Journal of Contaminant Hydrology 134–135 (2012) 54–68
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Journal of Contaminant Hydrology
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Analytical solutions of one-dimensional multispecies reactive transport in apermeable reactive barrier-aquifer system
John Mieles a, Hongbin Zhan a,b,⁎a Department of Geology and Geophysics, Texas A&M University, College Station, TX77843-3115, USAb Faculty of Engineering and School of Environmental Studies, China University of Geosciences, Wuhan, Hubei 430074, PR China
a r t i c l e i n f o
⁎ Corresponding author at: Department of GeologyA&M University, College Station, TX77843-3115, US7961; fax: +1 979 845 6162.
E-mail addresses: [email protected] ([email protected] (H. Zhan).
0169-7722/$ – see front matter © 2012 Elsevier B.V. Adoi:10.1016/j.jconhyd.2012.04.002
a b s t r a c t
Article history:Received 6 October 2011Received in revised form 31 March 2012Accepted 2 April 2012Available online 13 April 2012
The permeable reactive barrier (PRB) remediation technology has proven to be more cost-effective than conventional pump-and-treat systems, and has demonstrated the ability torapidly reduce the concentrations of specific chemicals of concern (COCs) by up to severalorders of magnitude in some scenarios. This study derives new steady-state analyticalsolutions to multispecies reactive transport in a PRB–aquifer (dual domain) system. Theadvantage of the dual domain model is that it can account for the potential existence of naturaldegradation in the aquifer, when designing the required PRB thickness. The study focusesprimarily on the steady-state analytical solutions of the tetrachloroethene (PCE) serialdegradation pathway and secondly on the analytical solutions of the parallel degradationpathway. The solutions in this study can also be applied to other types of dual domain systemswith distinct flow and transport properties. The steady-state analytical solutions are shown tobe accurate and the numerical program RT3D is selected for comparison. The results of thisstudy are novel in that the solutions provide improved modeling flexibility including: 1) everyspecies can have unique first-order reaction rates and unique retardation factors, and2) daughter species can be modeled with their individual input concentrations or solely asbyproducts of the parent species. The steady-state analytical solutions exhibit a limitation thatoccurs when interspecies reaction rate factors equal each other, which result in undefinedsolutions. Excel spreadsheet programs were created to facilitate prompt application of thesteady-state analytical solutions, for both the serial and parallel degradation pathways.
© 2012 Elsevier B.V. All rights reserved.
Keywords:Multispecies reactive transportIn situ remediationPRB design equationsModeling
1. Introduction
In recent years, the permeable reactive barrier (PRB)technology has proven capable of rapidly reducing theconcentration of some chemicals of concern (COCs) by upto several orders of magnitude (EPA, 1998). The purpose of aPRB is not to treat large volumes of an impacted aquifer, butrather to manage plume concentrations as groundwaterflows towards receptors or off-site. In its simplest form, a
and Geophysics, TexasA. Tel.: +1 979 862
Mieles),
ll rights reserved.
continuous PRB is a vertically emplaced rectangular porousmedium in which influent groundwater passively enters atreatment zone. The reactive media contained in thetreatment zone varies depending on the COC being treated,however the most common media is zero-valent iron(Gavaskar et al., 2000). As illustrated in Fig. 1, a continuousPRB is constructed with only the treatment zone, whereas thefunnel-and-gate PRB (not illustrated here) is constructedwith flanking impermeable walls that funnel capturedgroundwater to the treatment zone (or the gate).
As the groundwater flows through the treatment zone, thedissolved COCs come in contact with the reactive material andare rapidly degraded (EPA, 1998). The effluent groundwatercontains significantly lower concentrations as it re-enters theaquifer and flows towards the plane of compliance (POC), as
Fig. 1. Schematic of installed continuous PRB, solute plume, and aquifer. The xaxis is along the groundwater flow direction and the PRB is of thickness B. Thedown-gradient PRB–aquifer interface is located at x=0and xcomp is the plane ofcompliance (POC), which could be a property boundary or a predeterminedlocation where solute concentrations must achieve a target concentration.Figure adapted from Park and Zhan (2009) (with permission from AmericanGeophysical Union).
55J. Mieles, H. Zhan / Journal of Contaminant Hydrology 134–135 (2012) 54–68
illustrated in Fig. 1. It should be noted that effective implemen-tation of a PRB relies on accurate site characterization toidentify the existing COCs, their interactions, and their requiredresidence time in the PRB and down-gradient aquifer. Note thatthe down-gradient aquifer may also contribute to additionalCOC degradation through natural attenuation processes. Ensur-ing adequate residence time in a PRB–aquifer system allowsCOCs sufficient time to degrade, improving the likelihood thatregulatory or target concentrations are achieved at the POC.Therefore, minimizing performance uncertainties (such asinadequate barrier thickness) in the preliminary design phaseis critical in avoiding underperformance of the PRB.
To address design uncertainties, prior research has focusedprimarily on better understanding 1) the geochemistry of thePRB material (Allen-King et al., 1997; Arnold and Roberts,2000; Johnson et al., 1996; Roberts et al., 1996; Tratnyek et al.,1997), 2) flow characteristics (Gupta and Fox, 1999), and 3) theminimum PRB thickness as determined from current designequations (Eykholt, 1997; Park and Zhan, 2009; Rabideau et al.,2005). The one-dimensional (1D) design equations presentedby Eykholt (1997), Rabideau et al. (2005), and Park and Zhan(2009) utilize the advection–dispersion equation (ADE) withthe first-order reaction as the governing equation(s) but differprimarily in their application of the boundary conditions. Forexample, the two equations of Eykholt (1997) were derivedusing van Genuchten's (1981) analytical solutions of the ADEwith a first-type boundary condition C(x, t)|inlet=Cin at theinfluent face of the PRB and a semi-infinite boundary condition∂C∂x ∞; tð Þ ¼ 0 at the effluent face of the PRB, where C representsconcentration [ML−3], t is time [T], x is the spatial coordinatealong the flow direction [L], and Cin is the concentration at theinlet boundary [ML−3]. The two solutions of Rabideau et al.(2005)were derived using the Sun et al. (1999) transformationprocedure: the first assumed a third-type influent boundarycondition and semi-infinite effluent condition, while thesecond solution assumed a first-type influent condition andzero concentration gradient ∂C x;tð Þ
∂x outlet ¼ 0j effluent condition.The solutions of Eykholt (1997) and the second solution of
Rabideau et al. (2005) tend to overestimate mass in the PRBsystem (particularly at early time) by assuming that theconcentration gradient across the influent boundary isinitially zero (van Genuchten and Parker, 1984; Wexler,1992). However, the second solution of Rabideau et al. (2005)with a finite PRBwidth forces a zero concentration gradient atthe exit face of the PRB; hence it is useful in that it yields thelargest PRB width (Park and Zhan, 2009). Alternatively, thefirst solution of Rabideau et al. (2005) with the third-type ortotal flux influent condition (Kreft and Zuber, 1978) is moreaccurate; however, it incorrectly implies that PRBs have largethicknesses due to the semi-infinite effluent condition.Additionally, the Eykholt (1997) and Rabideau et al. (2005)design equations are of limited use because they are unable toaccount for the entirely distinct flow and chemical processesoccurring in the down-gradient aquifer. For example, solutedegradation in a PRB is typically induced by a strong (abiotic)reaction while the reaction in aquifers tends to be weaker(biologically driven) natural attenuation (EPA, 1998). As such,these solutions are unable to model the solute concentrationat the down-gradient POC in the aquifer.
Both the first solution of Rabideau et al. (2005) andthe Park and Zhan (2009) solution consider the dissolved
influent solute to be well-mixed and therefore describedcompletely by the advective mass flux condition or flowingconcentration flux, which upon entering the PRB is subject todispersive and advective fluxes. Hence, the third-typeboundary condition is more physically sound and tends toconserve mass (van Genuchten and Parker, 1984) at the inletboundary when applied to the ADE, without reaction. Inreality, before entering the PRB, the influent solute is notwell-mixed due to the porous nature of the up-gradientaquifer. Unlike the Rabideau et al. (2005) and Park and Zhan(2009) studies, this study includes the influent boundarycondition that considers dispersive and advective fluxes inthe up-gradient aquifer.
The difference between the first solution of Rabideau et al.(2005) and that of Park and Zhan (2009) is that Park and Zhan(2009) assumed a finite PRB width, maintained total flux andconcentration continuity at the PRB–aquifer interface, andassigned a separate governing equation to the aquifer, whichpermitted modeling solute concentrations at the POC. The Parkand Zhan (2009) solution, however, is limited to one reactivespecies in the PRB–aquifer system. Most groundwater plumeshave multiple chemicals present and many plumes havereactive solutes which decay to produce daughter chemicals.A common example is tetrachloroethene (PCE), which degradesto produce trichloroethene (TCE), then dichloroethene (DCE),with vinyl chloride (VC) as the final chlorinated daughterproduct. Given this limitation, the objective of this study is toexpand the Park and Zhan (2009)model to handlemultispeciesreactive transport in the PRB–aquifer system. Furthermore, theinlet boundary condition of Park and Zhan (2009) (see Eq. (3)there) has been modified to include both advective anddispersive fluxes. The results will focus on the closed-formsteady-state analytical solutions of the aquifer, but it is notedthat the transient solutions can be extracted from thesupplemental derivations withminimal effort and programmedinto a numerical Laplace inversion algorithm. Due to the lengthof the solutions, an electronic supplement is provided. TwoExcel programs were preprogrammed with the steady-stateanalytical solutions of the serial and parallel degradationpathways and are available upon request.
56 J. Mieles, H. Zhan / Journal of Contaminant Hydrology 134–135 (2012) 54–68
2. Model development—serial degradation solutions
Fig. 1, adapted from Park and Zhan (2009), is a schematic diagram illustrating the modeled assumption of 1D solute transportperpendicular to the installed PRB and down-gradient aquifer. The PRB is continuous and fully saturated with the up-gradient anddown-gradient surfaces positioned normal to the x axis, the direction of groundwater flow. The thickness of the PRB is B, with thePRB and down-gradient aquifer interface at x=0 representing the coordinate system origin, and xcomp representing apredetermined location where solute concentrations must achieve a regulatory limit; i.e., the POC.
Although the PRB–aquifer interface exhibits a common specific discharge (q), the PRB has higher porosity than the adjacentaquifer, and as such the pore water velocity is lower in the PRB (Gavaskar et al., 2000). This study considers only first-orderreaction kinetics since experimental studies have found that halogenated COCs in contact with zero-valent iron tend to degradevia first-order kinetics (Johnson et al., 1996). The PRB's ability to induce rapid degradation is the key to its design. Therefore, whenexamining the results of this study one should focus on the large difference between the reaction rates of the PRB and the down-gradient aquifer. Natural degradation that may be occurring in the up-gradient aquifer is not critical to the model sinceconcentrations at the effluent face of the PRB and in the down-gradient aquifer will ultimately determine if compliance isachieved at the POC. Ideally, the designer will use field data with known COC concentrations to model the influent PRBconcentrations at the proposed location of the PRB.
The 1D assumption to model the depiction in Fig. 1 is based on a few considerations. First, the 1D system works best for a PRBsystem whose dimension in the transverse horizontal direction (perpendicular to the groundwater flow direction) is largeenough to warrant the 1D approximation. The PRB is also assumed to be fully penetrating to the bottom no-flow boundary of theaquifer. Examples of such PRBs have been installed in many countries, including the USA (EPA, 1999) and Germany (Birke et al.,2003). Second, the 1Dmodel is a conservative approach since transverse dispersion will dilute the effluent concentration. In otherwords, in the 2D or 3D model the concentration calculated at the POC will always be smaller than that calculated from the 1Dmodel, given the same chemical reaction conditions. This implies that if the COC concentrations achieve regulatory complianceusing a 1D model then regulatory compliance will also be achieved using 2D or 3D models. In this sense, the 1D model provides ahigher factor of safety for the design thickness of a PRB, but at the expense of increased construction costs. Third, the 1D modelsimplifies the mathematical complexity of dealing with 2D or 3D transport equations; this enables the analytical analyses ofmultispecies reactive processes.
The 1D model has been extended to 2D/3D models by several investigators but also with some new restrictions. For instance,Craig et al. (2006), and Klammler and Hatfield (2008, 2009) have used the analytical element method to deal with 2D flow inPRB–aquifer systems in which the flow direction did not need to be perpendicular to the face of the PRB. However, these studiesconsidered advective flux only, and thus cannot handle dispersion and multispecies reactions. Also, numerical models are usuallyrequired for dealing with multi-dimensional flow and transport in a heterogeneous RPB–aquifer system (Bilbrey and Shafer,2001; Starr and Cherry, 1994). When conducting the numerical simulations, one must pay great attention to the PRB–aquiferinterfaces which are most susceptible to numerical errors due to abrupt changes in physical and chemical properties such aspermeability, dispersivity, and reaction rate (Zhan et al., 2009a,b).
For the PRB–aquifer system modeled here, the governing equation for the first parent species in the PRB is
RBi∂CBi
∂t ¼ L CBið Þ−λBiCBi;L CBið Þ ¼ DB∂2CBi
∂x2−uB
∂CBi
∂x ; ð1Þ
where the subscript “B” denotes the PRB, L is the advection-dispersion operator, and i=1 represents the first parent species. Forsubsequent daughter species the governing equation in the PRB is
RBi∂CBi
∂t ¼ L CBið Þ−λBiCBi þ yiλBi−1CBi−1; ð2Þ
where i=2, 3,…,n represent the species chain reaction in Fig. 2, DB is the longitudinal dispersion coefficient [L2T−1] in the PRB, uBis the groundwater flow velocity [LT−1], RBi is the species-dependent retardation factor for linear sorption [-], CBi is the speciesconcentration [ML−3], λBi is the species-dependent first-order reaction rate constant [T−1], and yi is the stoichiometric yieldfactor [–]. It is worthwhile to note that the reaction rate constant λBi used in this study includes the first-order decay constants inboth the liquid and solid phases, as presented in van Genuchten (1981). van Genuchten (1981) provided a formula (see Eq. (5)there) to calculate the combined (effective) decay based on the decay constants in the liquid and solid phases, the bulk density of
Y4 Y3 TCE(SP2)
DCE(SP3)
VC(SP4)Y2
PCE(SP1)
Hydrogenolysis Pathway of PCE Decay
Fig. 2. Conceptual view of the serial degradation pathway of PCE. Y denotes the stoichiometric yield of the daughter product from the parent chemical.
57J. Mieles, H. Zhan / Journal of Contaminant Hydrology 134–135 (2012) 54–68
the media, the distribution coefficient, and the porosity. All parameters in Eqs. (1) and (2) pertain to the PRB region; i.e., thedistance −B≤x≤0.
At the influent face of the PRB, one has
nupDup
∂Ciup
∂x −qCiup
!jx¼−B−
¼ −qCiintð Þ ¼ nBDB
∂CBi
∂x −qCBi
� �jx¼−Bþ
ð3Þ
where nB is the PRB porosity, nup is the up-gradient aquifer porosity, Dup is the up-gradient aquifer longitudinal dispersion, andCiup is the up-gradient aquifer concentration which is a known function of x and t. Note that in Eq. (3) of Park and Zhan (2009),Ciin(t) is used as a time-dependent function, but in this study Ciin(t) is calculated from Ciup (see Eq. (3) above), where the latter isa function of x and t. In other words, if the source area concentration (Ciup) and the concentration gradient in the up-gradientaquifer are known, then the effective Ciin can be determined. Therefore, from a mathematical perspective, the solution derivingprocedures outlined in Park and Zhan (2009) are directly applicable here. If dispersion in the up-gradient aquifer is neglected,then Ciup=Ciin and the well-mixed inlet boundary condition used in Park and Zhan (2009) applies. In this case the modeler isassuming a well-mixed inlet concentration is appropriate. Lastly, it should be noted that in later sections of this study and inthe excel programs, it is assumed that Ciin has already been determined from Ciup.
Since groundwater exiting the PRB enters a medium with entirely different chemical and flow processes, a separate ADE isrequired. Therefore, the governing equations of multispecies reactive transport in the down-gradient aquifer are
RLi∂CLi
∂t ¼ L CLið Þ−λLiCLi;L CLið Þ ¼ DL∂2CLi
∂x2−uL
∂CLi
∂x ; i ¼ 1; ð4Þ
RLi∂CLi
∂t ¼ L CLið Þ−λLiCLi þ yiλLi−1CLi−1; i ¼ 2; 3;… ð5Þ
where DL is the longitudinal dispersion coefficient [L2T−1] in the aquifer, uL is the groundwater velocity [LT−1], RLi is the species-dependent retardation factor [–], λLi is the species-dependent first-order reaction ratewhich (similar to λBi described above) includesthe first-order decay constants in both the liquid and solid phases [T−1], and CLi is the species concentration [ML−3]. Note that Eqs. (4)and (5)—denotedwith subscript “L”—pertain to the aquifer region 0bxb∞, but only Eq. (5) governs the transport of daughter species.It is also noted that groundwater flow velocities in the PRB and the aquifer can be determined from a common specific discharge (q)such that uBnB=uLnL=q, where nB and nL are the porosities of the PRB and the aquifer, respectively. Additionally, themodel assumesthe PRB and the down-gradient aquifer are initially free of contamination, thus the initial condition is CBi(t=0)=CLi(t=0)=0.Lastly, since the PRB–aquifer interface physically represents the transition between two types of porous media, the proper boundaryconditions to maintain at x=0 are the continuity of solute concentration and the continuity of total mass flux; i.e.,
CBi x¼0 ¼ CLi x¼0;jj ð6Þ
nBDB∂CBi
∂x −qCBi
� �x¼0
¼ nLDL∂CLi
∂x −qCLi
� �x¼0
;
�������� ð7Þ
and at the positive infinity limit of the aquifer a no-flux boundary condition is assumed
∂CLi
∂x x¼∞¼ 0:
���� ð8Þ
To facilitate computation, the following dimensionless parameters are introduced:
CBDi ¼CBi
CS;CDi ¼
CLi
CS; xD ¼ x
B; tD ¼ DB
B2 t;uBD ¼ uBBDB
;uLD ¼ uLBDB
; qD ¼ qBDB
; δ ¼ DL
DB;λBDi ¼
B2
DBλBi;λLDi
¼ B2
DBλLi; θ ¼ nL
nB; andp ¼ B2
DBs;
ð9Þ
where the subscript “D” denotes a dimensionless term, s and p are the dimensional [T−1] and dimensionless Laplace transformvariables that will be used below, and CS is the solubility [ML−3] of the first species which serves simply as a referenceconcentration. An alternative reference concentration can be used if desired. Applying the dimensionless parameters defined inEq. (9) and the Laplace transform to Eqs. (1), (2), (4), and (5) results in the following governing equations:
d2 �CBDi
dxD2 −uBD
d�CBDi
dxD− RBipþ λBDið Þ�CBDi ¼ 0; i ¼ 1; ð10Þ
58 J. Mieles, H. Zhan / Journal of Contaminant Hydrology 134–135 (2012) 54–68
d2 �CBDi
dxD2 −uBD
d�CBDi
dxD− RBipþ λBDið Þ�CBDi ¼ −yiλBDi−1
�CBDi−1; i ¼ 2; 3;… ð11Þ
δd2 �CLDi
dxD2 −uLD
d�CLDi
dxD− RLipþ λLDið Þ�CLDi ¼ 0; i ¼ 1; ð12Þ
δd2 �CLDi
dxD2 −uLD
d�CLDi
dxD− RLipþ λLDið Þ�CLDi ¼ −yiλLDi−1
�CLDi−1; i ¼ 2; 3;… ð13Þ
where the overbar denotes variables in the Laplace domain. Notice that the Laplace transform procedure eliminated the timedependency of the original partial differential equations (PDEs). The resulting equations are second-order linear ordinarydifferential equations (ODEs), which depend only on the distance from the source concentration (Nagle et al., 2000). For example,if in Eqs. (11) and (13) i is the fourth species then i−1 denotes the third species, which carries the solutions of species one andtwo. Applying Eq. (9) and the Laplace transform to the boundary conditions yields
∂�CBDi
∂xD−uBD
�CBDi
� �xD¼−1
¼∂�CDiup
∂xD−uBD
�CDiup
!xD¼−1
¼ −uBD�CDiin
;
���������� ð14Þ
�CBDixD¼0
¼ �CLDixD¼0
;
�������� ð15Þ
d�CBDi
dxD−qD �CBDi
� �xD¼0
¼ θδd�CLDi
dxD−qD �CLDi
� �xD¼0
;
���������� ð16Þ
d�CLDi
dxDxD¼∞ ¼ 0:��� ð17Þ
From Park and Zhan (2009), the general solution of Eqs. (10) and (12) in the Laplace domain are
�CBDi ¼ expuBDxD2
� �a exp υDixDð Þ þ b exp −υDixDð Þ½ �; ð18Þ
�CLDi ¼ cexpuLD−2ωDi
2δxD
� �; ð19Þ
where υDi pð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuBD
2=4þ RBipþ λBDi
qand ωDi pð Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuLD
2=4þ δ RLipþ λLDið Þq
, and undetermined coefficients a, b, and c dependon the boundary conditions. Note that Eq. (19) neglects the positive root of the characteristic equation because the concentrationgradient at the far edge of the down-gradient aquifer must be zero, in accordance with boundary condition (17). After applyingthe boundary conditions, the solution of Eq. (19) for species i=1 is
�CLDi�CDiin
pð Þ ¼ 2υDiuBDEDi expuLD−2ωDi
2δxD þ uBD
2
� �; i ¼ 1; ð20Þ
where EDi=1/[(υDiuBD+2θωDiυDi)cosh(υDi)+(θωDiuBD+2υDi2)sinh(υDi)]. The complete derivation of Eq. (20) is included in Parkand Zhan (2009). Note that the transient solution of Eq. (20) can be inverted numerically to yield a solution in the time domain.The steady-state analytical solution, however, can be determined if CDiin is constant instead of time-dependent. This is doneby letting p→0 (equivalent to tD→∞), applying the final value theorem lim
tD→∞F tDð Þ ¼ lim
p→0pf pð Þ (Dyke, 1999), and assigning
�CDiinpð Þ→CDiin
=p. The resulting steady-state aquifer solution of Eq. (20) is
CLDi ¼ 2υ0DiuBDE0DiCDiinexp
uLD−2ω0Di
2δxD þ uBD
2
� �; ð21Þ
where E0Di now incorporates υ0Di ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuBD
2=4þ λBDi
qand ω0Di ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuLD
2=4þ δλLDi
q. The PRB solution is not included here, but is
presented in Park and Zhan (2009). Notice that the PRB solution — although necessary for derivational purposes — is not the endgoal. Ultimately, the designer desires to know the solute concentration at the location of the selected POC for a selected PRBthickness. Only the aquifer solution provides this flexibility.
59J. Mieles, H. Zhan / Journal of Contaminant Hydrology 134–135 (2012) 54–68
For design purposes Eq. (21) is converted to its dimensional form
CLi ¼ 2υiuBEiCi expuL−2DLωi
2DLxþ uBB
2DB
� �; ð22Þ
where Ei=1/[(υiuB+2DLθωiυi)cosh(Bυi)+(θδωiuB+2DBυi2)sinh(Bυi)], and υi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuB
2=4DB2 þ λBi=DB
qand ωi ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuL
2=4DL2þλLi=DL
q.
As noted by Park and Zhan (2009), Eq. (22) can be used to determine the species concentration at the down-gradient POC given a PRBof thickness B. In its present form Eq. (22) cannot be rearranged to explicitly solve for B. However, assuming that the strong first-orderreaction in a PRB dominates over dispersion, then one has λBi>>DB and Bvi>>1. This in turn simplifies the hyperbolic function suchthat cos h(Bυi)≈sin h(Bυi)≈exp(Bυi)/2, and therefore Ei=2/exp(Bυi)(υiuB+2DLθωiυi+θδωiuB+2DBυi2). The updated form ofEq. (22) (not presented here for brevity) can now be rearranged to solve explicitly for the required PRB thickness (Breq) given amaximum target concentration (CiMCL
) to be achieved at the POC:
Breq ¼2DB
uB−2DBυiln
CiMCL
Ciin
!þ αi−
uL
2DL−ωi
� �xcomp
" #; ð23Þ
where αi= ln((υiuB+2DLθωiυi+θδωiuB+2DBυi2)/4uBυi), as defined by Park and Zhan (2009). Utilizing van Genuchten's (1981)solution, Eykholt and Sivavec (1995) and Eykholt (1997) presented a similar design equation,
Breq ¼2DB
uB−2DBυiln
CiMCL
Ciin
!" #: ð24Þ
Note that Eq. (24) arises from applying a first-type boundary condition CBi(−B, t)=Ciin at the influent face of the PRB and asemi-infinite boundary condition ∂CBi
∂x ∞; tð Þ ¼ 0 at the effluent face of the PRB. Although Eq. (24) is convenient, it cannot accountfor the reaction in the aquifer and it is derived with the assumption that PRBs have large thicknesses. Site data compiled byGavaskar et al. (2000) indicated that most PRBs were 1–2 m thick. Lastly, it is emphasized that Eqs. (23) and (24) apply only tothe special case when i=1.
For the case when i=2, Eq. (13) becomes
δd2 �CLD2
dxD2 −uLD
d�CLD2
dxD− RL2pþ λLD2ð Þ�CLD2 ¼ −y2λLD1
�CLD1; ð25Þ
where �CLD1 is Eq. (20) but presented in the slightly different form
�CLD1 ¼ 2υD1uBD expuBD
2
� �ED1 �CD1in exp
uLD−2ωD1
2δxD
� �: ð26Þ
Hence the particular solution of Eq. (25) is of the form �CLD2 ¼ ALexpuLD−2ωD1
2δ xD� �
, where AL is the coefficient for the aquifer thatsatisfies the homogeneous part of Eq. (25); i.e.,
AL ¼ −2υD1y2uBD expuBD
2
� �HED1 �CD1in ; ð27Þ
whereH pð Þ ¼ λLD1RL1−RL2ð ÞpþλLD1−λLD2
is a dimensionless ratio of the reactivities in the aquifer. Therefore, the general solution of Eq. (25)
is comprised of the complementary and particular solution such that:
�CLD2 ¼ ϕ expuLD−2ωD2
2δxD
� �−2υD1y2uBD exp
uBD
2
� �HED1 �CD1in
expuLD−2ωD1
2δxD
� �; ð28Þ
where ϕ depends on the boundary conditions. Performing an analogous procedure for the PRB gives
AB ¼ y2uBD expuBD
2
� �θωD1−υD1ð ÞHBED1 �CD1in ; ð29Þ
BB ¼ −y2uBD expuBD
2
� �θωD1 þ υD1ð ÞHBED1 �CD1in ; ð30Þ
where HB pð Þ ¼ λBD1RB1−RB2ð ÞpþλBD1−λBD2
. Coefficients AB and BB are defined and derived in Eqs. (S1) through (S11) in the supplemental
materials. Notice that they pertain to the particular solution of the PRB when i=2 and must satisfy the homogeneous part of
60 J. Mieles, H. Zhan / Journal of Contaminant Hydrology 134–135 (2012) 54–68
Eq. (S1). Therefore, the general solution of the PRB when i=2 is
�CBD2 ¼ α expuBD þ 2υD2
2xD
� �þ β exp
uBD−2υD2
2xD
� �þ y2uBD exp
uBD
2
� �θωD1−υD1ð ÞHBED1 �CD1in exp
uBD þ 2υD1
2xD
� �
−y2uBDexpuBD
2
� �υD1 þ θωD1ð ÞHBED1 �CD1in exp
uBD−2υD1
2xD
� �;
ð31Þ
where α and β depend on the boundary conditions. After applying the boundary conditions to Eqs. (28) and (31) and determiningϕ, the steady-state analytical solution of species i=2 in dimensional form is
CL2 ¼ 2υ2uBE2C2in þ2υ1θy2uB H−HBð Þδ ω1−ω2ð Þ 1−μ2ð ÞE1
υ2 þ θδω2C1in þ 2y2uBC1in υ2HBE2 þ υ1 H−HBð ÞE1ð Þ
�
expuL−2DLω2
2DLxþ uBB
2DB
� �−2υ1y2uBHE1C1in exp
uL−2DLω1
2DLxþ uBB
2DB
� �;
or simply
CL2 ¼ 2υ2uBE2C2in þ2υ1θy2uB H−HBð Þδ ω1−ω2ð Þ 1−μ2ð ÞE1
υ2 þ θδω2C1in þ 2y2uBC1in υ2HBE2 þ υ1 H−HBð ÞE1ð Þ
�
expuL−2DLω2
2DLxþ uBB
2DB
� �−y2HCL1;
ð32Þ
where x is the distance to the POC from the PRB–aquifer interface, E2=1/[(υ2uB+2DLθω2υ2)cosh(Bυ2)+(θδω2uB+2DBυ22)sinh(Bυ2)], and μ2=E2υ2(uB−2DBυ2)exp(−Bυ2) is a dimensionless term. The ratio H=λL1/(λL1−λL2) is also dimensionless, andHB is of equivalent form to H, but with the reactivities of the PRB. It can be seen from the expression of H that λL1=λL2 will make Hsingular; thus the steady-state solution here is only valid when λL1 and λL2 are different. Notice that retardation values areeliminated when p→0. It should also be noted that if C1in
is zero, Eq. (32) reverts to the form of Eq. (22) for a single species; thissupports the validity of Eq. (32).
The procedures outlined above were used to derive the steady-state analytical solution of species three and four. For speciesi=3 the steady-state solution in dimensional form is
CL3 ¼
2υ3uBE3C3inþ y3γ I−IBð Þυ3 þ θδω3
υ3 þ θδω2−θδ ω2−ω3ð Þμ3½ � α þ βð Þ þ y3γIB υ2 þ θδω2ð Þυ3 þ θδω3
1−μ3ð Þψþ
2υ1y3γy2uBE1θδ 1−μ3ð ÞC1inυ3 þ θδω3
ω1−ω3ð Þ JBHB−JHð Þ þ ω2−ω3ð ÞI H−HBð Þ½ � þ 2υ3y3γIBuBE3C2inþ
2y3γy2uBC1in υ1E1 JBHB−JH þ IH−IHBð Þ−υ3E3HB JB−IBð Þ½ �
8>>>>>>><>>>>>>>:
9>>>>>>>=>>>>>>>;
expuL−2DLω3
2DLxþ uBB
2DB
� �−y3γIϕexp
uL−2DLω2
2DLxþ uBB
2DB
� �þ 2υ1y3γy2uBJHE1C1in exp
uL−2DLω1
2DLxþ uBB
2DB
� �;
ð33Þ
where α, β and ϕ were defined for species two, μ3=E3υ3(uB−2DBυ3)exp(−Bυ3), I=λL2/(λL2−λL3), J=λL2/(λL1−λL3), and IBand JB are of the same form but with the reactivities of the PRB. Lastly,
ψ ¼ 2υ1θy2uB H−HBð Þδ ω1−ω2ð ÞE1υ2 þ θδω2
C1in : ð34Þ
Notice that the branching ratio γ must equal one if serial degradation is assumed; i.e., when no parallel daughter products areproduced (Clement, 2001; Tratnyek et al., 1997).
For species i=4 the steady-state solution in dimensional form is
CL4 ¼
2υ4uBE4C4inþ y4 K−KBð Þ
υ4 þ θδω4υ4 þ θδω3−θδ ω3−ω4ð Þμ4½ � ε þ ιð Þ þ y4KB υ3 þ θδω3ð Þ
υ4 þ θδω41−μ4ð Þσþ
y4y3γδυ4 þ θδω4
_Ω α þ βð Þ þ y4y3γLBIB υ2 þ θδω2ð Þ μ4−1ð Þυ4 þ θδω4
ψþ 2υ1y4y3γy2uBθδE1 1−μ4ð Þυ4 þ θδω4
€ΩC1inþ
2υ4y4uBKBE4C3inþ 2υ4y4y3γuBIBE4 KB−LBð ÞC2in
þ 2y4y3γy2uBΩC1in
8>>>>>><>>>>>>:
9>>>>>>=>>>>>>;
expuL−2DLω4
2DLxþ uBB
2DB
� �−y4Kζexp
uL−2DLω3
2DLxþ uBB
2DB
� �þ y4y3γLIϕexp
uL−2DLω2
2DLxþ uBB
2DB
� �
−2υ1y4y3γy2uBJHME1C1in expuL−2DLω1
2DLxþ uBB
2DB
� �;
ð35Þ
61J. Mieles, H. Zhan / Journal of Contaminant Hydrology 134–135 (2012) 54–68
where μ4=E4υ4(uB−2DBυ4)exp(−Bυ4), K=λL3/(λL3−λL4), L=λL3/(λL2−λL4), M=λL3/(λL1−λL4) , and KB, LB, andMB pertain tothe PRB. Also, ε, ι, and ζ are terms pertaining to species three (see supplemental derivations). Lastly, the following are defined forspecies four:
σ ¼ y3γθ I−IBð Þδ ω2−ω3ð Þυ3 þ θδω3
α þ βð Þ þ y3γIB υ2 þ θδω2ð Þυ3 þ θδω3
ψþ 2υ1y3γy2uBθδE1υ3 þ θδω3
C1in
½ ω1−ω3ð Þ JBHB−JHð Þ þ ω2−ω3ð ÞI H−HBð Þ�; ð36Þ
_Ω ¼ LBIB−LIð Þ υ4
δþ θω2−θ ω2−ω4ð Þμ4
� �þ K IB−Ið Þ θ ω3−ω4ð Þμ4−
υ4
δ−θω3
� �; ð37Þ
€Ω ¼ ω1−ω4ð Þ JHM−JBHBMBð Þ þ LI ω2−ω4ð Þ HB−Hð Þ þ K ω3−ω4ð Þ HB JB−Ið Þ−H J−Ið Þð Þ; ð38Þ
Ω ¼ υ1E1 JBHB K−MBð Þ þ JH M−Kð Þ þ I H−HBð Þ K−Lð Þð Þ þ υ4E4HB IB KB−LBð Þ þ JB MB−KBð Þð Þ: ð39Þ
It should benoted that Eqs. (32), (33), and (35) cannot be solved explicitly for B, hence a coded programor spreadsheet is themostefficientway to utilize the steady-state solutions. Also the denominators ofH,HB, I, IB, J, JB,K,KB, L, LB,M, andMB cannot be zero in orderfor the equations to be defined; i.e., a unique solution is not possible in the case when interspecies reaction rate factors are equal toeach other. This restriction, however, does not apply to the transient solutions, unless the interspecies retardation factors are alsoequal to each other. Detailed derivations of species 2, 3, and 4 are provided in the supplemental derivations.
3. Comparison of the serial degradation solutionswith VMOD
In this section the steady-state analytical solutionspresented above are compared to the sequential decayreaction module in the RT3D v2.5 transport engine of VisualMODFLOW (VMOD) 2009. In applying the above equations itis reminded that the transport and flow parameters assignedto the aquifer depend on accurate site characterization whilethe transport parameters assigned to the PRB are determinedusing the normalization method proposed by Johnson et al.(1996) and subsequently adopted by the US EPA (1998). Abrief review of the normalization method is provided below.
In an extensive review of prior studies on dechlorination byzero-valent iron (ZVI), Johnson et al. (1996) demonstrated thatobserved first-order reaction rates (λobs) of any one chemicalvaried by up to three orders of magnitude. To reduce thisapparent variability, Johnson et al. (1996) expanded the first-order rate model to account for the amount of available metalin the reaction, and normalized the reaction rates to the ironsurface area concentration (ρa); i.e.,
r ¼ −λsAasρmC;orr ¼ −λsAρaC
ð40Þ
where r is the normalized first-order reaction rate, λsA is thespecific reaction rate constant [L hr−1m−2], as is the specificsurface area [m2g−1] determined by the Brunauer, Emmett,and Teller (BET) gas adsorptionmethod,ρm is the ironmass persolution volume or mass concentration [g L−1], and ρa is theiron surface area per solution volume or surface areaconcentration [m2L−1]. Therefore, λobs=λsAρa where λsA isdetermined from the linear regression of λobs and ρa. Byrepeating this normalization procedure for individual chemi-cals, Johnson et al. (1996) provided a list of λsA values, wherethe λsA of each chemical varied only by one order ofmagnitude,instead of three orders of magnitude. The list of λsA constants
were not differentiated between column and batch tests. Forthe purpose of modeling, all subsequent uses of λ imply thatλobs was properly derived from λsA.
To compare themultispecies steady-state analytical solutionswith a numerical solution required creating both a spreadsheetprogram and a finite-difference grid in VMOD to simulate thedegradation of PCE → TCE → DCE → VC. The finite-differencegrid modeled 1D flow along a 20 m long domain, of which thefirst 0.5 m was assigned as the thickness (B) of the up-gradientPBR zone. The domain also consisted of a 10 mup-gradient head,a 9.9 m down-gradient head, and a 5 m width. Other flowparameters assigned to the domain included an aquifer effectiveporosity (nL) of 0.3, a PRB hydraulic conductivity (kB) of 60 m/d,and an aquifer hydraulic conductivity (kL) of 20 m/d. Using theabove flow parameters, themagnitude of discharge is calculatedby Darcy's law as Q=|kLA∇h|=5m3/d, where A and ∇hrepresent the flow cross-section and hydraulic gradient,respectively. Also, according to the VMODgeometricmulti-grid(GMG) flow solver, the seepage velocity in the aquifer is3.38×10−1 m/d. This value can also be confirmed analyticallyby using the above parameters; i.e., uL=kL∇h/nL= 0.33 m/d.Additionally, since specific discharge at the PRB–aquifer in-terface is uniform, the seepage velocity in a PRB with effectiveporosity of nB=0.5 is uB=uLnL/nB=0.2 m/d. This value alsoagrees with the numerical output of 2.03×10−1 m/d. Table 1illustrates the portion of the attached spreadsheet programwhere the flow parameters (including dispersivity) aredesignated and others (such as dispersion) are calculated.
The transport parameters such as yield, reaction rate, andinfluent concentration of each species are presented inTable 2 as they would be entered into the spreadsheetprogram. At this stage all necessary information is definedand the program conveniently calculates all other constantsutilized in the steady-state analytical solutions. For example,in this scenario the PRB reaction rate constants of species 1, 2,3, and 4 are 2 d−1, 1.2 d−1, 1.1 d−1, and 0.9 d−1 while in theaquifer the weaker reaction rates are 0.2 d−1, 0.05 d−1,
Table 1Modeled flow parameters.
Domain parameters PRB parameters Aquifer parameters
θ (–) 0.6 nB (–) 0.5 nL (–) 0.3Breq (m) 0.5 x (m) 0
Width (m) 5hup-gradient (m) 10hdown-gradient (m) 9.9L (m) 20Hydraulic gradient(–)
0.005
kB (mday−1)
60 kL (mday−1)
20
uB (mday−1)
0.20 uL (mday−1)
0.33
Q (m3 day−1) 5αB (m) 0.05 αL (m) 2
δ (–) 66.67 DB (m2
day−1)0.01 DL (m2
day−1)0.67
Fig. 3. Comparison of the analytical solutions and VMOD: PCE and TCE.
62 J. Mieles, H. Zhan / Journal of Contaminant Hydrology 134–135 (2012) 54–68
0.03 d−1, and 0.02 d−1, respectively. The trend of decreasingreaction rates is intended to simulate the decreasingdechlorination potential of each subsequent daughter speciesin the aquifer (Wiedemeier et al., 1999) and in the PRB(Matheson and Tratnyek, 1994). The solutions are not limitedto decreasing or increasing rates in one direction, variousreaction rates can be utilized at any stage of the reaction.After inputting the reaction rate constants, the programcalculates the values of υi, ωi, Ei, and the dimensionless ratiosof the reactivities. Lastly, the user inputs the pertinentstoichiometric yields and concentrations Ciin of each species.The program then quickly calculates the concentration ofeach species at the designated POC (the x in Table 1), which isdown-gradient of the PRB–aquifer interface.
The RT3D numerical solution concentrations for eachspecies were plotted along with selected concentrations fromthe steady-state analytical solutions. As seen in Figs. 3 and 4,
Table 2Modeled transport parameters.
Species 1 (PCE)E1 (day) 1.53E−05 λB1 (day−1)
ν1 (m−1)
Species 2 (TCE)E2 (day) 8.16E−05 λB2 (day−1)Y2 (–) 0.792 ν2 (m−1)
HB (–)
Species 3 (DCE)E3 (day) 1.05E−04 λB3 (day−1)Y3 (–) 0.738 ν3 (m−1)γbranch (–) 1 IB (–)
JB (–)
Species 4 (VC)E4 (day) 1.65E−04 λB4 (day−1)Y4 (–) 0.644 ν4 (m−1)
KB (–)LB (–)MB (–)
C1in (mg L−1) 10C2in (mg L−1) 15C3in (mg L−1) 5C4in (mg L−1) 3
the two methods agree particularly well in the aquifer zonewhere the discrete concentration points coincide with thelines representing the numerical solution. Two additionalsets of figures comparing the analytical solutions with VMODare included in the supplemental materials.
4. Application of the serial degradation solutionsfor PRB design
Applying the spreadsheet programs in a real scenario isstraightforward: the designer inputs the site gradient and thehydraulic conductivity of the aquifer (or just the groundwatervelocity), the porosity and dispersivity of each medium, andthe distance (x) to the down-gradient POC. The transportparameters required include the reaction rate constants (inboth media), stoichiometric yields, and concentrations Ciin(t).To determine Breq, the designer simply increases the value ofB until all species meet the required concentration at the POC.Alternatively, the Microsoft Excel tool Goal Seek performs the
2.0 λL1 (day−1) 0.21.73E+01 ω1 (m−1) 6.02E−01
1.2 λL2 (day−1) 0.051.48E+01 ω2 (m−1) 3.71E–012.50E+00 HL (–) 1.33E+00
1.1 λL3 (day−1) 0.031.45E+01 ω3 (m−1) 3.28E−011.20E+01 IL (−) 2.50E+001.33E+00 JL (–) 2.94E−01
0.9 λL4 (day−1) 0.021.38E+01 ω4 (m−1) 3.04E−015.50E+00 KL (–) 3.00E+003.67E+00 LL (–) 1.00E+001.00E+00 ML (–) 1.67E–01
CL1 (mg L−1) 1.58E−01CL2 (mg L−1) 2.14E+00CL3 (mg L−1) 3.80E+00CL4 (mg L−1) 3.92E+00
Fig. 4. Comparison of the analytical solutions and VMOD: DCE and VC.
63J. Mieles, H. Zhan / Journal of Contaminant Hydrology 134–135 (2012) 54–68
same function. A minimum input value of Breq=0.5 isrecommended for goal seek to return a non-negative width. Itis worthwhile to note that these equations can be applied toany other four species reacting in a serial chain reaction and toother types of dual domain systems. For example, theequations can also be used for two adjacent aquifer zoneswhere an oxidizing zone transitions into a reducing zone. Asanother example, if the reaction involves only three species,then both y4 and C4in
are zero, and so on if only two speciesreact. Overall, themultispecies steady-state analytical solutionspresented in this study can be quickly implemented using thespreadsheet programs which are cost-effective compared tothe standard numerical program. Hence the solutions areuseful as PRB design equations.
Once the barrier width is calculated, the required mass ofreactive material in the PRB is currently determined using theEPA (1998)mass estimation calculation,which is fundamentallyderived from Eq. (24) without dispersion. For example, ifdispersion is neglected, Eq. (24) becomes the common plugflow reactor equation
Breq ¼ − uB
λobsln
CMCL
Cin
� �¼ − uB
λSAasρmln
CMCL
Cin
� �
¼ − uBVw
λSAasMSln
CMCL
Cin
� �; ð41Þ
whereMS is the mass of the solid iron grains and Vw is the porevolume in the PRB. In the case of a continuous PRB, the totalvolume is simply VT=BreqA, where A is the designed area of thePRB normal to the groundwater flowdirection. Also, n=Vw/VT isthe PRB porosity, assuming installation in the saturated zonewhere the volume of air is negligible. Therefore, Vw=nBreqAand Eq. (41) becomes
MS ¼ − uBnAλSAas
lnCMCL
Cin
� �⇒
MS
A¼ − uBn
λSAasln
CMCL
Cin
� �; ð42Þ
where Eq. (42) is the exact EPA (1998) mass estimationformula and MS/A is the iron mass per unit area. In this study,an alternative mass estimation calculation is derived. Thealternative calculation relies on first determining Breq from the
multispecies reactive transport spreadsheet programs andsubsequently utilizing the relationship
VT ¼ Breq A ¼ Vw þ Vs; ð43Þ
where Vs is the volume of iron solids and the volume of air isneglected. Substituting the iron grain (or particle) densityρg=Ms/Vs and iron mass per pore volume ρm=Ms/Vw intoEq. (43) yields
MS
A¼ Breq
ρgρm
ρg þ ρm¼ Breqnρm;orsimply
MS
A¼ Breqρb;
ð44Þ
where ρb=Ms/VT is the dry bulk density of ZVI, of whichseveral values are reported by Gavaskar et al. (2000) and theEPA (1998). According to EPA (1998), ZVI bulk density istypically 2.56 g/cm3. As an example of Eq. (44), the continuousPRB installed in the U.S. Coast Guard (USCG) Support Centernear Elizabeth City, North Carolina was designed with areactive cell height of 5.5 m, length of 46 m, thickness of0.6 m, and 450 tons (408,233 kg) of granular iron (EPA, 1999).
Based on this data MSABreq
¼ ρb ¼ 2.69 g/cm3, which is well within
the normal range of reported iron bulk densities. Therefore,Eq. (44) can be used to estimate MS/A once Breq and ρb areknown. Lastly, PRBs tend to be designed with a factor of safetyapplied to Breq (Eykholt, 1997) and a reaction rate correctionadjustment factor due to lower subsurface temperatures(Gavaskar et al., 2000). Note that Eqs. (44) and (40) providethe necessary flexibility to incorporate the factor of safety andthe reaction rate adjustment.
5. Model development—parallel degradation solutions
Several authors including Wiedemeier et al. (1999) havenoted the parallel pathway by which TCE degrades tosimultaneously produce 1,1-DCE, cis-1,2-DCE, and trans-1,2-DCE. In practice, however, these isomers are typically lumpedtogether into one DCE species. It is important to note that inreality each DCE isomer may have a unique reaction rate andtherefore acquire mass and contribute mass at a different rate.The hydrogenolysis degradation model of PCE → TCE → DCEisomers → VC is the sequential replacement of one chlorineatomwith one hydrogen atom inwhich VC acquiresmass fromconvergent multi-parent reactions; i.e., the three DCE isomersdegrade to VC. In examining the abiotic reaction between thecommon chlorinated ethenes and ZVI, Roberts et al. (1996) andArnold and Roberts (2000) expanded the hydrogenolysismodel to include a separate β-elimination pathway in whichPCE → dichloroacetylene → chloroacetylene → acytelene.Degradation in this expanded model is a highly complexnetwork of various multi-parent and multi-daughter reactionswhere even the hydrogenolysis pathway and the β-elimination(or chlorinated acetylene) pathway exchange mass. Complex-ity of this level is not practical for the purpose of analyticalmodeling; hence this study considers the degradation networkillustrated in Fig. 5 to be a reasonable simplification. Theabsence of the dichloroacetylene intermediary from Fig. 5 isjustified since it tends to react very rapidly with ZVI (Arnold
Y6
Y2
Y31
Y32
Y33
5
32
33
31
Y5
Y41
Y42
Y43 PCE(SP1)
DCE1 (SP31)
DCE2 (SP32) VC(SP4)
Acetylene(SP6)
TCE(SP2)
Chloroacetylene (SP5)
DCE3 (SP33)
Fig. 5. Conceptual view of the parallel degradation pathway of PCE. Y denotes the stoichiometric yield of the daughter product from the parent chemical.γ denotes the branch factor. Note: the sum of all branch factors should equal 1 to appropriately account for mass.
64 J. Mieles, H. Zhan / Journal of Contaminant Hydrology 134–135 (2012) 54–68
and Roberts, 2000; Roberts et al., 1996). After considering themodel in Fig. 5, the PRB and aquifer governing equations ofspecies 1 and 2 are unchanged (Rodiguin and Rodiguina, 1964).However, the governing equations of species DCE1 (SP31),DCE2 (SP32), DCE3 (SP33), and chloroacetylene (SP5) arerevised as follows:
RB3i∂CB3i
∂t ¼ L CB3ið Þ−λB3iCB3i þ γ3iy3iλB2CB2; ð45Þ
RB5∂CB5
∂t ¼ L CB5ð Þ−λB5CB5 þ γ5y5λB2CB2; ð46Þ
for the PRB, and for the aquifer:
RL3i∂CL3i
∂t ¼ L CL3ið Þ−λL3iCL3i þ γ3iy3iλL2CL2; ð47Þ
RL5∂CL5
∂t ¼ L CL5ð Þ−λL5CL5 þ γ5y5λL2CL2; ð48Þ
where i is a counter from 1 to 3 and γ is the branching ratiopreviously defined, which is now less than one for each speciesbut should satisfy γ31+γ32+γ33+γ5=1 (Clement, 2001).
Table 3Modeled transport parameters of SP3i and SP5.
Species 3-1 (1,1-DCE)E3-1 (day) 1.05E−04 λB3-1 (day−1)Y3-1 (–) 0.738 ν3-1 (m−1)γ3-1 (–) 0.25 IB3-1 (–)
JB3-1 (–)
Species 3-2 (Cis-1,2-DCE)E3-2 (day) 1.05E−04 λB3-2 (day−1)Y3-2 (–) 0.738 ν3-2 (m−1)γ3-2 (–) 0.25 IB3-2 (–)
JB3-2 (–)
Species 3-3 (Trans-1,2-DCE)E3-3 (day) 1.05E−04 λB3-3 (day−1)Y3-3 (–) 0.738 ν3-3 (m−1)γ3-3 (–) 0.25 IB3-3 (–)
JB3-3 (–)
Species 5 (Chloroacetylene)E5 (day) 1.05E−04 λB5 (day−1)Y5 (–) 0.460 ν5 (m−1)γ5 (–) 0.25 IB5 (–)
JB5 (–)
With the exception of the branching ratio not equaling 1,Eqs. (45) through (48) are equivalent in form to Eqs. (2)and (5) of the serial degradation scenario i=3. Therefore,the steady-state analytical solution to Eqs. (45) and (47) isequivalent in form to Eq. (33), where α, β and ϕ areunchanged. As seen in Table 3, the major difference arises inthe values of the dimensionless reactivity ratios; i.e., I3i=λL2/(λL2−λL3i), J3i=λL2/(λL1−λL3i), where IB3i and JB3ipertain to the PRB. Similarly, the steady-state dimensional
forms of υ and ω are υ3i ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuB
2=4DB2 þ λB3i=DB
qand ω3i ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
uL2=4DL
2 þ λL3i=DL
qwhere i is the counter defined above.
The same analyses apply to I5, J5, IB5, JB5, υ5, and ω5 whichpertain to species 5.
The current version of RT3D preprogrammed in VMOD doesnot include this type of parallelmultispecies degradationmodel.To formally compare the analytical solutions of Eqs. (45)–(48)with a numerical model requires creating a program in theuser-defined RT3D reaction module programmed by Clementet al. (1998). This study presents an alternative method oftesting the validity of the (parallel degradation) solutions thatis based on the conservation of total mass between the serialand parallel chains. For example, if each of the four branch
1.10 λL3-1 (day−1) 0.031.45E+01 ω3-1 (m−1) 3.28E−011.20E+01 IL3-1 (–) 2.50E+001.33E+00 JL3-1 (–) 2.94E−01
1.10 λL3-2 (day−1) 0.031.45E+01 ω3-2 (m−1) 3.28E−011.20E+01 IL3-2 (–) 2.50E+001.33E+00 JL3-2 (–) 2.94E−01
1.10 λL3-3 (day−1) 0.031.45E+01 ω3-3 (m−1) 3.28E−011.20E+01 IL3-3 (–) 2.50E+001.33E+00 JL3-3 (–) 2.94E−01
1.10 λL5 (day−1) 0.031.45E+01 ω5 (m−1) 3.28E−011.20E+01 IL5 (–) 2.50E+001.33E+00 JL5 (–) 2.94E−01
65J. Mieles, H. Zhan / Journal of Contaminant Hydrology 134–135 (2012) 54–68
factors equals 0.25 and all interspecies stoichiometric yieldsand first-order reaction rates are equal (as designated inTable 3), then SP3i and SP5 gain and lose moles at the samerate. That is, SP3i and SP5 react (degrade) at the same rate.Therefore, it is expected that: 1) at a selected POC, SP3i and SP5exhibit identical concentrations, and 2) the total of theirsummed concentrations is equal to an effective SP3 concentra-tion. This effective SP3 concentration should exactly equal theSP3 concentration from the serial degradation pathway (whichwas discussed in Section 2).
This test was performed on several discrete points andwithseveral decay constants and found to be exact as predicted. Afinal assumption (which facilitates this test but is not required)is to treat SP3i, SP5, and SP3 solely as daughter products ofweathered TCE; i.e., their influent concentrations are zero.Additionally, it should be pointed out that the spreadsheetprogram is sufficiently flexible to model only the parallelhydrogenolysis pathway by neglecting the β-eliminationpathway altogether. Such an assumption is conservative indealing with the chlorinated ethenes and can be accomplishedby letting y5, C5in
, and C6inequal zero and ensuring that γ31+
γ32+γ33= 1. However, caution should be taken in establish-ing this assumption as the norm. Chlorinated acetylenes arereported to be toxic (Roberts et al., 1996). Although theydegrade rapidly in laboratory studies (Arnold and Roberts,2000; Roberts et al., 1996), it is uncertain to what extent theyaccumulate as PRB effluents in the field.
The next chemical in Fig. 5, SP4, gainsmass from convergentmulti-parent reactions. In this case the updated governingequations of SP4 are (Rodiguin and Rodiguina, 1964):
RB4∂CB4
∂t ¼ L CB4ð Þ−λB4CB4 þ y41λB31CB31 þ y42λB32CB32
þ y43λB33CB33;
ð49Þ
RL4∂CL4
∂t ¼ L CL4ð Þ−λL4CL4 þ y41λL31CL31 þ y42λL32CL32
þ y43λL33CL33;
ð50Þ
where y41, y42, and y43 can be distinctive stoichiometric yields,but in this case they are equivalent due to the DCE isomers.Applying the dimensionless parameters in Eq. (9) and theLaplace transform to Eqs. (49) and (50) gives
d2 �CBD4
dxD2 −uBD
d�CBD4
dxD− RB4pþ λBD4ð Þ�CBD4 ¼ −y41λBD31
�CBD31
−y42λBD32�CBD32−y43λBD33
�CBD33; ð51Þ
δd2 �CLD4
dxD2 −uLD
d�CLD4
dxD− RL4pþ λLD4ð Þ�CLD4 ¼ −y41λLD31
�CLD31
−y42λLD32�CLD32−y43λLD33
�CLD33: ð52Þ
Recall that in the case of the serial degradation pathway, thesteady-state analytical solution of SP4 was presented asEq. (35). Also recall that in the serial degradation pathway,the boundary conditions were applied once. After comparingthe right-hand side of Eqs. (11) and (13) with the right-handside of Eqs. (51)–(52), it becomes apparent that the generalprocedure used to determine Eq. (35) can also be used to solveEqs. (51)–(52), with a slight variation. To solve Eqs. (51)–(52),the boundary conditions are now applied two additional times
to the two additional terms on the right-hand side; i.e., theterms associated with �CD32 and �CD33. This realization allowsone to quickly deduce the analytical solution of SP4 for theparallel degradation pathway and avoid the laboring derivationrequired to manually solve Eqs. (51)–(52). Therefore, thesolution to a daughter product that results from a convergentmulti-parent reaction can be determined by applying theboundary conditions once to each additional term on the right-hand side. Inevitably, this procedure results in additional termsthat are (at every algebraic step) added to the derivation,resulting in a much longer analytical solution. The resultingsteady-state analytical solution of SP4 is included in thesupplemental derivations.
To test the validity of this solution, the mass conservationtechnique introduced above is revisited. In this case thefollowing are assumed: let γ5 (or y5) and C3iin equal zero, letall y4i equal each other, let γ31=γ32=γ33=1/3, and let allstoichiometric yields and first-order reaction rates pertainingto SP3i equal each other. The physical representation of theseassumptions is that SP4 gains the same amount of moles fromeach SP3i isomer at the same rate. In this case, the SP3ivariants again mimic one effective SP3 chemical. Therefore,the SP4 concentration from the parallel degradation pathwayshould exactly equal the SP4 concentration from the serialdegradation pathway. This test was performed on severaldiscrete points and with several decay constants and alsofound to be exact as predicted. Table 4 presents the modifiedinputs of SP4 in the case of the parallel degradation reaction.As seen in Table 4, the rate at which SP4 degrades isunchanged from the serial degradation model; i.e., the valuesof υ4 and ω4 do not change from Table 2 to Table 4. However,the additional terms on the right-hand-side of Eqs. (49) and(50) require that the analytical solution of SP4 accounts forpotentially different rates of mass contribution from each ofthe three DCE isomers. Hence K3i, L3i, and M3i are defined forthe aquifer and KB3i, LB3i, and MB3i pertain to the PRB, where iis the counter from 1 to 3.
The final chemical in Fig. 5, SP6, results only from thedegradation of chloroacetylene, hence its governing equationsare:
RB6∂CB6
∂t ¼ L CB6ð Þ−λB6CB6 þ y6λB5CB5; ð53Þ
RL6∂CL6
∂t ¼ L CL6ð Þ−λL6CL6 þ y6λL5CL5: ð54Þ
Notice that Eqs. (53) and (54) are equivalent in form toEqs. (2) and (5) of the serial degradation scenario i=4.Therefore, the steady-state analytical solution of SP6 can bevalidated by treating the β-elimination pathway as the soleserial degradation pathway and neglecting the hydrogeno-lysis pathway. That is, let γ3i (or y3i)=0, γ5=1, allow y5 andy6 to equal y3 and y4, respectively, and assign the samereaction rates of SP4 from Table 2 to SP6. In this case, theanalytical solution for SP6 is validated if SP6 exactly matchesthe concentration of SP4 in the serial degradation section ofthis study. This final test was performed on several discretepoints and with several decay constants and found to beexact as predicted.
Table 4Modeled transport parameters of SP4.
Species 4 (VC)E4 (day) 1.65E−04 λB4 (day−1) 0.90 λL4 (day−1) 0.02Y4/3-1 (–) 0.644 ν4 (m−1) 1.38E+01 ω4 (m−1) 3.04E−01Y4/3-2 (–) 0.644 KB3-1 (–) 5.50E+00 KL3-1 (–) 3.00E+00Y4/3-3 (–) 0.644 LB3-1 (–) 3.67E+00 LL3-1 (–) 1.00E+00
MB3-1 (–) 1.00E+00 ML3-1 (–) 1.67E−01KB3-2 (–) 5.50E+00 KL3-2 (–) 3.00E+00LB3-2 (–) 3.67E+00 LL3-2 (–) 1.00E+00MB3-2 (–) 1.00E+00 ML3-2 (–) 1.67E−01KB3-3 (–) 5.50E+00 KL3-3 (–) 3.00E+00LB3-3 (–) 3.67E+00 LL3-3 (–) 1.00E+00MB3-3 (–) 1.00E+00 ML3-3 (–) 1.67E−01
66 J. Mieles, H. Zhan / Journal of Contaminant Hydrology 134–135 (2012) 54–68
6. Summary and conclusions
In this study,wederived steady-state analytical and transientsemi-analytical solutions to multispecies reactive transport in aPRB–aquifer (dual domain) system based on a 1D approach. Thestudy focuses primarily on the steady-state analytical solutionsof the PCE serial degradation pathway and secondly on theanalytical solutions of the parallel degradation pathway. Thesesolutions can also be applied to other types of dual domainsystems with distinct flow and transport properties and canhandle up to four decaying species in the serial reaction andeight species in the parallel reaction. In addition, these solutionsmaintain flux and concentration continuity at the PRB–aquiferinterface and account for the potential existence of naturaldegradation in the down-gradient aquifer. Degradation in theaquifer can be incorporated into the solutions when consideringthe required PRB thickness so that all species involved in thedegradation pathway achieve the regulatory limits at the POC.The steady-state analytical solutions exhibit a limitation thatoccurs when interspecies reaction rate factors equal each other,which result in undefined solutions. This restriction, however,does not apply to the transient solutions, unless the interspeciesretardation factors are also equal to each other.
In addition to the study, an extensive electronic supple-ment is included to facilitate review of the derivations andprompt application of the steady-state solutions. The derivedsolutions provide increased modeling flexibility. For exam-ple: every species can have unique reaction rates andretardation factors, and daughter species can be modeledwith their individual input concentrations or solely asbyproducts of the parent species by assigning the daughter'sinput concentration as zero. Note that species-specificretardation factors cannot be used in the solutions presentedby Sun et al. (2004). Additionally, it should be noted thatalthough the transient solutions are not discussed in detailhere, they can be deduced from the supplemental derivationswith minimal effort. However, a numerical Laplace inversionprogram is needed to obtain a solution in the time domain.
List of major symbols
Generalt Timex Spatial coordinate along the 1D flow directionB, Breq PRB thickness, and required (modeled) PRB thicknessCLi,CBi Concentrations of the i-th species in the aquifer and
PRB
Ciup Concentration of the i-th species up-gradient of thePRB
Ciin Concentration of the i-th species calculated fromEq. (3) based on Ciup
kL,kB Hydraulic conductivities in the aquifer and PRBp Dimensionless Laplace variables Dimensional Laplace variablenL,nB Effective porosities in the aquifer and PRBαL,αB Dispersivities in the aquifer and PRBuL,uB Groundwater seepage velocities in the aquifer and
PRBDL,DB Longitudinal dispersion coefficients in the aquifer
and PRBλLi,λBi First-order reaction rates of the i-th species in the
aquifer and PRBRLi,RBi Retardation factors of the i-th species in the aquifer
and PRByi Stoichiometric yield of the i-th speciesγi Branch factor of the i-th speciesEi Ratio of hyperbolic functions for the i-th speciesq Specific dischargeQ DischargeλSA Specific reaction rate constantaS Specific surface areaA Area normal to 1D flowMS Mass of the solid iron grainsVT,VW,VS Total volume, water volume, and solids volumeρm,ρa Iron mass per pore volume and surface area
concentrationρg,ρb Grain (particle) density and dry bulk density
Species 2α,β PRB coefficientsϕ Aquifer coefficientH,HB Dimensionless ratios of the reactivities in the
aquifer and PRB
Species 3ε,ι PRB coefficientsζ Aquifer coefficientI,IB First set of dimensionless ratios of the reactivities
in the aquifer and PRBJ,JB Second set of dimensionless ratios of the reactiv-
ities in the aquifer and PRB
Species 4κ,ξ PRB coefficients
67J. Mieles, H. Zhan / Journal of Contaminant Hydrology 134–135 (2012) 54–68
ρ Aquifer coefficientK,KB First set of dimensionless ratios of the reactivities
in the aquifer and PRBL,LB Second set of dimensionless ratios of the reactiv-
ities in the aquifer and PRBM,MB Third set of dimensionless ratios of the reactivities
in the aquifer and PRB
Parallel Species ReactionI3i,IB3i,J3i,JB3i Dimensionless ratios of the reactivities in the
aquifer and PRB for SP3iI5,IB5,J5,JB5 Dimensionless ratios of the reactivities in the
aquifer and PRB for SP5K3i,L3i,M3i Dimensionless ratios of the reactivities in the
aquifer for SP4KB3i,LB3i,MB3i Dimensionless ratios of the reactivities in the
PRB for SP4K6,L6,M6 Dimensionless ratios of the reactivities in the
aquifer for SP6KB6,LB6,MB6 Dimensionless ratios of the reactivities in the
PRB for SP6
Acknowledgements
This research was partially supported by the AdvancedResearch Program (ARP) from the Texas Higher EducationCoordination Board, the National Basic Research Program ofChina (973) (2011CB710600), the National Natural ScienceFoundation of China (41172281), and by the W.G. MillsScholarship Program from the Texas Water ResourcesInstitute. Special thanks go to the anonymous reviewers andeditor for their critical comments which improved the qualityof the manuscript.
Appendix A. Supplementary data
Supplementary data to this article can be found online athttp://dx.doi.org/10.1016/j.jconhyd.2012.04.002.
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