# Analytical solutions of one-dimensional multispecies reactive transport in a permeable reactive barrier-aquifer system

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, TX77of Geos

Article history:Received 6 October 2011Received in revised form 31 March 2012Accepted 2 April 2012Available online 13 April 2012

Keywords:Multispecies reactive transport

Journal of Contaminant Hydrology 134135 (2012) 5468

Contents lists available at SciVerse ScienceDirect

Journal of Cont mi

j ourna l homepage: www else1. 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, but

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 therather to manage plume concentrationsflows towards receptors or off-site. In its

Corresponding author at: Department of GeologyA&M University, College Station, TX77843-3115, US7961; fax: +1 979 845 6162.

E-mail addresses: mielesjm@aggienetwork.com (J.zhan@geos.tamu.edu (H. Zhan).

0169-7722/$ see front matter 2012 Elsevier B.V. Adoi:10.1016/j.jconhyd.2012.04.002 2012 Elsevier B.V. All rights reserved.a b s t r a c t

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 PRBaquifer (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.In situ remediationPRB design equationsModelinga r t i c l e i n f oAnalytical solutions of one-dimensiopermeable reactive barrier-aquifer sy

John Mieles a, Hongbin Zhan a,b,a Department of Geology and Geophysics, Texas A&M University, College Stationb Faculty of Engineering and School of Environmental Studies, China Universityas groundwatersimplest form, a

and Geophysics, TexasA. Tel.: +1 979 862

Mieles),

ll rights reserved.a

.multispecies reactive transport in am

843-3115, USAciences, Wuhan, Hubei 430074, PR China

nant Hydrology

v ie r .com/ locate / jconhydfunnel-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

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 PRBaquifer 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 reactive

55J. Mieles, H. Zhan / Journal of Contaminant Hydrology 134135 (2012) 5468illustrated 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 PRBaquifer 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 advectiondispersion 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 conditionCx ; t 0 at the effluent face of the PRB, where C representsconcentration [ML3], t is time [T], x is the spatial coordinatealong the flow direction [L], and Cin is the concentration at theinlet boundary [ML3]. 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;t x outlet 0j effluent condition.

The solutions of Eykholt (1997) and the second solution ofRabideau 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 dissolvedspecies in the PRBaquifer 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 PRBaquifer 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.

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 PRBaquifer 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).

2. Model developmentserial 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.

tes the stoichiometric yield of the daughter product from the parent chemical.

56 J. Mieles, H. Zhan / Journal of Contaminant Hydrology 134135 (2012) 5468Fig. 2. Conceptual view of the serial degradation pathway of PCE. Y denoAlthough the PRBaquifer 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 inPRBaquifer 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 RPBaquifer system (Bilbrey and Shafer,2001; Starr and Cherry, 1994). When conducting the numerical simulations, one must pay great attention to the PRBaquiferinterfaces 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 PRBaquifer system modeled here, the governing equation for the first parent species in the PRB is

RBiCBit L CBi BiCBi;L CBi DB

2CBix2

uBCBix ; 1

where the subscript B denotes...

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