Benchmarking a Visual-Basic based multi-component one-dimensional reactive transport modeling tool Jagadish Torlapati 1 , T. Prabhakar Clement n Department of Civil Engineering, Auburn University, Auburn, AL 36849, USA article info Article history: Received 22 May 2012 Received in revised form 9 August 2012 Accepted 16 August 2012 Available online 24 August 2012 Keywords: Groundwater models Reactive transport Bioremediation Geochemical transport Numerical model abstract We present the details of a comprehensive numerical modeling tool, RT1D, which can be used for simulating biochemical and geochemical reactive transport problems. The code can be run within the standard Microsoft EXCEL Visual Basic platform, and it does not require any additional software tools. The code can be easily adapted by others for simulating different types of laboratory-scale reactive transport experiments. We illustrate the capabilities of the tool by solving five benchmark problems with varying levels of reaction complexity. These literature-derived benchmarks are used to highlight the versatility of the code for solving a variety of practical reactive transport problems. The benchmarks are described in detail to provide a comprehensive database, which can be used by model developers to test other numerical codes. The VBA code presented in the study is a practical tool that can be used by laboratory researchers for analyzing both batch and column datasets within an EXCEL platform. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction Groundwater is an important source of water supply used by populations all around the world (Todd, 1980). In the United States, groundwater accounts for approximately twenty one percent of the annual water supply budget and hence it is considered as an important natural resource (Perlman, 2011). Inadvertent discharge of harmful contaminants including metals and organics into groundwater aquifers poses a significant threat to this resource. Groundwater systems could be contaminated by leachates emanating from several anthropogenic sources includ- ing landfills, mines, leaking underground storage tanks (LUSTs), and other industrial waste sites. Also, the extensive use of various forms of chlorinated solvents for dry cleaning and metal degreas- ing activities has resulted in widespread contamination of groundwater and soil systems (Coleman et al., 2002). In addition to heavy metal and chlorinated solvent issues, contamination of aquifers by petroleum products released from LUSTs has been reported at several field sites in different continents including North America, Europe, and Australia (Lu et al., 1999; Moreau, 1987; Prommer et al., 1998). In the US alone, about 10 to 20% of the estimated total of 2 million underground storage tanks are expected to be leaking (Atlas and Cerniglia, 1995). Therefore, contamination of groundwater by LUSTs and other sources poses a significant threat to groundwater quality. Once contaminated, groundwater aquifers require some type of remediation strategy to restore its water quality to safe drinking water levels. The type of remediation system used and the associated costs would depend on the type of contaminant, extent of contamination and the local geological conditions. Typically, most contaminated field sites are first remediated using some form of conventional pump-and-treat systems. However, due to solubility limitations, conventional pump-and-treat meth- ods have been ineffective at several sites. Therefore, in recent years, engineers have attempted to use innovative in-situ tech- nologies such as the bioremediation methods to transform the contaminants into non-toxic daughter products (Beeman and Bleckmann, 2002; Clement et al., 2004). Active and passive (or natural attenuation) bioremediation methods can be used to treat petroleum and chlorinated solvent plumes (Clement, 2011). Design and application of these bioremediation methods require tools that can model the fate and transport of the contaminants and the associated site-specific biogeochemical reactions. The modeling step is particularly important when natural attenuation methods are employed for managing petroleum and/or chlori- nated solvent plumes (Clement et al., 2000; Lu et al., 1999; Prommer et al., 1998; Rolle et al., 2008). Laboratory-scale experiments are routinely used to develop a better understanding of various biogeochemical transport pro- cesses expected to occur under field conditions. Both batch and column studies have been employed for establishing the feasi- bility of proposed remediation methods (Schaefer et al., 2009 , , Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/cageo Computers & Geosciences 0098-3004/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cageo.2012.08.009 n Corresponding author. Tel.: þ1 334 844 6268; fax: þ1 334 844 6290. E-mail addresses: [email protected] (J. Torlapati), [email protected] (T. Prabhakar Clement). 1 Tel.: þ917 266 5900. Computers & Geosciences 50 (2013) 72–83
Transcript
Computers & Geosciences 50 (2013) 72–83
Contents lists available at SciVerse ScienceDirect
Computers & Geosciences
0098-30
http://d
n Corr
E-m
clement1 Te
journal homepage: www.elsevier.com/locate/cageo
Benchmarking a Visual-Basic based multi-component one-dimensionalreactive transport modeling tool
Jagadish Torlapati 1, T. Prabhakar Clement n
Department of Civil Engineering, Auburn University, Auburn, AL 36849, USA
a r t i c l e i n f o
Article history:
Received 22 May 2012
Received in revised form
9 August 2012
Accepted 16 August 2012Available online 24 August 2012
We present the details of a comprehensive numerical modeling tool, RT1D, which can be used for
simulating biochemical and geochemical reactive transport problems. The code can be run within the
standard Microsoft EXCEL Visual Basic platform, and it does not require any additional software tools.
The code can be easily adapted by others for simulating different types of laboratory-scale reactive
transport experiments. We illustrate the capabilities of the tool by solving five benchmark problems
with varying levels of reaction complexity. These literature-derived benchmarks are used to highlight
the versatility of the code for solving a variety of practical reactive transport problems. The benchmarks
are described in detail to provide a comprehensive database, which can be used by model developers to
test other numerical codes. The VBA code presented in the study is a practical tool that can be used by
laboratory researchers for analyzing both batch and column datasets within an EXCEL platform.
& 2012 Elsevier Ltd. All rights reserved.
1. Introduction
Groundwater is an important source of water supply used bypopulations all around the world (Todd, 1980). In the UnitedStates, groundwater accounts for approximately twenty onepercent of the annual water supply budget and hence it isconsidered as an important natural resource (Perlman, 2011).Inadvertent discharge of harmful contaminants including metalsand organics into groundwater aquifers poses a significant threatto this resource. Groundwater systems could be contaminated byleachates emanating from several anthropogenic sources includ-ing landfills, mines, leaking underground storage tanks (LUSTs),and other industrial waste sites. Also, the extensive use of variousforms of chlorinated solvents for dry cleaning and metal degreas-ing activities has resulted in widespread contamination ofgroundwater and soil systems (Coleman et al., 2002). In additionto heavy metal and chlorinated solvent issues, contamination ofaquifers by petroleum products released from LUSTs has beenreported at several field sites in different continents includingNorth America, Europe, and Australia (Lu et al., 1999; Moreau,1987; Prommer et al., 1998). In the US alone, about 10 to 20% ofthe estimated total of 2 million underground storage tanks areexpected to be leaking (Atlas and Cerniglia, 1995). Therefore,
ll rights reserved.
: þ1 334 844 6290.
ti),
contamination of groundwater by LUSTs and other sources posesa significant threat to groundwater quality.
Once contaminated, groundwater aquifers require some typeof remediation strategy to restore its water quality to safedrinking water levels. The type of remediation system used andthe associated costs would depend on the type of contaminant,extent of contamination and the local geological conditions.Typically, most contaminated field sites are first remediated usingsome form of conventional pump-and-treat systems. However,due to solubility limitations, conventional pump-and-treat meth-ods have been ineffective at several sites. Therefore, in recentyears, engineers have attempted to use innovative in-situ tech-nologies such as the bioremediation methods to transform thecontaminants into non-toxic daughter products (Beeman andBleckmann, 2002; Clement et al., 2004). Active and passive(or natural attenuation) bioremediation methods can be used totreat petroleum and chlorinated solvent plumes (Clement, 2011).Design and application of these bioremediation methods requiretools that can model the fate and transport of the contaminantsand the associated site-specific biogeochemical reactions. Themodeling step is particularly important when natural attenuationmethods are employed for managing petroleum and/or chlori-nated solvent plumes (Clement et al., 2000; Lu et al., 1999;Prommer et al., 1998; Rolle et al., 2008).
Laboratory-scale experiments are routinely used to develop abetter understanding of various biogeochemical transport pro-cesses expected to occur under field conditions. Both batch andcolumn studies have been employed for establishing the feasi-bility of proposed remediation methods (Schaefer et al., 2009,,
J. Torlapati, T. Prabhakar Clement / Computers & Geosciences 50 (2013) 72–83 73
2010). Numerical models are also routinely utilized at thisfeasibility assessment stage to interpret the laboratory data andto develop a better understanding of underlying treatmentprocesses (Clement et al., 1998; Phanikumar et al., 2002;Torlapati et al., 2012). The modeling exercises can greatly helpthe scaling and design steps that are required for deploying field-scale remediation technologies.
Currently, there are several models available that are capableof simulating multi-component, multi-dimensional reactivetransport processes. Zheng and Wang (1999) developed MT3DMS,which is capable of simulating three-dimensional advective–dispersive multi-component transport processes. Clement et al.(1998) developed a reactive transport code RT3D, which is basedon MT3DMS, that can simulate bioreactive transport scenariosinvolving kinetic reactions. Prommer et al. (1998) combinedMT3DMS with PHREEQC (Parkhurst and Appelo, 1999) tosimulate both equilibrium and kinetic reactions. UTCHEM(de Blanc et al., 1996) and BIOPLUME-II (Rafai et al., 1987) aretwo generic reactive transport models that are capable of simu-lating bioreactive transport processes. BIOPLUME-II uses a mod-ified version of the USGS code MOC and can simulate aerobicbiodegradation of petroleum plumes. A later version, known asBIOPLUME-III (Rafai et al., 1998), can simulate both aerobic andanaerobic reactions involved in petroleum biodegradation.BIOCHLOR (Aziz et al., 2000) is an EXCEL-based tool whichimplements a sequential decay analytical solution described inSun and Clement (1999), Sun et al. (1999), and Clement (2001) tosimulate natural attenuation processes occurring at chlorinatedsolvent contaminated sites. However, BIOCHLOR is an analyticalmodel and is limited by the capabilities of the underlyingsolution procedure; some of these limitations are explained inQuezada et al. (2004), Srinivasan and Clement (2008) andSrinivasan et al. (2007).
Most biological processes that degrade organic contaminantssuch as hydrocarbons and chlorinated solvents are kinetic-limitedreactions, and they are described using a set of ordinary differ-ential equations (ODEs). Within a numerical reactive transportformulation, these ODEs are normally implemented as a reactionpackage using the operator-split strategy and are independentlysolved by an ODE solver (Clement et al., 1998). The geochemicalprocesses (which mediate the fate and transport of inorganiccontaminants such as metals) on the other hand are mostlyequilibrium-controlled reactions that require solution to a set ofnon-linear algebraic equations. These non-linear equilibriumequations can be solved by an independent geochemistry routineand they can also be integrated into a transport formulation usingthe operator-split strategy (Cederberg et al., 1985; Prommer et al.,1998). There are several computer codes available in the litera-ture that can be used to solve chemical speciation problems.WATEQ was one of the first geochemical models that uses aniteration scheme to solve the system of non-linear equations; thecode, however, cannot explicitly handle heterogeneous reactionssuch as precipitation and dissolution processes (Truesdell andJones, 1974). MINEQL is a commonly used model that employsthe well-known tableau approach to define and solve the chemi-cal equilibrium problem. MINEQL is capable of handling reactionssuch as mineral precipitation and dissolution (Westall, 1976).MICROQL-I is a simplified version of the MINEQL code which cansolve chemical equilibrium without sorbed or solid phase species(Westall, 1979a). MICROQL-II is an updated version that includesroutines for modeling adsorption equilibrium using constantcapacitance, diffuse layer, and triple layer adsorption models(Westall, 1979b). The chemical speciation code MINTEQA2 isthe most comprehensive software (also developed based on theideas espoused in MINEQL and MICROQL codes) for modelingdilute aqueous solutions (Allison et al., 1990). PHREEQC is another
widely used geochemical code that can perform a variety ofgeochemical speciation calculations (Parkhurst and Appelo,1999). PHAST is a three-dimensional reactive-transport modelderived from coupling the geochemical model PHREEQC with thesolute-transport model HST3D. The flow and transport model isrestricted to constant density, constant temperature, and satu-rated ground–water flow conditions. Chemical reactions consid-ered in HST3D include mineral and gas equilibrium, ion exchange,surface complexation, solid solutions, and kinetic reactions(Parkhurst, 2004).
Most of the multi-dimensional, numerical transport codesdiscussed above require considerable experience and expertiseto model coupled reactive transport problems. Therefore, in thepublished literature, laboratory researchers have developed sev-eral simpler one-dimensional reactive transport tools to modelcolumn-scale datasets. For example, Miller and Benson (1983)developed a numerical model CHEMTRN to simulate the transportof solutes in saturated porous media. The model simulatedadvection, dispersion, ion exchange, and formation of aqueous-phase complexes. Engesgaard and Kipp (1992) developed a one-dimensional geochemical code to simulate precipitation-dissolution and oxidation-reduction reactions and used it tomodel pyrite oxidation processes at a field site in Denmark.Zysset et al., (1994) presented a numerical model for describingreactive transport processes occurring within a biofilm. Clementet al. (1996) developed a one-dimensional model to simulatebioremediation patterns occurring near an injection well. Clementet al. (1997) developed a one-dimensional code to simulatebacterial transport and denitrification processes observed in acolumn experiment. Prommer et al. (1998) developed a one-dimensional numerical model for predicting biodegradationoccurring at a petroleum hydrocarbons site. Islam and Singhal(2002) presented a one-dimensional multi-component reactivetransport model coupled with geochemical equilibrium reactionsto simulate the interactions between the microbial redox reac-tions and inorganic geochemical reactions. Amos et al. (2009)developed a numerical model to study the enhanced dissolutionof PCE-DNAPL in presence of dechlorinating bacterial cultures.Clement et al. (2004) developed a code for modeling DNAPL-dissolution and rate-limited sorption occurring in a biologicallyreactive one-dimensional porous media system. Schaefer et al.(2009) and Torlapati et al. (2012) developed one-dimensionalmodels which were used to simulate laboratory studies thatexplored the effects of bioaugmentation on chlorinated solventcontaminants.
Unfortunately, most of the one-dimensional tools discussedabove, which are primarily developed for solving a specificresearch problem, have little or no documentation. Moreover,none of these codes are user friendly tools that can be used byother laboratory researchers. Also, these tools support either akinetic formulation or an equilibrium formulation; none of thesemodels provide a flexible framework, such as an EXCEL interface,which would allow users to modify the code to their specificneeds. The objective of this effort is to develop a comprehensive,one-dimensional reactive transport model within a user-friendly,EXCEL-based Visual Basic environment. Our goal is to provide aunified EXCEL tool that can be easily adapted by others to modellaboratory-scale experiments involving different types of biologi-cal (kinetic) and geochemical (equilibrium) reactions. In thispaper, we present the details of this tool and demonstrate itsuse by solving five benchmark problems. The benchmark pro-blems illustrate the characteristics of a variety of bio-geochemicalproblems that would be of interest to a broad range of environ-mental scientists. The problems are described in detail to providea comprehensive benchmarking database which can be used fortesting other reactive transport codes.
J. Torlapati, T. Prabhakar Clement / Computers & Geosciences 50 (2013) 72–8374
2. Model details
The multi-component one-dimensional reactive transportmodel developed in this study, designated as RT1D, solves acoupled set of advection–dispersion-reaction equations for a totalof ‘‘n’’ components. The model simulates the transport of ‘‘m’’mobile components that are either fully or partially coupled to aset of ‘‘n–m’’ immobile components. The reactions between thesecomponents could be mediated by biological/geochemical kineticreactions, or geochemical equilibrium reactions. The governingset of equations solved by the model can be written in a generalform:
where V is the velocity (m/day); D is the hydrodynamic dispersioncoefficient (m2/day), Ci is the aqueous phase concentration (mg/L)of a mobile component i; Sj is the solid phase concentration (mg/mg) of an immobile component j; m is the total number of mobilecomponents; n is the total number of components (note that atotal of (n–m) immobile components are numbered sequentiallyafter numbering all the mobile components); Ri is the linearretardation factor of the ith mobile component [R¼ 1þðrKd=jÞ
� �,
where r is the bulk density (mg/L); j is the porosity; Kd is thelinear sorption constant (L/mg)]; and bi and bj are the reactionsinvolving mobile and immobile components, respectively. Theexpressions used for bi and bj terms would vary depending on thetype of reactions involved in the system. Note the immobilecomponent equations do not have advection dispersion terms, butwill have reaction terms that will be coupled to some of themobile component reaction terms. Also, the mobile-componentreaction terms themselves could be coupled to each other. Thebenchmark problems described in the next section illustrate thesecoupling effects.
The coupled set of reactive transport equations, represented byEqs. (1) and (2) are solved using the operator split approach(Clement et al., 1998). Using this approach, the governing set oftransport equations can be written as:
@Ci
@t¼�
V
Ri
@Ci
@xð3Þ
@Ci
@t¼
D
Ri
@2Ci
@x2ð4Þ
dCi
dt¼
bi
Rið5Þ
dSj
dt¼ bj ð6Þ
In the numerical code, the advection terms in all the mobilecomponents (Eq. 3) are first solved using an advection solvermodule. Next, the advected concentrations (C) are used to solvefor dispersion terms (Eq. 4) in the mobile component transportequations using a dispersion solver module. Finally, the dispersedconcentrations (C ) are used to solve a set of coupled reactionterms involving both mobile and immobile components, repre-sented by Eqs. (5) and (6). For transport problems involvingkinetic reactions, the reaction part of the transport equationsEqs. (5) and (6) would yield a set of coupled ODEs. These ODEs,referred as the kinetic reaction package, are solved using an ODEsolver. For transport problems involving geochemical equilibriumreactions, the reaction terms would yield a set of coupled non-linear equations. These non-linear equilibrium equations are
represented using the tableau approach (Westall, 1976) and aresolved using the geochemical equilibrium solver, MICROQL,developed by Westall (1979a, b).
The advection module provides two explicit solver options: atotal variation diminishing (TVD) solver and an explicit finitedifference solver that uses backward difference approximation.The advected concentrations are then used to solve the dispersionequation, within the dispersion module, using the implicit finitedifference method. In addition to these explicit and implicitsolvers, there is also a fully-implicit option that solves theadvection–dispersion terms together using a fully-implicitapproach. Appendix-A provides the numerical details of thesetransport solvers. The set of ODEs described within a kineticreaction package can be solved using two different ODE solvers: astandard 4th order Runge–Kutta (RK) solver, or a more robustRunge–Kutta–Fehlberg (RKF) solver (Chapra and Canale, 1998).The RK solver uses a constant reaction time step, whereas the RKFsolver will automatically subdivide the reaction time step intosub steps to minimize the local error.
The geochemical equilibrium reaction problem is formulated inthe form of a tableau that represents the interactions between allthe components and species involved in the chemical system. Asdefined by Westall (1976), a species is a chemical entity of interestpresent in the system whereas a component is a basic building blockused for forming various chemical species in the system. Thestoichiometric relationship between the components and the spe-cies can be represented in the form of a matrix known as thetableau. The chemical speciation problem, defined by the tableau, issolved using an EXCEL-VBA version of MICROQL code. The details ofthe numerical solution schemes employed by MICROQL are dis-cussed in Westall (1979b). Within RT1D, the transport equationsthat involve geochemical (or equilibrium) reactions are solved usingan approach proposed by Cederberg et al. (1985), which is slightlydifferent from the approach used for the solving the transportequations involving kinetic reactions. As discussed in Cederberget al. (1985), first the aqueous concentration of a component ofinterest will be transported using the transport module. The aqu-eous concentration of a component at a particular node is calculatedby subtracting the concentrations of all the sorbed species asso-ciated with that component from its total concentration. After thetransport step, the updated (advection–dispersed) aqueous compo-nent concentration is added back to the sorbed concentrations of therespective component to compute the total component concentra-tion at that node. This total component concentration is thentransferred to MICROQL to solve the geochemical speciation pro-blem. The equilibrated species concentrations are used to update thevalues of aqueous component concentrations for the next timetransport step. Further details of this transport algorithm arediscussed in Cederberg et al. (1985).
The RT1D model was designed to provide a unified platformfor simulating transport problems involving a variety of geo-chemical and kinetic reactions. Fig. 1 shows various features ofthe code. As shown in the figure, RT1D supports two independentmodules, namely kinetic module and equilibrium module, formodeling kinetic and equilibrium reactions, respectively. If thekinetic module is selected, the user should also provide aproblem–dependent reaction package. The kinetic module sup-ports several standard reaction models that are already codedwithin a set of pre-programmed reaction packages. In addition tothese preprogrammed packages, RT1D also supports a user-defined reaction package via which the user can input any typeof kinetics. The geochemical module, on the other hand, does notrequire a problem–dependent reaction package, and instead usesMICROQL to solve the equilibrium problem. As describe before,the information required for formulating a specific geochemicalproblem are input using the reaction tableau.
Fig. 1. Schematic of RT1D simulations options.
J. Torlapati, T. Prabhakar Clement / Computers & Geosciences 50 (2013) 72–83 75
The current version of RT1D is also capable of running inbatch-kinetic or batch-equilibrium modes. When running in abatch mode, the program automatically sets the total number ofnodes to 1, and the number of mobile components to zero. In thebatch mode, reactions (both aqueous and solid phase reactions)are viewed as immobile reactions occurring within a hypotheticalnode. Overall, RT1D supports four simulation options (see Fig. 1).The first option can be used to solve batch kinetic problems. Thesecond option can be used to solve one-dimensional reactivetransport problems involving kinetic reactions. The third optioncan solve batch equilibrium problems. The fourth option can solveone-dimensional reactive transport problems involving geochem-ical equilibrium reactions.
RT1D can simulate constant or pulse-type boundary conditionsat the inlet. The outlet boundary is defined using a standard free-flow boundary condition (Clement et al., 1997). The initial con-centrations are defined through the spreadsheet input interface.Also, if required, spatially variable initial concentration arrays canbe read into the code by modifying the initial concentrationsubroutine. Further details of input and output data structuresfor different types of problems are provided in Torlapati andClement (2012). In the following sections, we discuss solutions tofive different reactive transport benchmark problems that wereused to test the RT1D code. These benchmarks vastly vary in thedegree of reaction complexity and are chosen to demonstratevarious features of the code. All the simulations were completedon a personal desktop computer that supported Microsoft Win-dows XP operating system and Excel 2007 software. The code isfully compatible with the newer version of Windows and Excel.
Table 1Model parameters used for benchmark problem-1.
Length (cm) 40
Total time (days) 50
Dx 0.4
Dt 0.01
Pore velocity (cm/day) 0.53
Longitudinal dispersion coefficient
(cm2/day)
0.08
Mobile components 1
Immobile components 1
3. Benchmarking results
3.1. Benchmark problem 1—Rate-limited sorption reaction in
porous media
Adsorption reaction is one of the most important transportprocesses occurring in groundwater systems. While modelingsorption, most models assume the aqueous phase concentrationis in equilibrium with the solid phase concentration. However,this equilibrium assumption might not be valid under severalpractical conditions. For example, concentrated stresses inducedby pumping and injection wells could lead to disequilibriumconditions. Such non-equilibrium problems require a kinetic
formulation to describe the rate-limited sorption effects. Thefollowing two transport equations were used to model thetransport processes in a one-dimensional porous media systeminvolving rate-limited linear sorption reactions:
@C
@t¼�V
@C
@xþD
@2C
@x2�x C�
S
Kd
� ��kC ð7Þ
rj@S
@t¼ x C�
S
Kd
� �ð8Þ
where C is the concentration of the aqueous phase component(mg/L); S is the concentration of the solid phase component(mg/mg); r is the bulk density (mg/L); j is the porosity; Kd isthe linear sorption constant (L/mg); k is the first order decayconstant (day�1); and x is the mass transfer coefficient (day�1).Clement et al. (1998) used a similar type of formulation to modelrate-limited reactions, although their study ignored the first orderdecay term. This reactive transport problem involves two com-ponents: a mobile component that represents the aqueous phaseconcentration (C), and an immobile component that representsthe solid phase concentration (S). Using the operator split strat-egy, the reaction kinetics for this problem can be formulated as:
dC
dt¼�x C�
S
Kd
� ��kC ð9Þ
dS
dt¼jxr C�
S
Kd
� �ð10Þ
Eqs. (9) and (10) are coded into a reaction package. Thecolumn was assumed to be initially clean and the left handboundary condition was fixed at 1 mg/L. The porosity of thecolumn was assumed to be 0.3, the bulk density of the porousmedia (r) was set to 1600 g/L, and the sorption constant (Kd) wasset at 1.875�10�4 L/g. The model was run using three differentmass transfer coefficients (x): 0.00015, 0.015, and 2 (day�1).Other model parameters used in this benchmark problem aresummarized in Table 1. Note when the value of mass transfercoefficient is low, the solute is expected to behave like a tracerwith R¼1; on the other hand, when the mass transfer coefficientis high, the solute is expected to behave like a retarded plumewith R¼2. The scenarios in-between these two extreme condi-tions would result in rate limited, non-equilibrium transportconditions. Toride et al. (1993) presented a set of analyticalsolutions for transport equations involving non-equilibrium sorp-tion and first-order decay terms. Valocchi and Werth (2004)developed a web-based Java applet to implement the analyticalsolutions developed by Toride et al. (1993). This Java applet wasused to benchmark the results of the RT1D code. The definition ofmodel parameters used in the analytical solution vary slightlyfrom the model definitions described above; in order to comparethe results, the mass transfer coefficient to be used in theanalytical solutions must be calculated using the formulax0 ¼ ðjx=rKdÞ where x0 is the mass transfer coefficient used inthe analytical solution.
J. Torlapati, T. Prabhakar Clement / Computers & Geosciences 50 (2013) 72–8376
Fig. 2a shows the aqueous phase concentrations simulated byRT1D for different values of mass transfer coefficients; the figurealso shows the analytical solution results. Similar results for solidphase concentrations are shown in Fig. 2b–d. Simulations werealso completed using a constant decay co-efficient (k) of 0.03day�1 and Fig. 3 compares the numerical results with analyticalresults. The total computer time required for solving this bench-mark problem was about 18 s. The figures show that the resultsfrom the RT1D simulations were able to match the analyticalresults. Furthermore, as expected, Fig. 2a shows that the aqueousphase concentration profile was retarded by a factor of R¼2,when the mass transfer coefficient was set to an arbitrarilyhigh value.
0.0
0.2
0.4
0.6
0.8
1.0
0 10 20 30 40
Con
cent
ratio
n (m
g/L)
Length (cm)
Fig. 3. Comparison of the RT1D results with the analytical solutions for the
benchmark problem-1 with a decay rate constant value of 0.03 day�1.
3.2. Benchmark problem 2—Microbial transport and growth under
denitrification conditions
Clement et al. (1997) studied the effects of denitrifyingconditions on the growth and transport of bacteria in a porousmedia column under two substrate loading conditions. A numer-ical model was developed to generate the breakthrough profilesof bacterial cells and substrates. A first order attachment anddetachment model was used to describe the exchange processesbetween mobile and immobile-phase bacterial cells. This bench-mark problem considered three mobile components namelynitrate, acetate and aqueous-phase bacteria, and one immobilecomponent namely immobile bacteria. The reaction package usedin the problem is given below:
dCN
dt¼�rNXa�
rNXsrn
ð11Þ
0.0
0.2
0.4
0.6
0.8
1.0
Aqu
eous
con
cent
ratio
n (m
g/L)
Distance (cm)
0.00E+00
2.00E-05
4.00E-05
6.00E-05
8.00E-05
1.00E-04
1.20E-04
Solid
Con
cent
ratio
n (m
g/m
g)
Distance (cm)
0 5 10 15 20 25 30 35 40
0 10 20 30 40
Fig. 2. Comparison of RT1D simulations with analytical solutions for benchmark probl
phase concentrations for x¼0.00015; (c) solid phase concentrations for x¼0.015 and (
dCA
dt¼�rAXa�
rAXsrn
ð12Þ
dXa
dt¼ rXXa�KatXaþ
KdeXsrn
ð13Þ
dXs
dt¼ rXXs�KdeXsþ
nKatXa
r ð14Þ
where CN, CA, Xa and Xs are concentrations (mg/L) of nitrate,acetate, aqueous-phase bacteria and immobile-phase bacteria(mg/mg), respectively. The parameters Kat (day�1) and Kde
(day�1) are the attachment and detachment coefficients ofmobile and immobile phase bacteria, respectively; n is the
em-1: (a) aqueous concentration for different mass transfer coefficients; (b) solid
d) solid phase concentrations for x¼2.
J. Torlapati, T. Prabhakar Clement / Computers & Geosciences 50 (2013) 72–83 77
porosity of soil; and r is the bulk density of the soil (mg/L). Therate expression rN is the nitrate utilization rate described usingMonod kinetics as: rN ¼ qmaxðCN= KNþCNÞð ÞðCA= KAþCAÞð Þ, where
Table 2Model parameters used for benchmark problem-2.
Fig. 4. Comparison of the RT1D model results (solid line) with the published model res
nitrate for low substrate conditions; (b) effluent biomass for low substrate conditions;
substrate conditions.
qmax is the maximum nitrate utilization rate (mg nitrate/mgbiomass-day), KN is the half saturation coefficient for nitrate(mg/L); KA is the half saturation coefficient for acetate (mg/L).The specific utilization rate of acetate (rA) and biomass growthrate (rX) are given by the expressions: rA¼YA/NrN and rX ¼ YX=N
rN�Kd, where YA/N and YX/N are the yield coefficients for acetateand biomass, respectively, and Kd is the cell decay rate coefficient(day�1). A finite difference grid of size 1 cm and a time step of0.001 day were used in this problem. Other model parametersused are summarized in Table 2. Further details of the experi-ments are available in Clement et al. (1997). The total amount ofcomputer time required for solving this benchmark problem wasabout 590 s. Fig. 4a–d compare RT1D simulation results withpublished model results and data available in the literature.Fig. 4a and c compare effluent concentrations of nitrate atdifferent times and Fig. 4b and d show mobile phase bacteriaconcentrations in the effluent. The results show that the RT1Dmodel simulations closely matched published data.
3.3. Benchmark problem 3—Carbon tetrachloride biodegradation
Phanikumar et al. (2002) developed a bioremediation model topredict carbon tetrachloride (CT) degradation processes observedin sequential column experiment. In this study, we have used oneof their experiments, identified as once-fed (OF) column, as abenchmark problem. The laboratory experiment used a 200 cmlong column fitted with an 11-cm long slug injection zone at adistance of 34 cm away from the column inlet. The injection zonewas fitted with an inlet and an outlet to circulate flow within thiszone. This injection–extraction setup was used to inoculate thecolumn with nutrients and mobile bacteria for about 16 min by
ults (dotted line) and published data (dots) for benchmark problem-2: (a) effluent
(c) effluent nitrate for high substrate conditions and (d) effluent biomass for high
J. Torlapati, T. Prabhakar Clement / Computers & Geosciences 50 (2013) 72–8378
circulating flow at rate of 20 ml/min. The inoculation step wascompleted just once at the beginning of the experiment. It wasassumed that the inoculation step completely replaced the initialcontents of the slug injection zone and hence the concentrationsin the inoculant solution were used as the initial conditions forthe 11 cm zone. Table 3 summarizes the details of the boundaryand initial conditions used in this problem for the entire column.The transport problem considered four mobile components:carbon tetrachloride, acetate, nitrate, and mobile-phase bacteria;and two immobile components: sorbed carbon tetrachloride andimmobile-phase bacteria. The reaction package used for modelingthis bioremediation problem, as provided in Phanikumar et al.(2002) is:
1þrf Kd
j
� �dCCT
dt¼�k0CCT ðXMþXIMÞ�
rkj
1�fð ÞKdCCT�SCT
� �ð15Þ
RadCa
dt¼�
mmaxMaMn
YaðXMþXIMÞ ð16Þ
RndCn
dt¼�
mmaxMaMn
YnðXMþXIMÞ�
bKC
Ynbð1�MaÞþgMn
� ðXMþXIMÞ
ð17Þ
dXM
dt¼ mmaxMaMn�bKCð1�MaÞ�Kat
� �XMþKdeð1�MaÞXIM ð18Þ
Table 3Model parameters used for benchmark problem-3.
Note: carbon tetrachloride has been abbreviated as CT.
dSCT
dt¼ k 1�fð ÞKdCCT�SCT
� �ð19Þ
dXIM
dt¼ mmaxMaMn�bKCð1�MaÞ�Kdeð1�MaÞ� �
XIMþKatXM ð20Þ
where f is the fraction of equilibrium sites, bKC is the microbialdecay rate (day�1), Kat is the attachment coefficient (day�1), Kde isthe detachment coefficient (day�1), k0 is the CT reaction rate(day�1), g is the nitrate reaction rate (day�1), k is the kineticdesorption rate (day�1), mmax is the maximum specific growthrate (day�1), Ya, Yn and Ynb are the yield rates of acetate, nitrateand biomass, respectively; CCT, Ca, Cn and SCT are the aqueousconcentrations of carbon tetrachloride, acetate, nitrate and thesorbed concentration of carbon tetrachloride, respectively; XM
and XIM are the concentrations of mobile and immobile bacteria,respectively. Also, Ma and Mn are the Monod terms for acetate andnitrate reactions, respectively, and given by the expressions:Ma ¼ ðCa= KsaþCaÞð Þ and Mn ¼ ðCn= KsnþCnð ÞÞ where Ksa and Ksn
are the half saturation coefficients of acetate and nitrate utiliza-tion reactions, respectively. The kinetic Eqs. (15) to (17) describebiodegradation of carbon tetrachloride, utilization of an electrondonor (acetate), and an electron acceptor (nitrate). Eq. (18)describes the growth, decay, and attachment of the mobile phasebacteria, Eq. (19) describes the sorption of carbon tetrachlorideusing a two-site sorption model, and Eq. (20) describes thegrowth, decay, and detachment of immobile-phase bacteria. Thegrid size used was 1 cm and time step was 0.001 day. Other modelparameters are summarized in Table 4.
Fig. 5 compares RT1D simulation results with the publishedmodel results. Fig. 5a shows the biodegradation patterns ofcarbon tetrachloride within the column after 4 day. The totalamount of computer time required for simulating this benchmarkproblem was 28 s. As expected, the model results show increasedbiodegradation activity near the slug injection zone which wasinoculated with active bacterial cells. It can be observed from thefigures that the results from the RT1D simulations match wellwith the published model results.
3.4. Benchmark problem 4—Geochemical transport involving a
constant capacitance model
Cederberg et al. (1985) developed a research code, TRANQL, tosimulate geochemical multi-component transport in a saturatedgroundwater system. The TRANQL code was used to studycadmium transport in the presence of chloride and bromide ions.The one-dimensional reactive transport model considered advec-tion, dispersion, surface complexation of cadmium ion, andsorption of free cadmium to solids in the column. They used the
Table 4Stoichiometry of chemical reactions (tableau) for the benchmark problem-4.
Cl– Br- Cdþ2 SOH Psi Hþ log K
1 H[þ] 0 0 0 0 0 1 0
2 Cdþ2 0 0 1 0 0 0 0
3 Cl– 1 0 0 0 0 0 0
4 Br– 0 1 0 0 0 0 0
5 CdClþ 1 0 1 0 0 0 1.8
6 CdCl2 2 0 1 0 0 0 2.6
7 CdBrþ 0 1 1 0 0 0 2.2
8 CdBr2 0 2 1 0 0 0 3
9 CdOHþ 0 0 1 0 0 �1 �12.69
10 OH– 0 0 0 0 0 �1 �13.91
11 SOH 0 0 0 1 0 0 0
12 SOH2 0 0 0 1 1 1 7.4
13 SO– 0 0 0 1 �1 �1 �9.24
14 SOCdþ 0 0 1 1 �1 �1 �7
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18C
once
ntra
tion
(ppm
)
Distance (cm)
CT (RT1D)CT (Model)CT (Data)
0
300
600
900
Con
cent
ratio
n (p
pm)
Distance (cm)
Acetate (Model)
Acetate (RT1D)
0
5
10
15
20
25
30
35
40
45
Con
cent
ratio
n (p
pm)
Distance (cm)
Nitrate (Model)
Nitrate (RT1D)
1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
Con
cent
ratio
n (C
FU/m
L)
Distance (cm)
Mobile KC (RT1D)
Mobile KC (model)
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
Con
cent
ratio
n (C
FU/m
L)
Distance (cm)
Mobile KC (model)Mobile KC (RT1D)
1.00E+03
1.00E+04
1.00E+05
1.00E+06
1.00E+07
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200 250
0 50 100 150 200 250 0 50 100 150 200
Con
cent
ratio
n (C
FU/m
L)
Distance (cm)
Mobile KC (model)Mobile KC (RT1D)
Fig. 5. Comparison between the published model data (dots) and the RT1D simulations (line) for benchmark problem-3: (a) carbon tetrachloride after 4 days; (b) acetate
after 13 days; (c) nitrate after 4 days; (d) mobile KC after 7 days; (e) mobile KC after 11 days and (f) mobile KC after 14 days (Note: KC is the strain of the mobile bacteria).
J. Torlapati, T. Prabhakar Clement / Computers & Geosciences 50 (2013) 72–83 79
finite-element approach to solve the governing transport equa-tions. The geochemical problem was defined using the tableaunomenclature similar to the method presented by Westall(1979a). The chemical equilibrium problem considered a total of6 components and 14 species. Table 4 provides the reactiontableau for the problem and the log K values for all the chemicalreactions. Using the RT1D code, we tracked the concentrations ofthe following three mobile components: cadmium, bromide, andchloride. The remaining three components in this problemincluding hydrogen ion (Hþ), surface hydroxyl group (SOH) andelectrostatic potential (Psi) were not tracked in the transportmodule for the following reasons: pH and total SOH values are
fixed in this problem and hence remained constant throughoutthe simulation. The last component, electrostatic potential (Psi), isa hypothetical component which is only used within MICROQLcalculations. In order to compute the aqueous cadmium compo-nent concentration, we subtracted the concentration of sorbedcadmium species SOCdþ (species 14 in the tableau) from the totalcadmium component concentration. Similar calculations can alsobe made for other components of interest, as described inCederberg et al. (1985). The grid size used in this problem was0.03 cm and the time step was 0.06 h. Cederberg et al. (1985)solved a total of six cases with different initial and boundaryconcentration levels. These six cases were divided into two
J. Torlapati, T. Prabhakar Clement / Computers & Geosciences 50 (2013) 72–8380
groups of three cases based on the total initial and boundaryconcentrations of cadmium, chloride and bromide ions. Furtherdetails about each of these cases are available in Cederberg et al.(1985). In this benchmark exercise, we solved two cases namely,Case-1 and Case-5, described in Cederberg et al.’s study. Thesetwo cases were chosen because Case-1 is a base case scenariowhere the initial and boundary conditions chloride and bromideion concentrations remained the same at the background levels.Case-5, on the other hand, shows the system’s response whenbromide and chloride ion concentration levels were allowed to behigher than the background concentration levels. The transportparameters the initial and boundary conditions used for these twocases are summarized in Table 5. The amount of computer timerequired for solving this benchmark problem was about 30 s.
Table 5Model parameters used for benchmark problem-4.
Fig. 6. Comparison of the RT1D results (solid lines) with the published model r
and (b) simulation results for Case-V.
The results predicted by the RT1D are compared againstTRANQL model results, reported in Cederberg et al., in Fig. 6aand b. Fig. 6a shows that the aqueous-phase and sorbed-phasecadmium profiles predicted by RT1D are in good agreement withTRANQL results. Since concentrations of chloride and bromideremained constant during Case-1, they are not presented inFig. 6a. However, when the concentration of chloride wasincreased (Case-5 problem), it interacted with sorbed-phasecadmium species and this, resulted in reduced chloride ion levels;these results are shown in Fig. 6b. Overall, the results from RT1Dsimulations are in excellent agreement with published results.
3.5. Benchmark problem 5—Multiple sequential batch reactor
Jeppu et al. (2012) proposed a sequential equilibration reactor(SER) system to investigate transport problems involving geo-chemical equilibrium reactions. They studied adsorption ofAs(V) on goethite-coated sand using three sequentially linkedreactors, identified as a multiple sequential equilibration reactors(MSER). We simulated the results of their MSER experiment asour fifth benchmark problem. In this experiment, as an initialstep, the first reactor was filled with As(V) solution while reactors2 and 3 were filled with clean water. This is the initial conditionfor the problem. After equilibrating for 24 h, the aqueous solutionfrom the first reactor was transferred to the second reactor andwas allowed to equilibrate with the solids in the second reactor.During the same time period, new arsenic laden solution wastransferred to the first reactor, the solution in the second reactorwas transferred to the third reactor, and the solution in the thirdreactor was discharged for chemical analysis. The volume ofwater discharged from a single reactor was designated as the‘‘reactor volume.’’ The experiment had two distinct phases;during the first phase, a total of 14 reactor volumes weredischarged from the system while simultaneously renewing thesolution in the first reactor with new arsenic-laden solution.During the second phase, a total of 4 reactor volumes weredischarged while simultaneously replacing the solution in thefirst reactor with clean water. To simulate this experiment, ahypothetical column with 4 finite-difference nodes was used. Thelength of the each finite difference grid is set to 1 cm, and thetotal distance between node-1 to node-4 was 3 cms, representingthe 3 reactors. The velocity was assumed to be 1 cm/day andthe time step used was 1 day, maintaining a Courant number 1.Note, although there were 4 nodes in the system, the boundarynode was used as an hypothetical node to define the boundarycondition; geochemical reactions are allowed to occur only in
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
0 2 4 6 8 10
Con
cent
ratio
n (M
)
Distance, cm
Aqueous Cd (RT1D)Chloride (RT1D)Sorbed Cd (RT1D)Chloride (data)Sorbed Cd (data)Aqueous Cd (data)
esults (dots) for the benchmark problem-4: (a) simulation results for Case-I
J. Torlapati, T. Prabhakar Clement / Computers & Geosciences 50 (2013) 72–83 81
nodes 2, 3, and 4, which represented the three sequentiallycoupled reactors. There is no hydrodynamic dispersion in thissequential batch problem and hence the dispersion module wasnot used. The Courant number was set to 1 in the numericalmodel to allow one node explicit advection that exactly mimickedthe batch transfer process without any numerical dispersioneffects. The total period of simulation was 18 days. The aqueousphase As(V) concentration at the inlet boundary node was set at1.25 mM for 14 days, followed by zero concentration for 4 moredays. Other model parameters and the tableau for representing
Table 6Model parameters used for benchmark problem-5.
Length (cm) 4
Total time (days) 18
Pulse time (days) 14
Dx 1
Dt 1
Velocity (cm/day) 1
Mobile components 1
Immobile components 1
Boundary condition:
As(V) concentration (mM)
0–14 day 1.25
15–18 day 0
pH (constant/fixed) 7
Ionic strength 0.01
Surface site density (sites/nm2) 1.04
Surface area (m2/g) 1.08
Sorbent concentration (mg/L) 1.0
Table 7Stoichiometry of chemical reactions (tableau) used for benchmark problem-5.
AsO4– SOH Psi Hþ Log K
Hþ 0 0 0 1 0
AsO4[�3]Aq 1 0 0 0 0
FeOH 0 1 0 0 0
OH– 0 0 0 �1 �13.91
HAsO4[�2] 1 0 0 1 11.23
H2AsO4[�1] 1 0 0 2 18.01
H3AsO4 1 0 0 3 20.16
4FeH2AsO4 1 1 0 3 31.44
4FeHAsO4- 1 1 �1 2 26.18
4FeAsO4-2 1 1 �2 1 20.1
4FeOH2[þ] 0 1 1 1 7.17
4FeO[�] 0 1 �1 �1 �9.32
0.0
0.4
0.8
1.2
1.6
0 5 10 15 20
Con
cent
ratio
n (
Reactor volume
RT1D
PHREEQCI Model
Jeppu et al (2012) Data
Fig. 7. Comparison of the RT1D results with the published model results for
Benchmark problem-5.
the chemical reactions are given in Tables 6 and 7, respectively.The total amount of computer time required for solving thisbenchmark problem was 2 s. RT1D simulation results are com-pared against PHREEQCI (Charlton and Parkhurst, 2002) results(reported in Jeppu et al., 2012) and the experimental data (alsoreported in Jeppu et al., 2012) in Fig. 7. It can be observed fromthe figure RT1D matched the published data well.
4. Summary and conclusions
In this study, we have presented the details of a numericalmodeling tool for solving a variety of biochemical and geochem-ical reactive transport problems. The code was developed withinthe EXCEL Visual Basic platform and it can be run within thestandard EXCEL without any additional software. The tool iscapable of solving a wide range of kinetic-limited reactive trans-port problems that could be defined through a reaction package.RT1D can also solve a variety of equilibrium-controlled geochem-ical transport problems defined through a chemical reactionmatrix (also known as the tableau). The capabilities of the toolwere demonstrated by solving five benchmark problems ofvarying level of complexity. The results show that RT1D simula-tions closely matched previously published results. RT1D is aflexible tool that allows users to add their own routine to defineany type of user-defined kinetic reactions. The geochemistrypackage can be used to define and solve transport problemsinvolving a variety of surface complexation reactions. The tool isequipped with a robust TVD advection solver, an implicit disper-sion solver, and an adaptive time stepping ODE solver to handleany complex problem. RT1D code is a useful tool for laboratoryresearchers who are interested in analyzing batch and columndata within a user-friendly EXCEL platform.
Acknowledgements
This research was, in part, supported by the office of science(BER), U.S. Department of Energy Grant No. DE-FG02-06ER64213.The RT1D tool can be either downloaded from Dr. Clement’swebsite: http://www.eng.auburn.edu/users/clemept/ or can beobtained by contacting the authors.
Appendix A. Numerical details
A1. Advection module–Explicit solution
The advection part of the transport equation can be solvedusing the explicit backward difference approximation as shownbelow:
Cnþ1i �Cn
i
Dt
!¼�
V
Ri
Cni �Cn
i-1
Dx
� �ðA1Þ
where Cnþ1i is the concentration of the component at the current
time step at the current node; Cni is the concentration of the
mobile component at the previous time step at the current nodeand Cn
i�1 is the concentration of the mobile component at theprevious time step at the preceding node. After further simplifica-tion, we can solve for the concentration of mobile component atthe current node (Cnþ1
i ) as shown below:
Cnþ1i ¼�
Cr
RiCn
i �Cni�1
� �þCn
i ðA2Þ
where Cr¼ ðVDt=DxÞ is known as the grid Courant number.
J. Torlapati, T. Prabhakar Clement / Computers & Geosciences 50 (2013) 72–8382
A2. Advection module—Explicit TVD solution
Numerical dispersion is a major concern while solving theadvection dominated problems. RT1D includes a robust totalvariation diminishing (TVD) scheme that minimizes numericaldispersion errors. Details of this scheme are given below. Usingthe Taylor series expansion, the standard Lax–Wendroff (LW)scheme for the advection term can be written as (LeVeque, 2002):
Cnþ1i ¼ Cn
i �Cr
2ðCn
iþ1�Cni�1Þþ
Cr2
2ðCn
iþ1�2Cni þCn
i�1Þ ðA3Þ
The above equation can be rearranged and written in a flux-balance format as:
Cnþ1i �Cn
i
Dt¼�
Fniþ1=2�Fn
i�1=2
DxðA4Þ
where Fniþ1=2 ¼ VCn
i þV2 ð1�CrÞðCn
iþ1�Cni Þ and Fn
i�1=2 ¼ VCni�1þ
V2 ð1�CrÞðCn
i �Cni-1Þ, respectively. Note the flux terms defined above
consist of a lower and higher order flux terms. The lower orderflux term in Fn
iþ1=2is VCni and the higher order term in Fn
iþ1=2 isV2 ð1�CrÞðCn
iþ1�Cni Þ.
Furthermore in TVD schemes, a flux limiter will be used tominimize the potential numerical oscillations induced by thehigher order term as shown below:
Fniþ1=2 ¼ VCn
i þV
2ð1�CrÞðCn
iþ1�Cni Þ �F ðA5Þ
Different types of flux limiters are available in the literatureand in this study we have used the Van-leer flux limiter given as(LeVeque, 2002):
F¼yþ y
1þ y ðA6Þ
where y¼ ðCni �Cn
i�1Þ=ðCni�1�Cn
i�2Þ
A3. Dispersion module—Implicit solution
The dispersion part of the transport equation can be numeri-cally discretized using a central difference approximation.
RiCnþ1
i �Cni
Dt¼D
Cnþ1i�1 �2Cnþ1
i þCnþ1iþ1
Dx2ðA7Þ
where Ci�1 is the concentration at the previous node for thecurrent time step, Ciþ1 is the concentration at the next node forthe current time step. The above equation can be further simpli-fied as follows
�Cni ¼
lRi
Cnþ1i�1 þCnþ1
i �2lRi�1
� �þ
lRi
Cnþ1iþ1 ðA8Þ
where l¼ ðDDt=Dx2Þ. Assembling Eq. (A8) on a node-by-nodebasis would yield a following tri-diagonal matrix of the form:
1 0 0 0 0 0 0
a b c 0 0 0 0
0 a b c 0 0 0
0 0 a b c 0 0
0 0 0 a b c 0
:: :: :: :: :: :: :
0 0 0 0 0 0 1
2666666666664
3777777777775
Cnþ1
1
Cnþ1
2
Cnþ1
3
Cnþ1
4
Cnþ1
5
::
Cnþ1
n
2666666666666664
3777777777777775
¼
C0
d2
d3
d4
d5
::
dn
2666666666664
3777777777775
ðA9Þ
where a¼ ðl=RiÞ, b¼ �ð2l=RiÞ�1� �
, c¼ ðl=RiÞ and d¼�Cn
i ; Co isthe concentration at the boundary node. The above matrix can besolved using a tridiagonal matrix solver to solve for all theunknown concentrations at the new time level.
A4. Implicit solution to advection and dispersion terms
In this option, we solve the advection and dispersion togetherimplicitly. We use a central difference approximation for theadvection term and the numerically discretized form for theadvection–dispersion equation is as follows:
Cnþ1i �Cn
i
Dt
!¼�
V
Ri
Cnþ1iþ1 �Cnþ1
i�1
2Dx
!þ
D
Ri
Cnþ1i�1 �2Cnþ1
i þCnþ1iþ1
Dx2
!
ðA10Þ
The above equation can be further simplified as follows:
�Cni ðRi �CÞ ¼ ðaþ1ÞCnþ1
i�1 þð�2�Ri �CÞCnþ1i þð1�aÞCnþ1
iþ1 ðA11Þ
where C¼ ðDx2=DDtÞ� �
and a¼ ðVDx=2DÞ. Expanding the Eq.(A11) for all the nodes would yield a tridiagonal matrix similarto A9. For this problem, the values of a, b, c and d are given asaþ1, 2�Ri �C, 1�a and �Cn
i ðRi �CÞ, respectively.
References
Allison, J.D., Brown, D.S., Novo-Gradac, K.J., 1990. MINTEQA2/PRODEFA2, a geo-chemical assessment model for environmental systems: version 3.0. Environ-mental research laboratory, office of research and development, U.SEnvironmental Protection Agency, Athens. Georgia, 106.
Amos, B.K., Suchomel, E.J., Pennell, K.D., Loffler,, E.,., 2009. Spatial and temporaldistributions of Geobacter lovleyi and Dehalococcoides sp. during bioenhancedPCE-NAPL dissolution. Environmental Science & Technology 43, 1977–1985.
Atlas, R.M., Cerniglia, C.E., 1995. Bioremediation of petroleum pollutants: diversityand environmental aspects of hydrocarbon biodegradation. Bioscience 45,332–338.
Aziz, C.E., Newell, C.j., Gonzales, J.R., Haas, P., Clement, T.P., Sun, Y., 2000.BIOCHLOR: Natural Attenuation Decision Support System: User’s ManualVersion 1.0. National Risk Management Research Laboratory, Office ofResearch and Development, US Environmental Protection Agency.
Beeman, R.E., Bleckmann, C.A., 2002. Sequential anaerobic-aerobic treatment of anaquifer contaminated by halogenated organics: field results. Journal of Con-taminant Hydrology 57, 147–159.
Cederberg, G.A., Street, R.L., Leckie, J.O., 1985. A groundwater mass transport andequilibrium chemistry model for multicomponent systems. Water ResourcesResearch 21, 1095–1104.
Chapra, S.C., Canale, R.P., 1998. Numerical Methods for Engineers with Program-ming and Software Applications, third ed. WCB/McGraw-Hill.
Charlton, S.R., Parkhurst, D.L., 2002. PHREEQCI: a graphical user interface to thegeochemical model PHREEQC. US Department of the Interior, US GeologicalSurvey.
Clement, T.P., 2001. Generalized solution to multispecies transport equationscoupled with a first-order reaction network. Water Resources Research 37,157–163.
Clement, T.P., 2011. Bioremediation of contaminated groundwater systems. In:Aral, M.M., Taylor, S. (Eds.), Groundwater Quantity and Quality Management.American Society of Civil Engineers (ASCE), pp. 522–559.
Clement, T.P., Hooker, B.S., Skeen, R.S., 1996. Numerical modeling of biologicallyreactive transport near nutrient injection well. Journal of EnvironmentalEngineering 122, 833–839.
Clement, T.P., Johnson, C.D., Sun, Y., Klecka, G.M., Bartlett, C., 2000. Naturalattenuation of chlorinated ethene compounds: model development andfield-scale application at the Dover site. Journal of Contaminant Hydrology42, 113–140.
Clement, T.P., Peyton, B.M., Skeen, R.S., Jennings, D.A., Petersen, J.N., 1997.Microbial growth and transport in porous media under denitrification condi-tions: experiments and simulations. Journal of Contaminant Hydrology 24,269–285.
Clement, T.P., Sun, Y., Hooker, B.S., Petersen, J.N., 1998. Modeling multispeciesreactive transport in ground water. Ground Water Monitoring & Remediation18, 79–92.
Coleman, N.V., Mattes, T.E., Gossett, J.M., Spain, J.C., 2002. Biodegradation of cis-dichloroethene as the sole carbon source by a {beta}-proteobacterium. Appliedand Environmental Microbiology 68, 2726–2730.
de Blanc, P., McKinney, D., Speitel, G., Delshad, M., Pope, G.A., Sepehrnoori, K.,1996. UTCHEM biodegradation model description and capabilities. Center forPetroleum and Geosystems Engineering, 1–17.
Engesgaard, P., Kipp, K.L., 1992. A geochemical transport model for redox-controlled movement of mineral fronts in groundwater flow systems: a caseof nitrate removal by oxidation of pyrite. Water Resources Research 28,2829–2843.
J. Torlapati, T. Prabhakar Clement / Computers & Geosciences 50 (2013) 72–83 83
Islam, J., Singhal, N., 2002. A one-dimensional reactive multi-component landfillleachate transport model. Environmental Modelling & Software 17, 531–543.
Jeppu, G.P., Clement, T.P., Barnett, M.O., Lee, K.-K., 2012. A modified batch reactorsystem to study equilibrium-reactive transport problems. Journal of Contami-nant Hydrology 129-130, 2–9.
Lu, G., Clement, T.P., Zheng, C., Wiedemeier, T.H., 1999. Natural attenuation ofBTEX compounds: model development and field-scale application. GroundWater 37, 707–717.
Miller, C., Benson, L., 1983. Simulation of solute transport in a chemically reactiveheterogeneous system: model development and application. Water ResourcesResearch 19, 381–391.
Moreau, M., 1987. Some European perspectives on prevention of leaks fromunderground petroleum storage systems. Ground Water Monitoring & Reme-diation 7, 45–48.
Parkhurst, D.L., 2004. PHAST—A Program for Simulating Ground-Water Flow,Solute Transport, and Multicomponent Geochemical Reactions. US Depart-ment of the Interior, US Geological Survey.
Parkhurst, D.L., Appelo, C.A.J., 1999. User’s Guide to PHREEQC (version 2)—AComputer Program for Speciation, Batch-Reaction, One-Dimensional T5ran-sport, Reaction Path, and Inverse Geochemical Calculations,. U.S GeologicalSurvey Water-Resources Investigations Report, p. 312 pp.
Perlman, H. 2011. Groundwater Use, The USGS Water Science School. /http://ga.water.usgs.gov/edu/wugw.htmlS (Last accessed on May 9th, 2012).
Phanikumar, M.S., Hyndman, D.W., Wiggert, D.C., Dybas, M.J., Witt, M.E., Criddle,C.S., 2002. Simulation of microbial transport and carbon tetrachloride biode-gradation in intermittently-fed aquifer columns. Water Resources Research 38,1–13.
Prommer, H., Barry, D.A., Davis, G.B., 1998. A one-dimensional reactive multi-component transport model for biodegradation of petroleum hydrocarbons ingroundwater. Environmental Modelling and Software 14, 213–223.
Quezada, C.R., Clement, T.P., Lee, K.K., 2004. Generalized solution to multi-dimensional multi-species transport equations coupled with a first-orderreaction network involving distinct retardation factors. Advances in WaterResources 27, 507–520.
Rafai, H.S., Bedient, P., Borden, R.C., Haasbeek, J.F., 1987. BIOPLUME II—ComputerModel of Two-Dimensional Contaminant Transport Under the Influence ofOxygen Limited Biodegradation in Ground Water. Robert S. Kerr Environ-mental Research Laboratory, Ada, Oklahoma, pp. 17–19.
Rafai, H.S., Newell, C., Gonzales, J., Dendrou, S., Dendrou, B., 1998. BIOPLUME III:Natural Attenuation Decision Support System, User’s Manual Version 1. 0.National Risk Management Research Laboratory, Office of Research andDevelopment, U.S Environmental Protection Agency, Cincinnati, OH.
Rolle, M., Clement, T.P., Sethi, R., Di Molfetta, A., 2008. A kinetic approach forsimulating redox-controlled fringe and core biodegradation processes ingroundwater: model development and application to a landfill site in Pied-mont, Italy. Hydrological Processes 22, 4905–4921.
Schaefer, C.E., Towne, R.M., Vainberg, S., McCray, J.E., Steffan, R.J., 2010. Bioaug-mentation for treatment of dense non-aqueous phase liquid in fracturedsandstone blocks. Environmental Science & Technology 44, 4958–4964.
Srinivasan, V., Clement, T., 2008. Analytical solutions for sequentially coupled one-dimensional reactive transport problems-Part I: Mathematical derivations.Advances in Water Resources 31, 203–218.
Srinivasan, V., Clement, T., Lee, K., 2007. Domenico solution—Is it valid? GroundWater 45, 136–146.
Sun, Y., Clement, T.P., 1999. A decomposition method for solving coupled multi-species reactive transport problems. Transport in Porous Media 37, 327–346.
Sun, Y., Petersen, J., Clement, T., 1999. Analytical solutions for multiple speciesreactive transport in multiple dimensions. Journal of Contaminant Hydrology35, 429–440.
Todd, D.K., 1980. Groundwater Hydrology, second ed. John Wiley & Sons.Toride, N., Leij, F.J., Van Genuchten, M.T., 1993. A comprehensive set of analytical
solutions for nonequilibrium solute transport with first-order decay and zero-order production. Water Resources Research 29, 2167–2182.
Torlapati, J., Clement, T.P. 2012. RT1D v1.0 Multi-Species Reactive Transport inOne-Dimensional System Manual. /http://www.eng.auburn.edu/�clemept/RT1D/RT1D_documentation.pdfS (Last accessed on August 1, 2012).
Torlapati, J., Clement, T.P., Schaefer, C.E., Lee, K.-K., 2012. Modeling Dehalococcoidessp. Augmented Bioremediation in a Single Fracture System. Ground WaterMonitoring & Remediation.http://dx.doi.org/%2010.1111/j.1745-6592.2011.01392.x.
Truesdell, A.H., Jones, B.F., 1974. WATEQ, a computer program for calculatingchemical equilibria of natural water. U.S. Geological Survey, Journal ofResearch, 233–248.
Valocchi, A.J., Werth, C.J., 2004. Web-based interactive simulation of groundwaterpollutant fate and transport. Computer Applications in Engineering Education12, 75–83.
Westall, J.C., 1976. MINEQL: A Computer Program for the Calculation of theChemical Equilibrium Composition of Aqueous System. Department of CivilEngineering, Massachussets Insitute of Technology, Cambridge, p. 91.
Westall, J.C., 1979a. MICROQL. I. A Chemical Equilibrium Program in BASIC. SwissFederal Institute of Technology, EAWAG, Dubendorf, Switzerland.
Westall, J.C., 1979b. MICROQL. II. Computation of Adsorption Equilibria in BASIC.Swiss Federal Institute of Technology, Dubendorf, Switzerland.
Zheng, C., Wang, P.P., 1999. MT3DMS, A Modular Three-Dimensional MultispeciesTransport Model for Simulation of Advection, Dispersion and ChemicalReactions of Contaminants in Groundwater Systems; Documentation andUser’s Guide. U.S. Army Engineer Research and Development Center ContractReport SERDP-99-1, Vicsburg, MS, p. 202.
Zysset, A., Stauffer, F., Dracos, T., 1994. Modeling of reactive groundwatertransport governed by biodegradation. Water Resources Research 30,2423–2434.
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