Analytical Models for the Design of Iron-Based Permeable Reactive Barriers

Abstract

The preliminary design of iron-based permeable reactive barriers is often accomplished using analytic expressions for one-dimensional groundwater flow and contaminant transport. Typically, one or more of the governing processes is simplified or neglected to facilitate development of a tractable solution. This paper presents a set of improved design equations that include the effects of dispersion, finite domain boundary, sequential decay, and production processes, and increased flow through high conductivity barriers. When applied to realistic example problems, application of the expanded design equations typically results in the specification of a larger permeable reactive barrier thickness than obtained using conventional approaches.

Full text

Analytical Models for the Design of Iron-Based
Permeable Reactive Barriers
Alan J. Rabideau, P.E., M.ASCE
1
; Raghavendra Suribhatla
2
; and James R. Craig
3
Abstract: The preliminary design of iron-based permeable reactive barriers is often accomplished using analytic expressions for one-
dimensional groundwater flow and contaminant transport. Typically, one or more of the governing processes is simplified or neglected to
facilitate development of a tractable solution. This paper presents a set of improved design equations that include the effects of dispersion,
finite domain boundary, sequential decay, and production processes, and increased flow through high conductivity barriers. When applied
to realistic example problems, application of the expanded design equations typically results in the specification of a larger permeable
reactive barrier thickness than obtained using conventional approaches.
DOI: 10.1061/共ASCE兲0733-9372共2005兲131:11共1589兲
CE Database subject headings: Barriers; Design; Analytical techniques; Ground-water pollution
.
Introduction
A popular technology for remediating contaminated groundwater
is to emplace a permeable reactive barrier 共PRB兲 in the path of
the plume 共e.g., Starr and Cherry 1994; USEPA 1998兲. According
to Environmental Technologies Inc. 共ETI兲, over 125 PRB instal-
lations have been reported worldwide 共ETI 2005a兲. The large ma-
jority of these applications utilize zero-valent iron to promote the
degradation of chlorinated organic compounds.
For chlorinated ethenes such as tertrachloroethylene 共PCE兲,
the zero-valent iron induces rapid transformations within the PRB
that are typically described as a network of coupled first-order
decay reactions 共e.g., Roberts et al. 1996; Scherer et al. 2000兲.
One of the key design tasks is to determine the PRB thickness
共in the direction of groundwater flow兲 needed to provide the
residence time to reduce the concentrations of target compounds
to the desired effluent concentration. The most straightforward
approach to PRB design is to conduct bench or pilot testing to
directly determine the required residence times 共e.g., Gavaskar et
al. 1997, 1998, 2000兲. Alternatively, mathematical modeling can
be used to extrapolate the PRB performance using reaction rate
constants measured from small scale laboratory experiments
and/or obtained from the literature.
For PRB design, a common simplification is to apply a single-
solute transport model to the parent compound, with the produc-
tion of reaction products represented by approximate correction
factors and/or safety factors 共e.g., Tratnyek et al. 1997兲. In some
cases, complex geochemical models have been developed to con-
sider multiple degradation pathways and other chemical consid-
erations such as the impact of mineral precipitation on barrier
performance 共e.g., Yabusaki et al. 2001兲. However, for prelimi-
nary design, it is desirable to maintain an analytical framework
that can be implemented simply and easily, while maintaining a
suitably accurate treatment of the chemical degradation processes.
This paper presents several improvements to current models that
increase the degree of accuracy and conservatism while retaining
a simple analytic framework that can be implemented quickly and
easily.
Transport Models
Current Design Practice
For PRBs based on engineered transformation reactions, transport
through the barrier is typically modeled using a one-dimensional
form of the advective–dispersive–reactive equation 共ADRE兲. For
the transport of a single decaying solute in a one-dimensional
homogeneous porous medium, the governing ADRE is
⳵
c
⳵
t
=−␯
⳵
c
⳵
x
+ D
⳵
2
c
⳵
x
2
− k
⬘
共1兲
where c⫽aqueous phase contaminant concentration 关M/L
3
兴;
t⫽time 关T兴; x⫽distance from the entrance of the PRB 关L兴;
␯⫽interstitial fluid velocity 关L/T兴; D⫽dispersion coefficient
关L
2
/T兴; and k
⬘
⫽first-order decay constant 关l/T兴.
Application of Eq. 共1兲 to a PRB setting is commonly accom-
plished by neglecting the dispersion term and treating the PRB as
an ideal plug flow reactor, which leads to the following simple
design equation 共e.g., Gavaskar et al. 1998; USEPA 1998兲:
c共x = L兲
c
0
= exp
冉
− k
⬘
L
␯
冊
共2兲
where L⫽barrier thickness 关L兴; and c
0
⫽constant contaminant
concentration entering the barrier 关M/L
3
兴.
1
Associate Professor, Dept. of Civil, Structural, and Environmental
Engineering, Univ. at Buffalo, Buffalo, NY 14260 共corresponding
author兲. E-mail: rabideau@eng.buffalo.edu
2
Research Assistant, Dept. of Civil, Structural, and Environmental
Engineering, Univ. at Buffalo, Buffalo, NY 14260.
3
Assistant Professor, Dept. of Civil Engineering, Univ. of Waterloo,
Waterloo, Ontario, Canada N2L 3G1.
Note. Discussion open until April 1, 2006. Separate discussions must
be submitted for individual papers. To extend the closing date by one
month, a written request must be filed with the ASCE Managing Editor.
The manuscript for this paper was submitted for review and possible
publication on August 2, 2004; approved on April 6, 2005. This paper is
part of the Journal of Environmental Engineering, Vol. 131, No. 11,
November 1, 2005. ©ASCE, ISSN 0733-9372/2005/11-1589–1597/
$25.00.
JOURNAL OF ENVIRONMENTAL ENGINEERING © ASCE / NOVEMBER 2005 / 1589

In addition to neglecting dispersion, the form of Eq. 共2兲
implies a constant concentration entrance boundary condition,
and a semi-infinite domain. The velocity term 共␯兲 is frequently
assigned based on the local groundwater flow prior to barrier
construction, although it is sometimes reduced to reflect the
higher porosity associated with iron-based PRBs.
For chlorinated solvents, it is often necessary to consider the
production and transport of toxic reactions products. Retaining
the assumptions of Eq. 共2兲, the resulting problem can be
expressed as a set of coupled linear ordinary differential equations
共ODEs兲
⳵
c
i
⳵
␶
=−k
i
c
i
+
兺
j⫽1
y
ij
k
j
c
j
共3兲
∀i =1,2,¯ n
where n⫽number of solutes in the decay chain; ␶⫽PRB residence
time 共L /␯兲 关T兴; k⫽solute decay constant 关l/T兴; and y
ij
⫽fractional
yield of solute i produced by the decay of solute j.
In practice, the system represented by Eq. 共3兲 is often solved
numerically using a standard stiff ODE solver 共e.g., ETI 2005b兲,
with the barrier thickness L adjusted to reduce the contaminant
concentrations to the cleanup targets. This approach is rigorous
with respect to the PRB chemistry, but simplifies the hydrody-
namics of the PRB by neglecting dispersion and exit boundary
effects. Furthermore, the placement of a high-permeability PRB
may change the regional groundwater flow patterns in a manner
that results in increased flow through the barrier. While the
effect of these factors may be small, the common simplifications
are nonconservative with respect to the predicted concentrations
exiting the barrier 共e.g., Eykholt and Sivavec 1995兲.Inthe
following sections, several modifications are presented to address
these issues while retaining the simple analytic framework.
Governing Equations for Chlorinated Ethenes
Roberts et al. 共1996兲 proposed a conceptual model for the
transformation of chlorinated ethenes by zero-valent iron that has
been widely accepted. Although the general framework includes
both series and parallel reactions 共i.e., daughter products with
multiple parents兲, laboratory bench testing to characterize PCE/
trichloroethylence 共TCE兲 transformation has indicated that the
reactions involving the production of toxic products may be
considered in terms of a single series of transformations
共i.e., negligible “branching”兲共ETI 2005b兲. The resulting ADRE is
given by
⳵
c
i
⳵
t
=−␯
⳵
c
i
⳵
x
+ D
⳵
2
c
i
⳵
x
2
− k
i
c
i
+ y
i
k
i−1
c
i−1
共4兲
∀i =1,2,¯ ,n
where the yield coefficient y
i
contains a single subscript to indi-
cate that reaction product i is produced only by the single parent
compound i−1 关i.e., y
i
is equivalent to y
i,i−1
in Eq. 共3兲, with all
other y
i,j
terms= zero兴.
The restriction of Eq. 共4兲 to a single decay chain does not
imply that other decay pathways are absent, but only that they do
not lead to significant production of the toxic reaction products of
interest. For example, other end products of PCE degradation
could include acetylene and ethylene 共e.g., Roberts et al. 1996兲,
which are typically not of regulatory concern. The form of Eq. 共4兲
also implies that the dispersion coefficients are equal for all
solutes; i.e., the solute-specific contribution of molecular
diffusion is much less than the hydrodynamic component of
dispersion.
The solution to Eq. 共4兲 requires two boundary conditions and
an initial condition. For field conditions, the contaminant source
can again be represented by a constant-concentration 共first type兲
entrance condition
c
i
共x =0,t兲 = c
0i
共5兲
∀i =1,2,¯ ,n
where c
0i
⫽concentration of solute i entering the PRB.
For laboratory columns, which are often operated by measur-
ing “steady-state” resident concentrations along the column
length, the mass-conserving third-type condition may be preferred
c
i
共x =0,t兲 = c
0i
+
D
␯
⳵
c
i
⳵
x
共x =0,t兲
共6兲
∀i =1,2,¯ ,n
For the barrier exit, a popular choice is to treat the PRB as a
semi-infinite medium, which leads to the following boundary
condition:
⳵
c
i
⳵
x
共x = ⬁,t兲 =0
共7兲
∀i =1,2,¯ ,n
The application of Eq. 共7兲 is appealing because it facilitates the
development of closed form solutions for both transient and
steady-state conditions. However, it incorrectly implies that the
reactive properties of the PRB extend into the adjoining aquifer. A
more plausible condition for advection-dominated transport is the
zero-gradient exit condition, which is given by
⳵
c
i
⳵
x
共x = L,t兲 =0
共8兲
∀i =1,2,¯ ,n
where L=thickness of the reactive medium.
In addition to providing a more conceptually appealing
description of the transition occurring at the PRB exit, Eq. 共8兲
leads to higher predicted effluent concentrations when the con-
stant concentration influent condition is used, and thus provides
more conservative predictions for the purpose of PRB design.
For PRB applications, it is common to assign the initial con-
dition based on the assumption that the contaminant is initially
absent within the barrier
c
i
共x,t =0兲 =0
共9兲
∀i =1,2,¯ ,n
Solutions
For design purposes, the consideration of steady-state conditions
is customary and leads to conservative predictions. Closed-form
steady-state solutions to Eq. 共1兲 for several combinations of
boundary conditions are summarized in the Appendix.
Sun et al. 共1999兲 developed a simple transformation procedure
that can be used to extend solutions based on single solute
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transport to the case of multiple solutes subject to sequential first-
order decay. A transformed concentration variable 共a兲 is defined
as
a
i
= c
i
+
兺
j=1
i−1
冋
兿
l=j
i−1
冉
y
l+1
k
l
k
l
− k
i
冊
c
j
册
共10兲
∀i =2,3,¯ ,n
By applying this transformation to Eq. 共4兲, the governing ADRE
for each transformed variable a
i
is reduced to the single-solute
form 关Eq. 共1兲兴. The entrance boundary condition is similarly
transformed as
a
i0
= c
i0
+
兺
j=1
i−1
冋
兿
l=j
i−1
冉
y
l+1
k
l
k
l
− k
i
冊
c
j0
册
共11兲
∀i =2,3,¯ ,n
Once the appropriate single-solute solution to Eq. 共1兲 is identified,
a back transformation is performed to develop closed-form
solutions for each solute in the decay chain
c
i
= a
i
−
兺
j=1
i−1
兿
l=j
i−1
y
l+1
k
l
k
l
− k
i
c
j
共12兲
∀i =2,3,¯ ,n
The multisolute expansions for a four-solute chain are given in
the Appendix, and the resulting steady-state solutions to Eq. 共1兲
can be readily implemented in a spreadsheet. For known influent
concentrations and specified constraints on the effluent concentra-
tions, the design equations can be solved iteratively to determine
the minimum required barrier thickness L.
Example
Consider the case of a PRB designed to treat a plume containing
PCE. Within the PRB, the pathway of interest is the sequential
decay of PCE to TCE, cis1,2,dichlorethylene 共DCE兲, and vinyl
chloride 共VC兲. The federal drinking water standards for PCE,
TCE, DCE, and VC are 5, 5, 70, and 2 ␮g /L, respectively. For
this system, the third reaction product 共VC兲 often controls the
system design and a four-solute model is needed. Fig. 1 shows the
simulated effluent concentrations for a typical PRB system based
on the parameters summarized in Table 1, for a variety of bound-
ary conditions.
Although small differences are apparent between the semi-
infinite and finite boundary conditions, the most significant
difference in predicted PRB performance occurs when dispersion
is included in the model. For advection-dominated groundwater
systems, the dispersion coefficient is typically assumed to be
linearly proportional to the velocity, i.e.,
D ⬃ ␣␯ 共13兲
where ␣=D /␯⫽dispersivity of the medium 关L兴.
Field measurements of ␣ for zero-valent iron systems were not
located in the literature. In recognition of the scale dependence of
dispersion processes, dispersivity values are often estimated as a
fraction of the solute travel distance 共e.g., Gelhar et al. 1992兲. For
the example simulations shown in Fig. 1, ␣ was set at a value
equal to 2% of the domain length 共0.02L兲, based loosely on the
Fig. 1. 共a兲 Predicted permeable reactive barrier performance for
parameters listed in Table 1, with dispersion neglected and
semi-infinite exit boundary; 共b兲 predicted permeable reactive barrier
performance for parameters listed in Table 1, with dispersion
included and finite zero-gradient exit boundary; and 共c兲 predicted
permeable reactive barrier performance for parameters listed in
Table 1, with dispersion included and semi-infinite exit boundary
JOURNAL OF ENVIRONMENTAL ENGINEERING © ASCE / NOVEMBER 2005 / 1591

results of column tests conducted over length scales similar to an
installed PRB 共Sivavec et al. 1996; Casey et al. 2000; Farrell et al.
2000兲.
Comparison of the various solutions indicates that the model
in which dispersion is neglected 关Eq. 共2兲兴 predicts effluent con-
centrations that are significantly lower than the others. Applica-
tion of the finite exit boundary condition 关Eq. 共7兲兴 leads to higher
predicted effluent concentrations, but the resulting difference is
small compared to the effect of neglecting dispersion. To further
explore the impact of dispersion, an optimization was performed
to identify the smallest PRB thickness that would produce effluent
concentrations at or below the MCL values for the transport
parameters given in Table 1. The optimization approach may be
expressed as follows:
Minimize
L = barrier thickness
subject to
c
i
共L兲 艋 c
ei
∀i =1,2,¯ ,n 共14兲
L 艌 L
min
where c
ei
=design effluent concentration for solute i 共e.g., the
drinking water standard兲 and L
min
=minimum barrier thickness
共provided to facilitate convergence兲.
For the reaction parameters given in Table 1, the “baseline” L
value was calculated at 1.05 m for the advection-only model.
For the recommended finite thickness design model 关based on
Eq. 共7兲兴, the calculated L values were 1.10, 1.23, and 1.98 m for
the ␣ /L ratios of 0.005, 0.02, and 0.1, respectively 共other param-
eters retained at Table 1 values兲. Application of the semi-infinite
exit condition yielded slightly smaller L values of 1.09, 1.21, and
1.85 m. It is noteworthy that application of the commonly used
“10%” dispersivity relationship 共e.g., Aziz et al. 2000兲 resulted in
a calculated PRB thickness nearly twice that determined by the
model that neglected dispersion. Application of a more realistic
dispersivity ratio of 2% increased the design PRB thickness by
23% over the pure advection scenario.
Estimating the Velocity Term
Although the inclusion of dispersion significantly impacts the pre-
dicted PRB performance, the most sensitive variables, in general,
are the groundwater velocity 共␯兲 and the reaction constants. The
velocity within the PRB is often estimated based on the precon-
struction groundwater flow patterns. In some cases, a correction
may be applied to account for the higher porosity within the PRB,
which may be 2–3 times the aquifer porosity. A more rigorous
analysis would utilize a numerical model of groundwater flow to
model the local velocity patterns 共e.g., Starr and Cherry 1994;
Gupta and Fox 1999; Das 2002兲. This approach is appealing be-
cause it accounts for the tendency of the high-permeability barrier
to preferentially channel groundwater flow, leading to increased
interior velocities. However, implementation of a numerical
model for this purpose can be cumbersome, and the accuracy of
the computed velocities may be subject to significant errors from
numerical discretization.
For aquifers that are characterized by horizontal regional flow,
an appealing alternative is the analytic element method 共AEM兲,
which can simulate both regional and local hydrologic features
without discretization artifacts 共Strack 1989兲. By configuring a
model to appropriately capture the regional flow features that
control local flow patterns, the degree of flow distortion caused
by the placement of a PRB can be easily quantified. Applications
of the AEM are usually limited to steady-state flow in a single
layer aquifer, which is appropriate for the design of systems
that contain fully penetrating barriers. An advantage of the AEM
is its applicability to confined, unconfined, or partially confined
conditions.
Emplacement of a PRB that has a different hydraulic conduc-
tivity from the regional aquifer can be represented using
“inhomogeneity” elements of various geometries. Fitts 共1997兲
proposed the use of line elements to represent the influence of
emplaced barriers on regional flow field. However, to facilitate
the calculation of the velocity within the PRB, the simple solution
for an elliptical inhomogeneity in uniform flow developed inde-
pendently by Strack 共1989兲 and Obdam and Veling 共1987兲 is a
more appealing choice. The elliptical geometry is a good repre-
sentation of a typical fully penetrating PRB configuration and
avoids the singularities and/or numerical artifacts associated with
modeling barriers using line segments in an analytical or numeri-
cal model. While some distortion of the barrier geometry may
occur at the ends of the ellipse, these areas usually are outside the
reactive zone of interest. Also, when used in a regional model
where the uniform flow assumption is applicable, the elliptical
geometry captures the residence time distribution with good
accuracy, with shorter residence times calculated near the edges
of the PRB. The public domain AEM program Visual Bluebird
共Craig and Matott 2004兲 supports high-order elliptical elements
共as developed by Surlbhatla et al. 2004兲 and provides user-
specified velocity transects useful for characterizing the behavior
of barrier systems.
For preliminary design purposes, a simple equation can be
developed using the conceptual model of a single fully penetrat-
ing elliptical hydraulic conductivity inhomogeneity placed in a
uniform regional flow field. For the conceptual model shown in
Fig. 2, the discharge across the PRB is given by
Table 1. Parameters used in Permeable Reactive Barrier 共PRB兲
Simulation
Parameter Value
PRB thickness 共L兲 1.25 m
Groundwater velocity 共v兲 0.5 m/ day
Scaled dispersivity 共D /L
v
兲
0.02
Decay rates
a
—
PCE 共k
1
兲
3.61 day
−1
TCE 共k
2
兲
5.73 day
−1
cDCE 共k
3
兲
2.97 day
−1
VC 共k
4
兲
3.6101 day
−1
Conversion ratio
a
—
PCE–TCE 共 y
12
兲
0.40
TCE–DC 共 y
23
兲
0.02
DC–VC 共 y
34
兲
0.01
Entrance concentration
a
—
PCE 共c
01
兲 10,000 ␮g/L
TCE 共c
02
兲 1,000 ␮g/L
cDCE 共c
03
兲 100 ␮g/L
VC 共c
04
兲 10 ␮g/L
a
Based on ETI 共2005b兲.
1592 / JOURNAL OF ENVIRONMENTAL ENGINEERING © ASCE / NOVEMBER 2005

Q = 共a + b兲Q
0
冋
k
e
2
sin
2
␥
共bk + ak
e
兲
2
+
k
e
2
cos
2
␥
共ak + bk
e
兲
2
册
1/2
共15兲
where Q =flow per unit width inside the PRB 关L
2
/T兴;
Q
0
=regional flow oriented at an angle ␥ from the longitudinal
axis of the PRB 关L
2
/T兴; a and b= long and short semiaxes of the
elliptical PRB, respectively; and k
e
and k= hydraulic conductivi-
ties of the elliptical PRB and the aquifer, respectively 关L/T兴.
The derivation of Eq. 共15兲 is given in the Appendix. Although
the effect of aquifer recharge is not considered, the effects
of recharge are negligible if the regional flow is estimated from
field observations of the piezometric head, which is typically
accomplished using the following approximation:
Q
0
⬇
h
2
− h
1
⌬x
kh
ave
共16兲
where h
2
and h
1
=observed piezometric heads at upgradient and
downgradient locations 关L兴, respectively; ⌬x =separation distance
in the direction of flow; k=mean aquifer hydraulic conductivity
关L/T兴; and h
ave
=average saturated thickness.
Eq. 共15兲 can be used to examine the influence of PRB configu-
ration on velocity. The most significant increases in the PRB dis-
charge 共relative to the regional flow兲 occur when the PRB is not
oriented perpendicular to the regional flow and when the aspect
ratio 共a/ b兲 is small. However, when the PRB is not perpendicular
to the regional flow, the direction of flow within the PRB and
the resulting distribution of residence times is complex. For the
simpler case of flow perpendicular to the PRB, the velocity
normal to the PRB 共␯兲 can be determined from
Q = 共a + b兲Q
0
冋
k
e
bk + ak
e
册
共17兲
␯ =
Q
␧h
PRB
共18兲
where ␧= porosity of the PRB; h
PRB
=average saturated thickness
within the PRB 关L兴; and ␯= fluid velocity appropriate for use in
the ADRE.
Fig. 3 illustrates the effect of the relative hydraulic conductiv-
ity on the PRB discharge for various aspect ratios 共 a /b兲. For low
aspect ratios, as the relative hydraulic conductivity of the PRB
increases, the PRB exerts a “wicking” effect on the groundwater
flow field and the resulting velocity is approximately 20%
higher than the pre-PRB condition. However, as the PRB aspect
ratio increases, the asymptotic velocity approaches the regional
condition.
Conclusions
This paper presents a set of simple closed-form equations suitable
for the preliminary design of iron-based permeable reactive bar-
riers for commonly encountered scenarios 关e.g., Eqs. 共41兲–共44兲
combined with Eqs. 共16兲–共18兲兴. Sensitivity calculations indicate
that some of the assumptions commonly used in PRB analysis—
negligible dispersion, semi-infinite domain, and negligible flow
channeling—can lead to nonconservative designs.
By including dispersion and a finite reactive domain, the
model reflects a more realistic and conservative representation
than the commonly applied “batch” type equations. At the present
stage of development, the approach is applicable only to a single
reaction pathway. The understanding of PRB reaction chemistry
continues to evolve, and the single pathway approximation may
not be applicable to more complex systems containing other
contaminants such as chlorinated ethanes or other zero-valent
metals such as zinc 共e.g., Arnold and Roberts 1998, 2000兲.If
parallel reactions are included 共i.e., multiple parents contributing
to a reaction product兲, the common approach of neglecting
dispersion converts the problem into a set of coupled ordinary
differential equations that can be solved numerically using a stan-
dard ODE solver or, in some cases, analytically 共e.g., Eykholt
1999兲. However, for the decay of chlorinated ethylene by zero-
Fig. 2. Schematic of conceptual model for fully penetrating elliptical
permeable reactive barrier placed in homogeneous aquifer with
uniform regional flow
Fig. 3. Ratio of barrier flow rate to regional flow rate for various
permeable reactive barrier geometries, for barrier perpendicular to
regional flow. Contrast in conductivity 共k
e
/k兲 and barrier aspect ratio
共a /b兲 control degree of difference between flux within barrier and
regional flow conditions
JOURNAL OF ENVIRONMENTAL ENGINEERING © ASCE / NOVEMBER 2005 / 1593

valent iron, the current understanding of reaction pathways
共e.g., ETI 2005b兲 is consistent with the single pathway approach.
Sample calculations indicate that the predicted PRB perfor-
mance is quite sensitive to the assumed dispersion coefficient.
The potential influence of dispersion is well known in reactor
engineering 共e.g., Levenspiel 1999兲, but has not been a focus of
PRB-related research, perhaps due to the large safety factors
typically used to account for various parameter uncertainties.
Until actual measurements of dispersion coefficients are avail-
able, one promising approach would be to develop models that
explicitly account for dispersion in a probabilistic framework
such as Monte Carlo analysis 共e.g., Vidumsky and Landis 2001兲.
However, as the PRB technology matures, a reliable procedure
for estimating the PRB dispersivity could reduce the contribution
of dispersion to prediction uncertainty, potentially leading to
smaller safety factors.
The other contribution of this work is in the use of elliptical
analytic elements to compute the velocity through an idealized
PRB, in a manner that accounts for the potential increase in
flow through the barrier due to its higher hydraulic conductivity.
Future work will examine analytical approaches for analyzing
PRB hydraulics in more complex flow regimes.
Acknowledgments
This research was partially supported by Grant No. R82-7961
from the United States Environmental Protection Agency’s 共EPA兲
Science to Achieve Results 共STAR兲 program. This paper has
not been subjected to any EPA review and therefore does not
necessarily reflect the views of the Agency, and no official
endorsement should be inferred. James Craig’s graduate study
was supported by the National Science Foundation Integrated
Graduate Education and Research Training 共IGERT兲 program in
Geographic Information Science.
Appendix
Steady-State Solutions to One-Dimensional Multisolute
Advective–Dispersive–Reactive Equation
The application of the transformation developed by Sun et al.
共1999兲 to the multisolute form of the advective–dispersive–
reactive equation 关Eq. 共4兲兴 leads to
⳵
a
i
⳵
t
=−␯
⳵
a
i
⳵
x
+ D
⳵
2
a
i
⳵
x
2
− ka
i
共19兲
∀i =1,2,¯ ,n
To develop the solute-specific solutions, the entrance boundary
conditions for the reaction products are first transformed using
Eq. 共10兲 as follows: For i =2
a
20
= c
20
+
冉
y
2
k
1
k
1
− k
2
冊
c
10
共20兲
For i=3
a
30
= c
30
+
兺
j=1
2
兿
l=j
2
y
i+1
k
l
k
l
− k
3
c
j0
共21兲
a
30
= c
30
+
冉
y
2
k
1
k
1
− k
3
冊冉
y
3
k
2
k
2
− k
3
冊
c
10
+
冉
y
3
k
2
k
2
− k
3
冊
c
20
共22兲
For i=4
a
40
= c
40
+
兺
j=1
3
兿
l=j
3
y
l+1
k
l
k
l
− k
4
c
j0
共23兲
a
40
= c
40
+
冉
y
2
k
1
k
1
− k
4
冊冉
y
3
k
2
k
2
− k
4
冊冉
y
4
k
3
k
3
− k
4
冊
c
10
+
冉
y
3
k
2
k
2
− k
4
冊冉
y
4
k
3
k
3
− k
4
冊
c
20
+
冉
y
4
k
3
k
3
− k
4
冊
c
30
共24兲
The solutions for the solute concentration are then developed
using the back transformation given by Eq. 共11兲. For i =2
c
2
= a
2
−
兺
j=1
1
兿
l=j
1
y
l+1
k
l
k
l
− k
2
c
j
共25兲
c
2
= a
2
−
y
2
k
1
k
1
− k
2
c
1
共26兲
For i=3
c
3
= a
3
−
兺
j=1
2
兿
l=j
2
y
l+1
k
l
k
l
− k
3
c
j
共27兲
c
3
= a
3
−
兿
l=1
2
y
l+1
k
l
k
l
− k
3
c
1
−
兿
l=2
2
y
l+1
k
l
k
l
− k
3
c
2
共28兲
c
3
= a
3
−
冉
y
2
k
1
k
1
− k
3
冊冉
y
3
k
2
k
2
− k
3
冊
c
1
−
y
3
k
2
k
2
− k
3
c
2
共29兲
For i=4
c
4
= a
4
−
兺
j=1
3
兿
l=j
3
y
l+1
k
l
k
l
− k
4
c
j
共30兲
c
4
= a
4
−
兿
l=1
3
y
l+1
k
l
k
l
− k
4
c
1
−
兿
l=2
3
y
l+1
k
l
k
l
− k
4
c
2
−
兿
l=3
3
y
l+1
k
l
k
l
− k
4
c
3
共31兲
c
4
= a
4
−
冉
y
2
k
1
k
1
− k
4
冊冉
y
3
k
2
k
2
− k
4
冊冉
y
4
k
3
k
3
− k
4
冊
c
1
−
冉
y
3
k
2
k
2
− k
4
冊冉
y
4
k
3
k
3
− k
4
冊
c
2
−
y
4
k
3
k
3
− k
4
c
3
共32兲
For PRB design, it is useful to consider the simplified
case where the reaction products are not initially present
共i.e., c
20
=c
40
=c
30
=0兲. For this case, the transformed boundary
conditions are
a
40
=
冉
y
2
k
1
k
1
− k
4
冊冉
y
3
k
2
k
2
− k
4
冊冉
y
4
k
3
k
3
− k
4
冊
c
10
共33兲
a
30
=
冉
y
2
k
1
k
1
− k
3
冊冉
y
3
k
2
k
2
− k
3
冊
c
10
共34兲
a
20
=
冉
y
2
k
1
k
1
− k
2
冊
c
10
共35兲
1594 / JOURNAL OF ENVIRONMENTAL ENGINEERING © ASCE / NOVEMBER 2005

Steady-State Solutions
The solutions given above are applicable to transient and steady-
state transport, for a variety of boundary conditions. For the
design of barrier systems, several steady-state solutions are
potentially applicable 共e.g., van Genuchten 1981兲.
For the constant concentration entrance condition 关Eq. 共5兲兴 and
zero gradient exit condition 关Eq. 共8兲兴 the steady-state solution to
Eq. 共19兲 is
a
i
共x兲 = a
0i
exp
冋
共␯ − u
i
兲x
2D
册
+
冉
u
i
− ␯
u
i
+ ␯
冊
exp
冋
共␯ + u
i
兲x
2D
−
u
i
L
D
册
1+
冉
u
i
− ␯
u
i
+ ␯
冊
exp
冉
− u
i
L
D
冊
共36兲
where
u
i
=
冑
␯
2
+4k
i
D 共37兲
For the constant concentration entrance condition and semi-
infinite exit 关Eq. 共7兲兴, the solution is
a
i
= a
i0
exp
冋
x
␯ −
冑
␯
2
+4k
i
D
2D
册
共38兲
If dispersion is ignored, the solution for the constant concentra-
tion entrance condition and semi-infinite exit is
a
i
= a
i0
exp
冋
−
k
i
x
␯
册
共39兲
For the third-type entrance boundary condition and the semi-
infinite exit, the solution is
a
i
= a
0i
exp
冋
共␯ − u
i
兲x
2D
册
+
冉
u
i
− ␯
u
i
+ ␯
冊
exp
冋
共␯ + u
i
兲x
2D
−
u
i
L
D
册
u
i
+ ␯
2␯
−
共u
i
− ␯兲
2
2␯共u
i
+ ␯兲
exp
冉
− u
i
L
D
冊
共40兲
Expanded Solution for Permeable Reactive Barrier Design
Solutions for particular applications can be developed by combin-
ing the single-solute solutions given by Eqs. 共36兲–共40兲 with the
multisolute expansions given by Eqs. 共20兲–共35兲. The full
expansion is provided here for the recommended PRB design
Eq. 共36兲 and the special case where reaction products are initially
absent Eqs. 共33兲–共35兲. Application to other scenarios is straight-
forward. The resulting equations can be applied to PRB design by
evaluating the concentration at the barrier exit 共x=L 兲.
c
1
= c
10
exp
冋
共␯ − u
1
兲x
2D
册
+
冉
u
1
− ␯
u
1
+ ␯
冊
exp
冋
共␯ + u
1
兲x
2D
−
u
1
L
D
册
1+
冉
u
1
− ␯
u
1
+ ␯
冊
exp
冉
− u
1
L
D
冊
共41兲
c
2
=
y
2
k
1
k
1
− k
2
c
10
exp
冋
共␯ − u
2
兲x
2D
册
+
冉
u
2
− ␯
u
2
+ ␯
冊
exp
冋
共␯ + u
2
兲x
2D
−
u
2
L
D
册
1+
冉
u
2
− ␯
u
2
+ ␯
冊
exp
冉
− u
2
L
D
冊
−
y
2
k
1
k
1
− k
2
c
1
共42兲
c
3
=
冉
y
2
k
1
k
1
− k
3
冊冉
y
3
k
2
k
2
− k
3
冊
c
10
⫻
exp
冋
共␯ − u
3
兲x
2D
册
+
冉
u
3
− ␯
u
3
+ ␯
冊
exp
冋
共␯ + u
3
兲x
2D
−
u
3
L
D
册
1+
冉
u
3
− ␯
u
3
+ ␯
冊
exp
冉
− u
3
L
D
冊
−
y
3
k
2
k
2
− k
3
c
2
−
冉
y
2
k
1
k
1
− k
3
冊冉
y
3
k
2
k
2
− k
3
冊
c
1
共43兲
c
4
=
冉
y
2
k
1
k
1
− k
4
冊冉
y
3
k
2
k
2
− k
4
冊冉
y
4
k
3
k
3
− k
4
冊
c
10
⫻
exp
冋
共␯ − u
4
兲x
2D
册
+
冉
u
4
− ␯
u
4
+ ␯
冊
exp
冋
共␯ + u
4
兲x
2D
−
u
4
L
D
册
1+
冉
u
4
− ␯
u
4
+ ␯
冊
exp
冉
− u
4
L
D
冊
−
冉
y
2
k
1
k
1
− k
4
冊冉
y
3
k
2
k
2
− k
4
冊冉
y
4
k
3
k
3
− k
4
冊
c
1
−
冉
y
3
k
2
k
2
− k
4
冊冉
y
4
k
3
k
3
− k
4
冊
c
2
−
冉
y
4
k
3
k
3
− k
4
冊
c
3
共44兲
Solution for Groundwater Flow Influenced
by Elliptical Element
The solution for groundwater flow influenced by an elliptical
inhomogeneity may be found in Strack 共1989兲 or Obdam and
Veling 共1987兲, and is a simple case of the general solution devel-
oped by Suribhatla et al. 共2004兲. The development is based on the
AEM 共Strack 1989兲 and is applicable to steady-state horizontal
flow in a confined or unconfined aquifer with a uniform isotropic
regional hydraulic conductivity 共k兲 and a single elliptical zone
of a different hydraulic conductivity 共k
e
兲. Uniform flow 共Q
0
兲 is
oriented at an angle ␥ to the x axis as shown in Fig. 2. The
long and short semiaxes a ,b of the elliptical inhomogeneity are
oriented along the x and y axes, respectively. There is no recharge
anywhere in the model domain.
The complex potential ⍀
u
due to the uniform flow in elliptical
coordinates may be expressed as
⍀
u
=−Q
0
d cosh ␶ · e
−i␥
+ ⌽
0
共45兲
where d =focal distance of the elliptical inhomogeneity; and
⌽
0
=potential at the reference location. The transformed spatial
coordinate ␶ is defined in terms of the local elliptical coordinates
共Suribhatla et al. 2004兲 as
␶ = ␩ + i␸ 共46兲
where ␩= constant along ellipses with the same foci and
␸=constant along hyperbolas orthogonal to those ellipses.
The complex potential due to the elliptical inhomogeneity
itself is
JOURNAL OF ENVIRONMENTAL ENGINEERING © ASCE / NOVEMBER 2005 / 1595

⍀
e
=
冦
兺
n=0
⬁
␤
n
共e
n␶
+ e
−n␶
兲
inside 共␩⬍␩
0
兲
兺
n=1
⬁
共␤
n
−
␤
n
e
2n␩
0
兲 · e
−n␶
outside 共␩艌␩
0
兲
冧
共47兲
The complex coefficients ␤
n
=␤
n
R
+i␤
n
I
can be determined using
the following integrals:
␤
0
=
k
e
− k
4k␲
冕
−␲
␲
⌽
⫽e
共␩
0
,␸兲d␸ 共48兲
␤
n
R
=
k
e
− k
2␲关k cosh共n␩
0
兲 + k
e
sinh共n␩
0
兲兴
冕
−␲
␲
⌽
⫽e
共␩
0
,␸兲 · cos 共n␸兲d␸
共49兲
␤
n
I
=
k
e
− k
2␲关k sin共n␩
0
兲 + k
e
cosh共n␩
0
兲兴
冕
−␲
␲
⌽
⫽e
共␩
0
,␸兲 · sin共n␸兲d␸
共50兲
The quantity ⌽
⫽e
共␩
0
,␸兲 in the above equations is the potential
due to all the elements except for the inhomogeneity at the bound-
ary of the elliptical inhomogeneity where ␩ =␩
0
. For the case of
interest, ⌽
⫽e
共␩
0
,␸兲 equals the potential due to the uniform regional
flow only. Taking the real part of Eq. 共45兲 and solving for the
coefficients yields
␤
0
=
k
e
− k
2k
⌽
0
共51兲
␤
1
R
=
共k − k
e
兲Q
0
d cosh ␩
0
· cos ␥
2关k cosh ␩
0
+ k
e
sinh ␩
0
兴
共52兲
␤
1
I
=
− 共k − k
e
兲Q
0
d sinh ␩
0
· sin ␥
2关k sinh ␩
0
+ k
e
cosh ␩
0
兴
共53兲
Substituting the coefficients into Eq. 共47兲 and combining the
influence of uniform flow, the complex potential at any point
inside the elliptical inhomogeneity is
⍀ =−dQ
0
cosh ␶e
−i␥
+ ⌽
0
+2␤
0
+ ␤
1
共e
␶
+ e
−␶
兲共54兲
It is useful to consider the flow inside the PRB in terms of the
complex discharge function W, which is defined as the derivative
of the complex potential with respect to the complex Cartesian
coordinate z=x +iy 共Strack 1989兲
W =−
d⍀
dz
共55兲
The discharge function inside the ellipse becomes
W = Q
0
e
−i␥
+
1
a sinh ␶
兺
n=1
⬁
n␤
n
共e
−␩␶
− e
␩␶
兲 = Q
0
e
−i␥
+
−2
d
关␤
1
R
+ i␤
1
I
兴
= Q
0
冋
共k
e
− k兲cosh ␩
0
cos ␥
k cosh ␩
0
+ k
e
sinh ␩
0
− i
共k
e
− k兲sinh ␩
0
cos ␥
k sinh ␩
0
+ k
e
cosh ␩
0
+ e
−i␥
册
共56兲
This expression for complex discharge can be simplified using the
following geometric relationships:
a = d cosh ␩
0
共57a兲
b = d sinh ␩
0
共57b兲
leading to an expression for the discharge within the ellipse
W =
Q
0
k
e
共a + b兲cos ␥
ak + bk
e
− i
Q
0
k
e
共a + b兲sin ␥
ak
e
+ bk
共58兲
The magnitude of flow inside the elliptical inhomogeneity is
given as the absolute value of Eq. 共58兲
Q = 共a + b兲Q
0
冋
k
e
2
sin
2
␥
共bk + ak
e
兲
2
+
k
e
2
cos
2
␥
共ak + bk
e
兲
2
册
1/2
共59兲
When the ratio of the conductivities of the ellipse to the back-
ground 共k
e
/k兲 goes to infinity 共highly conductive barrier兲,
the total flow inside the ellipse reaches the value as given by
Q = 共a + b兲Q
0
冋
sin
2
␥
a
2
+
cos
2
␥
b
2
册
1/2
共60兲
For the case of flow perpendicular to the PRB 共␥=90°兲,
the discharge becomes
Q = 共a + b兲Q
0
冋
k
e
bk + ak
e
册
共61兲
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