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Bechtel National, Inc.
45 Fremont Street, San Francisco, Ca 94119-3965, USA
e-mail: anfindik@bechtel.com and nkcopty@bechtel.com
A statistical procedure for quantifying the uncertainty in the
evaluation of permeable reactive subsurface barriers is presented. The
hydraulic conductivity is defined as a random spatial variable whose
statistical structure is inferred from the available hydraulic conductivity
data. Monte Carlo simulations and
Bayes' Theorem on conditional probability are used to estimate
probability-based weights for each of the realizations of the hydraulic
conductivity using the hydraulic conductivity data, historical concentration
data and contamination source information.
Ground water flow and contaminant transport models are then used to
probabilistically evaluate the performance of the reactive barriers.
INTRODUCTION
Permeable
reactive subsurface barriers (PRSB) represent one of the most promising new
technologies for cleaning up contaminated ground water. A PRSB can be defined as "an emplacement of reactive materials in the
subsurface designed to intercept a contaminant plume, provide a preferential
flow path through the reactive media, and transform the contaminant(s) into
environmentally acceptable forms to attain remediation concentration goals at
the discharge of the barrier" (EPA, 1997).
The concept of using iron and other zero-valent or zero-oxidation-state
metals for contaminant remediation has been tested in the laboratory in pilot
scale field projects, and since 1995 the technology has been used in full-scale
field commercial applications for the treatment of plumes of chlorinated
hydrocarbons and chromate. The two most
common PRSB designs are the funnel and gate and the continuous trench.
In
addition to the uncertainty in different aspects of PRSB performance arising
from specific aspects of this technology, there is uncertainty in PRSB
performance due to the natural heterogeneity of subsurface formations and the
often limited data for their characterization and for the definition of the
contamination plume.
This paper
describes a probabilistic approach for the evaluation of the effectiveness of
reactive barriers. The approach uses Monte Carlo simulations and Bayes' Theorem
on conditional probability to generate multiple realizations of the hydraulic
conductivity field conditional on hydraulic conductivity measurements, and
consistent with other related information such as information on the
contamination source term and historical concentration data. There are similarities, but also
differences, between the present approach and other applications of the Monte
Carlo method for assessing the uncertainty in the prediction of the performance
of ground water remediation schemes (see for example Varljen and Shafer, 1991;
Bair et al. 1991; Guadagnini and Franzetti, 1997). Most of these applications were based on equiprobable multiple
realizations of the hydraulic conductivity field. The present approach uses additional information, often collected
as part of a remedial investigation, to estimate the probability that any
particular realization of the hydraulic conductivity field may represent the
actual conditions.
Probabilistic Evaluation of Remediation Schemes
Predictive
calculations for the evaluation of remediation measures can be conditioned on
the field data collected for the characterization of the site and the
determination of the present extent of contamination. In a typical remedial investigation. Typically available data include estimates of hydraulic
conductivity ki and
historical concentration data cjn
where i=1, ... I and I is the
total number of hydraulic conductivity data; j=1, ... J and J is the number of concentration
monitoring points; n = 1,...N and N is the number of times that
concentration data were collected at each monitoring point.
The
first step of the present procedure is to generate multiple, unconditional
realizations of the hydraulic conductivity field based solely on its ensemble
statistics. These realizations can be generated with the aid of a random field
generator such as the Turning Band Method (Mantoglou and Wilson, 1981). Conditional realizations of the
log-hydraulic conductivity field, Y c.m,
for m=1,... M can then be generated by conditioning each unconditional
realization on the available hydraulic conductivity data, y1, ... , yI.
where yi = ln[ki] and M is the total number realizations generated.
From
Bayes' Theorem on conditional probability (Ang and Tang, 1975), the probability
of occurrence of the mth
log-hydraulic conductivity realization given the concentration data, 
, can be written as:
(1)
Equation
(1) states that the probability of observing the hydraulic conductivity
realization Y c.m given
the concentration data cjn
is proportional to the product of the probability of observing the
concentration data cjn
given the hydraulic conductivity realization Y c.m, and the probability of that specific hydraulic
conductivity realization. Because the
generated hydraulic conductivity realizations Y c.m are equiprobable,
the terms 
appearing in the
first half of Equation (1) are equal for all values of m. Therefore, 
, can be expressed as shown in the second half of Equation
(1).
The
probability,
, of observing the concentration data cjn given the hydraulic conductivity
realization Y c.m, can be
evaluated if the non-stationary conditional multivariate concentration
distribution is known over the entire domain.
In practice though the following approximate procedure is proposed:
a. Simulate
with a numerical flow and transport model the concentration distribution 
given the
log-hydraulic conductivity realization Y c.m,
the probability density function of the source release term, and other input
parameters such as recharge, storativity, dispersivity and distribution
coefficient. This will produce a
different concentration distribution for each log hydraulic conductivity
realization m.
b. The
probability 
can be approximated
as proportional to the difference between the simulated and observed
concentrations, as follows:
(2)
where 
is the natural
log-transform of the measured concentration at location j and time n, and 
is the natural
log-transform of the simulated concentration at location j and time n and for
hydraulic conductivity realization m.
The
estimated probability of each hydraulic conductivity realization, based on the
available hydraulic conductivity data can be used to make probabilistic
predictions. For example, the
performance of a PRSB can be simulated for each of multiple generated hydraulic
conductivity realizations, and the results of these simulations can then be
weighted by the probability of occurrence of each realization, 
. The performance of
the PRSB can be evaluated probabilistically in terms of its ability to capture
and react with the contamination plume.
APPLICATION EXAMPLE
An example is used to illustrate the key elements of
the described procedure. The objective
in this example is to assess the uncertainty in the performance of a slurry
wall with a gate containing reactive material.
The example addresses the uncertainty associated with hydraulic
conductivity, the contamination source, and the performance of the reactive gate. All other parameters are treated
deterministically.
The
data used in this example are generated by creating a hydraulic conductivity
distribution and a contamination plume, which are treated as the "actual", but
unknown, site conditions. It is assumed
that the log-hydraulic conductivity is normally distributed (Y=log-k) with a
mean value of mY =5.3 (the
mean of hydraulic conductivity is 200 m/day) and a variance of 
. The semi-variogram
of the hydraulic conductivity is assumed to be an exponential function, i.e. 
where G(r) is the stationary, anisotropic
semi-variogram, function of distance r,
and I = 500 m is the integral
scale. A log-hydraulic conductivity
field is generated using the Turning Band Method over a 2500 by 3500 m portion
of the aquifer extending mostly downgradient of the contamination source. Similarly, a plume is generated by
simulating the migration of trichloroethylene (TCE) that entered the aquifer as
the result of a known continuous release rate of contaminants. It is assumed that all flow and transport
parameters, except the hydraulic conductivity, are known, and that recharge,
adsorption, dispersion and degradation and decay are negligible. Flow is simulated by assuming that there is
no flow through the north and south boundaries and assigning constant heads to
the western and eastern boundaries of the simulation domain which create ground
water flow with a general direction from west to east. The average hydraulic gradient is 0.002 m/m.
A
data set of hydraulic conductivity values is now created by sampling from the
"actual" hydraulic conductivity distribution, at twenty randomly selected data
points shown. These sampled values can
be viewed as hydraulic conductivity measurements from a field investigation
that would have measured the actual hydraulic conductivity at a limited number
of points. Similarly the "actual"
concentration distribution is sampled at twenty points located on a regular
grid to produce a data set of contamination data.
The Turning Bands Method was used to generate 1000
unconditional log-hydraulic conductivity realizations which were then
conditioned on the 20 available log-hydraulic conductivity data using ordinary
kriging. These realizations are all
equally probable.
Each
realization was used then to simulate numerically ground water flow and
historic contaminant migration. The
released mass at the contamination and the time of release were defined
probabilistically. It was assumed that
the time of release, T, is normally
distributed with mean to
and variance st2,
and that the rate of contaminant mass release is also normally distributed with
a mean 
and a variance sm2. The simulated concentrations for each
realization were then used in conjunction with the set of 20 "measured" values
to estimate 
, the probability of occurrence of the hydraulic conductivity
field of this realization. A comparison
of the 
values obtained for
different values of the variances st2
and sm2,
suggests that when the uncertainty in the contamination source (release rate
and release time) is large, the contamination data do not contribute
significantly in reducing the uncertainty in the definition of the hydraulic
conductivity distribution and that all hydraulic conductivity realizations are
nearly equiprobable. However, as the
uncertainty in the definition of the source term decreases, the concentration
data can be used to identify those realizations of the hydraulic conductivity
field which most probably are closer to the "actual" hydraulic conductivity
field. This is accomplished by giving
more weight to the realizations that produce better agreement with the measured
concentration data.
The
probability-based weights 
calculated for each hydraulic conductivity field are used
next to evaluate the effectiveness of the PRSB system. The effectiveness of the reactive material
in removing the contaminants is treated as a stochastic variable. It is expressed in terms of the fraction of
mass of TCE that is removed by the iron gate.
For simplicity, it is assumed that the effectiveness of the reactive
material is independent of the concentration level, i.e. Cdg
= f Cug where Cug and Cdg are the TCE concentration upgradient and
downgradient of the reactive gate, and f is the effectiveness of the gate defined in
terms of its mean value fmean
and standard deviation sf.
In this example it is assumed that f
is normally distributed with fmean
= 0.8 and sf = 0.10.
The location and dimensions of the wall were selected
by trial and error in such a manner that the reactive gate would intercept the
entire present contaminant plume according to deterministic model predictions
based on the "best estimate" hydraulic conductivity distribution. This would represent the optimal reactive
gate configuration using a deterministic modeling approach. The estimated
capture zone of the reactive gate, superimposed over the present plume, is
shown in Figure 1.
To evaluate the effectiveness of the system, the TCE
distribution downgradient of the system is simulated for each of the generated
1000 hydraulic conductivity realizations and each time selecting randomly the
normally distributed effectiveness parameter f of the reactive material.
Figure 2 shows the estimated distribution of the probability that a
water particle in a particular part of the aquifer may pass through the gate of
reactive material. The results
presented in Figures 1 and 2 illustrate the importance of accounting for
uncertainty in the definition of the hydraulic conductivity. According to
Figure 1, the entire plume would pass through the reactive gate. However, Figure
2 suggests that the probability of capturing a significant portion of the contaminant is less than 90
percent. In fact the probability of
capturing the most contaminated zone (shown in darker shade) is between 60 and
90 percent.
Figure 3
illustrates the variability of the concentrations of contaminated ground water
arriving at the gate as a function of time.
It shows the concentration vs. time for six of the 1000 simulations used
in this example.
Figure 4
shows the cumulative fraction of contaminant mass removed by the system as a function time. The uncertainty in the estimation of the
mass removed is illustrated in terms of the upper and lower 95 percentiles. In addition to these curves, Figure 4 shows
the mass-recovery curve based on the "actual" hydraulic conductivity and the
"best estimate" hydraulic conductivity distribution. These results indicate that the "weighted" mass recovery curve is
close to the "actual" curve.
Furthermore, a comparison between the "best estimate" curve and the
weighted curve suggests that the "best estimate" curve may overestimate the
rate of mass recovery. The "best estimate" curve does not provide a measure of
the uncertainty in the model predictions.
Such information is particularly useful in assessing the risks
associated with the failure of a
remediation scheme to perform as planned.
Figure
5 compares the range of uncertainty in estimating the contaminant mass removed
as described above with the uncertainty associated strictly with the performance
of the reactive materials. The latter
is estimated by estimating the weighted average of the contaminant mass
arriving at the gate and combining it with the upper and lower 95 percentiles
of the probability distribution of the effectiveness parameter f.
The difference between the two uncertainty ranges shown in Figures 5
reflects the uncertainty in the performance of the PRSB system due to the uncertainty in the definition of the
hydraulic conductivity field.
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