the interval that is expected to contain
the real value for the rate constant increases as well. As the level
of confidence increases, the lower boundary on the rate constant
decreases, and the projected time required to meet the clean-up
goal increases. In the examples presented above, the estimated
rate of natural attenuation of MTBE in MW-5 is 0.188 per year,
which requires 16 years to attain a concentration of 20 |ig/L. At a
90% confidence level, the lower boundary of the confidence
interval is 0.127 per year, which requires 24 years to meet the
goal. At a 95% confidence level, the lower boundary is 0.109 per
year, which requires 28 years to reach the goal. At the 95%
confidence level the upper bound of the time expected to reach
the clean-up goal has increased by a factor of almost two (from
16 years to 28 years). This does not necessarily mean that the
actual time to achieve cleanup will be 28 years; it simply means
that the length of time that will actually be required is estimated to
be no more than 28 years at a 95% level of confidence.
At many sites, the long-term monitoring data show that the
concentration of MTBE actually increases over time. At other
sites, the general trend in the concentration of MTBE may be
down, but there is a great deal of variation in the data. These
variations in concentrations over time are not necessarily errors
in sampling and analysis of ground water. In many cases they
reflect real changes in the plume caused by seasonal variations
in precipitation. These variations are a natural property of plume.
If the variation is large enough, one boundary of the "two tailed"
confidence interval will be a positive number and the other
boundary will be a negative number. When zero is included in
the confidence interval on the rate, there is no evidence in the
data that the true rate is different from zero. If this is the case, it is
possible that attenuation is occurring in that particular well over
time, but the monitoring data do not present evidence that
attenuation is occurring at the predetermined level of confidence.
At the predetermined level of confidence, it is impossible to predict
how long it will take to reach the clean-up goals.
The ability to extract a rate of attenuation from long-term monitoring
data is related to the number of measurements, and the time
interval over which they are collected. As an example, the rate of
attenuation extracted from the last three years of monitoring data
for well MW-5 (3/27/1998 to 6/22/2000) is 0.106 per year, but the
"one tailed" 90% confidence interval is all rates greater than
-0.125 per year. The confidence interval includes zero. If only
these three years of data were available, there would be no
evidence of natural attenuation of MTBE in well MW-5 at 90%
confidence. The rate extracted from the last four years of data
(5/17/1996 to 6/22/2000) is 0.130 per year. The 90% confidence
interval on the rate (0.0302 per year) would reach the clean-up
goal in 100 years. The rate extracted using all the seven years of
monitoring data is 0.188 per year. The 90% confidence interval
on the rate would reach cleanup in 24 years. A few extra years of
monitoring data have a strong influence on the ability to extract
useful rate constants.
Key Point:
The Point Decay Rate (kpo.J can be used to project the time
required for reaching a clean-up goal. However, there are a
number of points to keep in mind. First, an appreciable record of
long-term monitoring data must be available to make a statistically
valid projection of the rate of natural attenuation. As a practical
matter, it is difficult to extract rate constants that are statistically
significant with fewer than six sampling dates, or with a sampling
interval of less than three years. Second, it is unrealistic to expect
just a few years of monitoring data to accurately predict plume
behavior several decades into the future. Third, it is important to
realize that these estimates are merely estimates and that the
true rate may change over time.
Table I-3. Point Decay Rate (k o.J of Attenuation of MTBE in Monitoring Wells and the Projected Time Required to Reach a
Clean-Up Goal of 20 mg/L as Calculated from the Long-Term Monitoring Data for the Wells
Well
MW-5
MW-11
MW-6
MTBE (^g/L)
First
Sample
1993
1900
2200
270
Last
Sample
2000
420
146
51.2
Estimated rate and time
required
Rate
(per year)
0.188
0.453
0.29
Time
(years)
16
4.4
3.2
Rate and time significant
at 90% confidence
Rate
(per year)
0.127
0.365
0.246
Time
(years)
24
5.4
3.8
Rate and time significant
at 95% confidence
Rate
(per year)
0.109
0.337
0.231
Time
(years)
28
5.9
3.8
14
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Appendix II. Contaminant Concentration Attenuation Downgradient from Source Areas as a
Function of Dispersion, Sorption, and Biodegradation
INTRODUCTION: The Domenico solution to the advection-dispersion-biodegradation equation along the centerline of a plume was
applied to a hypothetical case to illustrate the impact of the different attenuation parameters on the overall bulk attenuation rate. The
Domenico solution is given by
erfc
x-vt. 1 + ^=
V v
erf
where Coisthe initial concentration, ax is the longitudinal dispersivity, ay is the transverse dispersivity Xisthe biodegradation rate, f is
time, xis distance from the source, v\s the retarded ground-water velocity (i.e., v=v/R), and Y\s source width.
DATA: The following are the parameters assumed for this example:
v = 100 ft/yr (median value from the HGDB database (Newell et al., 1990))
y=40ft
f = 10 years
ay=0.1 ax
b= 10 ft (source thickness used for the Bioscreen runs)
CALCULATION: Four different scenarios were considered to estimate the effect of the different parameters on the overall attenuation
rate: 1) the only process acting at the plume is dispersion (ax = 100 ft); 2) previous scenario plus the effect of sorption (R=5);
3) dispersion, sorption, and biodegradation (X=0.2 per yr) are acting; and 4) previous scenario plus the effect of source decay
(^ourea=°-139Peryr).
For each scenario, the Domenico solution was applied to obtain concentrations along the centerline of the plume. Next, concentrations
vs. distance were plotted and data were fit with an exponential equation (first-order model). The slopes of the C vs. D plots were
0.002, 0.0106, 0.0124, and 0.0237/ft for scenarios 1, 2, 3, and 4, respectively. Finally, the bulk attenuation rate constant, k, for each
scenario was calculated by multiplying the slope by the contaminant velocity (100 ft/yr/retardation factor). This calculation yielded
bulk attenuation rates equal to 0.2, 0.212, 0.248, and 0.474/year for scenarios 1, 2, 3, and 4, respectively. These values correspond
to dissolved-phase half-lives of 3.5, 3.3, 2.8 and 1.5 years after the contaminant leaves the source zone.
10
en
"c
03
O
C
O
O
Dispersion+Sorption+Biodegradation
//f=0.248/year
Dispersion
'/r=0.2/year
Dispersion+Sorption
/r=0.212/year
Dispersion+Sorption+Biodeqracation+
Source Decay
/r=0.474/year
O0OOO0
200
400 600 800
Distance from source (ft)
1000
1200
This example illustrates incremental attenuation impacts of the various attenuation processes and how the overall bulk rates change
as a result (i.e., the more processes present at a given site, the higher the bulk attenuation rate). The effect of individual parameters
on the attenuation rate is discussed below:
15
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Bulk Attenuation Rate (ft) as a Function of Longitudinal Dispersivity (ax)
The figures below show the calculation of k for different dispersivity values as well as a resulting plot of bulk attenuation rate as a
function of longitudinal dispersivity. The transverse dispersivity (a ) was set to 10% of the longitudinal dispersivity (ax), the vertical
dispersivity (aj was set to 10% of the transverse dispersivity (a ), and f = 30 years. The slopes of the concentration vs. distance plots
were multiplied by the contaminant velocity to obtain bulk attenuation rates. This type of calculation assumes that the plume is at
steady-state. The figures below suggest that the bulk attenuation rate (k) increases as dispersivity increases.
200
400 600 800
Distance from Source (ft)
1000
1200
0.25
00.2
ro
CD
feO.15-
Q.
0.1
=10.05
CO
20
40
60 80
a., feet
100 120
16
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Bulk Attenuation Rate (k) as a Function of Sorption prior to Equilibrium.
When a plume comes to a steady state, sorption no longer removes contaminants from ground water, and there is no effect of
sorption on the bulk attenuation rate (k). Prior to equilibrium, sorption removes contaminants from the ground water and contributes
to the bulk attenuation rate. The effect of sorption on the bulk attenuation rate was evaluated by calculating kfor different retardation
factors and plotting the resulting k values as a function of R as illustrated in the figures below. For this analysis a longitudinal
dispersivity of 100 ft was assumed, and f = 10 years. In this case, the slopes of the concentration vs. distance plots were multiplied by
the seepage velocity rather than the contaminant velocity to obtain bulk attenuation rates, since retardation was already included in
the Domenico calculation. It can be concluded that with all the other parameters constant, the bulk attenuation rate is roughly
proportional to the retardation factor.
R=1 y= 12.261e
-0.0031X
R=2 y = 43.744e
-0.0082X
R=5 y= 150.085e
-0.0185x
R=10 y= 120.133e
-0.0268X
R=50 y =169.45776
-0.061X
0
500 1000 1500
Distance from source (ft)
2000
2500
6-
>< 5
i_
0
Q. 4
CD
-I-J
CO
3-
^ 2-
m 1
o
0 10 20 30 40
Retardation factor
50
60
17
-------
Bulk Attenuation Rate (ft) as a Function of Biodegradation Rate (K)
Bulk attenuation rates for first-order biodegradation rates within the range 0 to 0.5/year were estimated and a plot of /(versus X was
prepared to illustrate the impact of this parameter on the overall attenuation rate. For this analysis a longitudinal dispersivity of and a
retardation factor equal to 1 (no sorption) were assumed. As shown in the following figures, with all the other parameters being
constant, the bulk attenuation rate increases as the biodegradation rate increases.
0
-0.0031 X
=0 y=12.26e
=0.1/yr y=11e
-0.0035X
y=9.1343e
-0.0043X
=0.5/yr y =7.9563e
-0.0051 X
=1/yr y=6.5575e
-0.0071 X
=3/yr y=5.9249e
-0.0137x
500
1000 1500 2000
Distance from source (ft)
2500
3000
o>
Q.
0)
DO
1 2 3
Biodegradation rate (A, per yr)
18
-------
Bulk Attenuation Rate (ft) as a Function of Source Decay Rate (ksource)
The figures below show the calculation of /(for source decay rates varying between 0 and 0.69/yr as well as the resulting plot of bulk
attenuation rate as a function of /(source.The effect of source decay was evaluated using the Bioscreen model (Newell et al., 1996). For
this scenario, a longitudinal dispersivity of 100 ft and no sorption nor biodegradation were assumed. It can be inferred that the bulk
attenuation rate decreases as source decay rate increases.
, \ , -0.0038X
' \ ^source =0 y =5.99296
\ * /
^source =0.0139/yr y =5.231 eaoo37x
s
/fsource =0.069/yr y =3.038e°0031x
/
~-A-.
Source =0.139/yr y=1.5404eaoo24x
^ /fsource =0.346/yr y =0.2007e-°0004x
-:L-t---r^- /
50 100 150 200 250 300
Distance from source (ft)
350 400 450
k oc
source
0.1 0.2 0.3
Source decay rate (/fsource peryr)
0.4
19
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Appendix III. Effect of Dispersion, Sorption, Biodegradation, and Source Decay on
Concentration vs. Time Profiles
INTRODUCTION: Concentration versus time profiles for a hypothetical case were generated using the Domenico solution to the
advection-dispersion-biodegradation equation along the centerline of a plume to illustrate the impact of the different attenuation
parameters on the point attenuation rate at two different locations, one near the source area and the other 200 ft downgradient from
the source.
DATA: The parameters assumed for this example are as follows:
vs= 100 ft/yr (median value from the HGDB database (Newell et al., 1990)), Y=40i\, a =0.1 ax,b= 10ft (source thickness used
for the Bioscreen runs)
CALCULATION: Four different scenarios were considered to estimate the effect of the different parameters on the overall attenuation
rate: 1) the only process acting at the plume is dispersion (ax= 100 ft); 2) previous scenario plus the effect of sorption (R=5); 3)
dispersion, sorption, and biodegradation (X=0.2 per yr) are acting; and 4) previous scenario plus the effect of source decay (ksourcg=
0.139 per yr).
For each scenario, the Domenico solution was applied to obtain concentrations at two locations: one near the source area (X=20 ft)
and the other at a point located 200 ft downgradient from the source as a function of time. As illustrated in the figures below, when
running Concentration vs. Time profiles, a decline in concentration near the source is not observed unless the source is decaying.
Without source decay, the concentrations increase until they reach a steady-state maximum value and thereafter remain constant
even when dispersion, sorption, and biodegradation are present at a site (scenarios 1, 2, and 3). On the other hand, when source
decay is included, concentrations increase up to a maximum and decrease with time. (Note the two graphs have different scales).
Near source location
Dispersion+SorptJOTi
W =75 yr/
' Dispersion+Sorption+Biodegradation ^ax ~ m
t,s=12yr,
ng/L
Dispersion+Sorption+Biodegradation-
Source decay
-------
Effect of Longitudinal Dispersivity (oj on Concentration vs. Time Profiles
The figures below show concentration vs. time profiles for different dispersivity values for a source location (X=20 ft) and a downgradient
location (X=200 ft). The maximum concentration decreases as the longitudinal dispersivity increases and the time required to reach
steady-state increases as dispersivity increases.
-------
Effect of Sorption on Concentration vs. Time Profiles
Changes in Concentration vs.Time profiles as a result of sorption were evaluated by plotting the profiles at the source and downgradient
locations for different retardation factors. For this analysis a longitudinal dispersivity of 100 ft was assumed. As can be seen in the
figures below, the time required to reach steady-state increases as the retardation factor increases. Sorption, however, does not
change the steady-state concentration. (Note the two graphs have different scales.)
20
40
60
Time (yr)
80
100
120
Near source location
20
40
60 80
Time (yr)
100
120
140
Downgradient location
22
-------
Effect of Biodegradation (X) on Concentration vs. Time Profiles
The figures below show concentration vs. time profiles for different biodegradation rates for both the source and downgradient
locations. For this analysis a longitudinal dispersivity of 100 ft and a retardation factor equal to 1 (no sorption) were assumed. As
shown below, the higher the biodegradation rate, the lower the maximum concentration and the shorter the time required to reach
steady-state. (Note the two graphs have different scales.)
l=0
l=0.3yr'1
I =3 yr
10 15 20 25 30
Time (yr)
35
40
45
50
Near source location
l=0
l=0.3yr'1
l=0.5yr"
1=1 yr'1
I =3 yr "1
10 15 20 25 30 35 40 45 50
Time (yr)
Downgradient location
23
-------
Effect of Source Decay (ksource) on Concentration vs. Time Profiles
The figures below show concentration vs. time profiles for various source decay rates for both the source and downgradient locations.
This scenario was run using the Bioscreen model (Newell et al., 1996) assuming a longitudinal dispersivity of 100 ft, no sorption and
no biodegradation.The maximum concentration is shown to be inversely proportional to the source decay rate. (Note the two graphs
have different scales.)
"source =0.346 per yr
Source =0.693 per yr
10 15 20 25 30 35 40 45 50
Time (yr)
Near source location
Source ~
10 15 20 25 30 35 40 45 50
/^ource =0.069 peryr- .........^"- =°-0139 PerVr
Source =0.139 per yr
source =0.346 per yr
source =0.693 per yr
Downgradient location
24
-------
Point Attenuation Rate /fpoint as a Function of Source Decay (/fsource)
A further analysis of Concentration vs. Time profiles for different source decay rates was conducted to calculate kgojnt values. The
effect of source decay on the point attenuation rate was then evaluated by plotting the calculated kpa.nt as a function of /(source as
illustrated in the figure below. This example illustrates that the point attenuation rate is proportional to the source decay rate.
0.1
0.2 0.3 0.4 0.5
Source decay rate (/csource per yr)
0.6
0.7
25
-------
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27
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