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USER MANUAL FOR RT1D – Multi-Component

1. OG _ VG 3 t R ox OC DOC ot OR Ox 9 dc Lf 5 11 dS 6 dt 5P In the numerical code the advection terms in all the mobile components equation 3 are first solved using an advection solver module Next the advected concentrations C are used to solve for dispersion terms equation 4 in the mobile component transport equations using a dispersion solver module Finally the dispersed concentrations C are used to solve a set of coupled reaction terms involving both mobile and immobile components represented by equations 5 and 6 For transport problems involving kinetic reactions the reaction part of the transport equations equations 3 amp 6 would yield a set of coupled ODEs These ODEs referred as the kinetic reaction package are solved using an ODE solver For transport problems involving geochemical equilibrium reactions 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 are solved using the geochemical equilibrium solver MICROQL developed by Westall 1979a b 2 1 TRANSPORT MODULE The advection module provides two explicit solver options a total variation diminishing TVD solver and an explicit finite difference solver that uses backward difference approximation The advected concentrations are then used to solve the dispersion equation within the dispersion mo
2. a Cr C C and F VC gt Cyc C respectively Note the flux terms od above consist of a lower and higher N flux terms The lower order flux term in E is VpC and the higher order term in EF is Z U CY PC PC Furthermore in TVD schemes a flux limiter will be used to minimize the potential numerical oscillations induced by the higher order term as shown below F VC l cr c 1 C 11 i Different types of flux limiters are available in the literature and in this study we have used the Van leer flux limiter given as Leveque 2002 i 0 0 ra ee Where 9 oie Ci Ce 2 1 3 IMPLICIT FINITE DIFFERENCE METHOD FOR DISPERSION SCHEME 13 The dispersion part of the transport equation can be numerically discretized using a central difference approximation At Ax R Ca 72G e 13 Where Cj 1s the concentration at the previous node for the current time step Cj is the concentration at the next node for the current time step The above equation can be further simplified as follows Where A ne Assembling equation 14 on a node by node basis would yield a following tri diagonal N of the form 1 0000 0 0C JC a b c 0 0 0 0 C Jd 0 abc 00 of fd 0 0 a b c 0 0 C d 15 000 a bc OFC d a 00000 0 1 PN 24 A n Where a r 2 1 SR and d C C is the concentration at the boundary node The above matrix can be solved using a tridia
3. without quotes This will take the user to the UsrRxnPkg subroutine in the program where the reaction package can be set Four species sequential first order decay is set by default and this can be modified to the user s convenience The ordinary differential equation for the first species is dydt 1 and the concentration of the first species 1s Conc 1 the retardation is R 1 and RC 1 is the first reaction parameter and so on Therefore the equation for a first order decay of a single species can be written as dydt 1 RC 1 Conc 1 where RC 1 1s the first order decay constant of the first species set by the user in the spreadsheet Several example reaction packages are shown in the section above These could be used as an example to program your own reaction packages The reaction terms could be either defined inside the subroutine or outside the spreadsheet It 1s recommended that the reaction terms be defined in spreadsheet so they could be changed quickly without having to open the code each time to modify these variables 32 4 EXAMPLE PROBLEMS 4 1 EXAMPLE 1 REACTION PACKAGE 1 In this example problem we will learn to use the reaction package 1 This reaction package simulates the first order decay of a single mobile component We will use this reaction package to simulate a simple tracer transport and a reactive transport problem with a first order decay constant of 0 075 day 1 Input the advection dispersion parameters
4. 2012 Benchmarking a Visual Basic based Multi Component One Dimensional Reactive Transport Modeling Tool Computers amp geosciences dol 10 1016 j cageo 2012 08 009 Valocchi A J Werth C J 2004 Web based interactive simulation of groundwater pollutant fate and transport Computer Applications in Engineering Education 12 75 83 van Genuchten M T Alves W 1982 Analytical solutions of the one dimensional convective dispersive solute transport equation Technical Bulletin Westall J 1976 MINEQL A computer program for the calculation of the chemical equilibrium composition of aqueous system Massachusetts Institute of Technology Cambridge MA Westall J C 1979a MICROQL I A Chemical Equilibrium Program in BASIC Swiss Federal 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 60 61
5. MEN O _ Pe 2 Explicit CR 0 10 5 0 8 CR 0 10 CR 0 50 S 0 6 CR 1 00 CI E 0 4 Analytical UV O 5 0 2 O o o 0 5 10 15 20 25 30 Length cm Figure 14 RT1D results for high Peclet number simulations with varying Courant number using the explicit advection and implict dispersion scheme v 0 4 cm day D 0 08 cm day dx 0 4 cm k 7 5E 2 day T 50 days 3 2 3 2 FULLY IMPLICIT ADVECTION AND DISPERSION SCHEME Simulations were performed with a time step of 1 0 5 0 1 and 0 01 for high Peclet number simulations and 26 5 13 25 2 65 and 0 265 days for low Peclet number simulations to generate Courant numbers of 1 0 5 0 1 and 0 01 respectively Figure 15 and 16 show the results for low and high Peclet number simulations respectively It was observed from the results that the fully implicit scheme performed well for both high and low Peclet numbers 100 3 Pe 0 5 Implicit CR 0 01 9 80 CR 0 10 60 CR 0 50 E CR 1 00 E 40 m Analytical vw O O 20 O 0 500 1000 1500 2000 2500 3000 Length cm Figure 15 RT1D results for low Peclet number simulations with varying Courant number using the fully implicit advection dispersion scheme v 1 cm day D 10 cm day dx 5 cm k 7 0E 4 day T 3000 days 24 MEN O s CR 0 01 gt Pe 2 Implicit 3 0 8 CR 0 10 E CR 0 50 o e CR 1 00 D 04 Analytical U O 6 0 2 O 0 0 0 9
6. Phy Seay Ca k R C k R C ot Ox Ox Where R is the retardation factor V the seepage velocity LT D is the hydrodynamic dispersion coefficient LTS k T is the first order decay constant Further details of the model are available in Bauer et al 2001 The code format for this reaction package is as follows dydt 1 RC 1 Conc 1 dydt 2 RC 1 Conc 1 R 1 RECO Conc 2 R 2 RO dydt 3 RC 2 Conc 2 R 2 RC 3 Conc 3 RGD R 3 dydt 4 RC 3 Conc 3 R RC 4 Conc 4 RKA RG The description for the variables is similar to the reaction packages explained above 3 3 4 MODIFIED MONOD KINETICS FOR TCE BIOAUGMENTATION RXNTYPE 4 Schaefer et al 2009 conducted batch experiments to study the bioaugmentation of TCE They used Modified Monod kinetics to model these reactions The reaction package for this model is as follows note that there are no advection dispersion terms as this is a batch system dCcg DR l Esa 28 dt Rice CrcE K ror Cock l Anc ACpcE de l Arcs ACTcE 29 dt Roce Ce K pcg 0 Cice R rce CrcE Krce Lice dCve En i qycXCvc l Apce C pce dt Ry Cc C RE fC 30 Cyo Ky 1 7 CocetK pop 1 IE lice DCE TCE 28 dX YX Arce ree l qdpceeCpce l AvcEve a RACER Re COR a a 8D Coce FK pce a E CutKyo ICE PCE TCE TCE DCE TCE TCE TCE Where C mM and X cells L are the concentration of ith compound and biomass respecti
7. 0 01 A constant boundary condition of 1 mg L was supplied at the inlet for the 17 complete duration of the experiment The results of the simulations for the Explicit and TVD schemes are shown in Figure 5 and 6 respectively It can be observed from these figures that at Courant number 1 the results from both the advection schemes produce sharp advective fronts However the numerical dispersion comes into effect as the Courant number is decreased The numerical dispersion is comparatively less in case of TVD schemes than the explicit schemes L gt O O O gt hb 56 O ncentration mg L 8 0 2 0 0 Length cm Figure 5 RT1D results for different Courant numbers for the explicit advection scheme with v 1 cm day dx 1 and a duration of 20 days i M a TVD Scheme gt O O O gt A o O ncentration mg L 8 0 2 0 0 0 10 20 30 40 50 Length cm Figure 6 RT1D results for different Courant numbers for the TVD advection scheme with v 1 cm day dx 1 and a duration of 20 days 18 3 2 2 ADVECTION DISPERSION MODULES The program was tested for a high and low Peclet numbers of 0 5 and 2 respectively for varying Courant numbers 0 01 0 1 0 5 and 1 A hydrodynamic dispersion coefficient of 0 05 cm day was used to perform high Peclet number simulations and a hydrodynamic dispersion coefficient of 0 2 cm day was used to perform low Peclet number simulations A grid size of 0 1 cm and a pore velocity of 1 cm da
8. 2 Geochemistry Type Problem 50 3 amp 4 Figure 49 Parameters for generating the input template in example problem 11 Species Name Conci me S e ASO4 3JAg_ 275606 FeOH 34E0 o e HASOA 2 SI7E0S H2ASO4 1 3 46E08 H3AsO4 E ST gt FeH2As04 355E09 gt FeHAsO4 6SIEO7 gt FeAs04 2 19BE0S 54 gt FeOH2 LAGE 05 gt FeO 5 74E 07 O LogX i ASOA 3 0 0000750000 5 561436606 males le moles L 5 453E 05 0 0000204731 Table 3 Results from RT1D simulations for example problem 11 A a a A O A A A A O A 13 91 11 23 15 01 20 16 31 44 26 18 20 1 7 17 9 32 O OoarrrRrF FFP GO ORFS lp OD ODO Ok DD eleimielolalalalalalala PPP NS wwe BB Oo O Oaqorrrerereradgero a ele ie lei lolealelalelalea ele imiel alalalalalalalea e NM WwW eee Ook Figure 50 Geochemistry parameters for example problem 11 55 Jeppu et al 2012 a RT1D Sorbed As V mg g geothite 0 1 2 3 4 Aqueous As V mg L Figure 51 As V adsorption curve for pH 7 4 12 EXAMPLE 12 CALCIUM CARBONATE SPECIATION In this test problem we will consider the speciation of a solution to which 10 moles L of CaCO has been added when it reaches equilibrium 1 Set up the spreadsheet as shown in Figure 52 The program ignores all the advection dispersion parameters when simulation option 3 is chosen as explained earlier B c D E Advection
9. 2 Ge istry Type blem 50 3 amp 4 Figure 43 Transport parameters for the example problem 9 49 4 The adsorption of arsenic 1s described by diffuse dual layer model short code 2 The tableau initial and boundary conditions can be set using the Figure 44 The sorbed arsenic concentration is composed of 3 species 8 9 and 10 6 The results for this simulation are shown in Figure 45 E O N ee es ee ee ee G 1H af 1 25 06 0 00E 00 28 29 31 32 33 35 36 37 38 39 20 21 42 43 45 26 47 48 49 50 51 52 ao 000000 Sor rP EP RP m kr a 2 F F F F CC CS CC CS FR CD a I j i M e i i RE Me Pa CO 53 54 55 56 57 ao oo fo 8 Oo 8 ca i OO hhh a a a a a e Rai ee ie oo ogo oe YF oe 9 Figure 44 Geochemistry parameters for example problem 9 10 90 1 6 dh N RT1D PHREEQCI Model Jeppuetal 2012 Data Concentration uM O 00 9 P 0 5 10 15 20 Reactor volume Figure 45 RT1D results for example problem 9 4 10 EXAMPLE 10 ION EXCHANGE Valocchi and Werth 2004 presented an analytical framework that allowed the characterization of certain key concentration profiles during the transport of ion exchange solutes based on the chromatography theory The validity of this theory was tested by applying 1t to a field situation in Palo Alto Baylands in California The field project involved the i
10. Number of components 1i if Species Number of species i Fixed component concentrations set the number of components whose concentrations are fixed Example fixed pH iv Aqueous components number of aqueous components that we are tracking v Sorbed concentrations the number of solid phase concentrations that are necessary for correcting the aqueous phase component concentrations vi SCM TYPE Type of surface complexation method 0 No surface complexation 1 Constant capacitance 2 Diffuse layer 3 Stern Layer 4 Triple Layer 5 Generalized Two Layer Modem Dzombak and Morel 1990 1 2 4 KINETIC REACTION PARAMETERS These labels are generated in the Section 4 of Figure 3 when either simulation option 1 or 2 are selected The spreadsheet displays four different columns requiring the input for the following parameters 1 Component name The input template automatically populates it with a default component name This can be changed to a more appropriate name by the user The program reads this name and uses for output in Sheets 2 and 3 11 111 IV R Retardation factor In case of linear sorption we have a retardation factor This retardation factor is given by C a where p is bulk density of the soil mg L Ka is p the linear sorption constant L mg and is the porosity This is equal to 1 when there is no sorption Cannot be less than 1 Initial This is the initial concent
11. b 1 00E 02 1 00E 03 1 00E 04 1 00E 05 Cubic Meter Injected Cubic Meter Injected 1 00E 03 Ca2 x Ca2 anjani E 1 00E 02 S Q 1 00E 01 AR lt _ gt C 100 1000 10000 100000 Cubic Meter Injected Figure 48 Results for example problem 10 a Breakthrough profile for Na b Breakthrough profile for Mg2 c Breakthrough profile for Ca2 4 11 EXAMPLE 11 ARSENIC ADSORPTION ISOTHERM In this example problem we will consider the As V adsorption on to the goethite coated sand as shown in Jeppu et al 2012 We will run RT1D with simulation option 3 to generate isotherm data In this exercise we will generate the isotherm data as 1f an experiment were being conducted We will input the initial aqueous As V concentration in the solution and the program will give the adsorbed concentration and the equilibrium aqueous As V concentration This can be done for different concentrations of As V and can be used to generate the isotherm data The equilibrium equations for setting up the problem can be found in Jeppu et al 2012 1 Set up the input sheet as shown in Figure 49 This is a batch geochemistry problem so the simulation option must be set to 3 and all the transport parameters can be ignored There are 4 components and 12 species in this example problem 2 We are generating isotherms so there is only one aqueous species of interest 1 e the aqueous concentration of As V There are 3 sorbed comp
12. for this problem as shown in Figure 19 1 Advection Dispersion Parameters 2 3 A 5 6 E 3 J Kinetics Type Problem 50 1 amp 2 Geochemistry Type Problem SO 3 amp 4 Figure 19 Advection dispersion parameters for the example problem 1 This reaction package is set up for one species only and therefore the number of mobile components in cell B16 is set to 1 Select the reaction package number as in B18 We will use the Runge kutta Fehlberg ODE solver for this solver and the short code for this solver is set in B19 We need to set the k value for this reaction package and hence we will set Reaction Parameters as Press the Generate Input Template button This will clear everything below and an input sheet will be generated as shown in Figure 20 Note that the program automatically sets the geochemistry type problem values to N A to show that we re running a kinetic type problem 33 eu 27 28 29 30 31 gt 1 ce ii Figure 20 Component initial and boundary conditions along with the kinetic parameters 7 The initial concentration in the column is O and a constant concentration pulse of 1 mg L is supplied at the inlet for the entire duration of the simulation Therefore we will set the boundary condition as The retardation factor for this example will be set as 1 8 The k value can be changed to any desired value in the B31 For this example we have used two different v
13. of fixed component concentrations for this example problem is 1 46 1 Advection Dispersion Parameters 2 3 4 3 6 T 8 9 Kinetics Type Problem PT 1 amp 2 Geochemistry Type Problem PT 3 amp 4 RR Figure 40 Advection dispersion parameters for example problem 8 There are 3 aqueous components of interest of the total 6 components These are Cd2 Br and Cl and one sorbed species a complex of cadmium formed by surface complexation that is used to correct the concentration of Cd2 to allow it to be transported along the column The SCM type for this model is constant capacitance The short code for this type of surface complexation model in MICROQL is 1 Press the generate input template button to enter the tableau mass action and mass balance as shown in Figure 41 For this problem both mass action and mass balance matrices are same Refer to the Figure 41 for the initial and boundary conditions and the surface complexation parameters for this problem The final table refers to the composition of the sorbed concentration In this case the sorbed concentration 1s only composed one surface complex of cadmium species 14 SoCd2 The results for this simulation are shown in Figure 42 Please refer to the Input guide provided with the RT1D package or Torlapati and Clement 2012 to obtain the parameters in case the Figure 41 1s unclear 47 MEE N A A UA O E A FEO TOO PUR 0 E SS JE gt B
14. problem dependent reaction package and instead uses a MICROQL based chemistry package to solve the equilibrium problem The information required for formulating a specific geochemical problem are input using the standard tableau Batch Kinetics RT1D Batch Geochemistry simulation option 1 simulation option 3 Reaction Module Kinetics Module simulation option 2 Reactions defined yc B dS Reactions defined using a Kinetic a E using a Tableau Reaction Package ODE Solver MICROQL II Geochemistry Module simulation option 4 Figure 1 Simulation options available in RT1D 1 1 UNDERSTANDING THE SPREADSHEET In this section we will briefly discuss the different sections of the spreadsheet and illustrate their use to set up various types of batch or reactive transport problems As explained in the earlier section RT1D has four different simulation options The data required for simulating these options are organized into three distinct sections within the input screen as shown in Figure 2 The parameters for the transport module should be input in Section l see Figure 2 These parameters are used by the model when a simulation option involving transport is selected 2 or 4 The left side of Section 2 is used for defining kinetic type problems whereas the right side is used for the geochemistry equilibrium problems Depending on the type of problem only one of these two sections will be used The progra
15. results for the above parameters are shown in Figure 33 41 co D 2 RT1D 0 015 RT1D 4 0 00015 RT1D A 2 Analytical e 0 015 Analytical m 0 00015 Analytical o k o ho Aqueous concentration mg L 0 5 10 15 20 25 30 35 40 Distance cm Figure 33 Results for example problem 5 for varying mass transfer coefficients 4 6 EXAMPLE 6 REACTION PACKAGE 6 This is the second benchmark problem in the Torlapati and Clement 2012 paper The code form of this reaction package 1s as follows l Input the advection dispersion parameters as shown in Figure 34 Please note that this is a computationally intensive problem and will take a substantial amount of time for the simulation to finish The excel program will be unresponsive during this process please don t close the program There are three mobile component and one immobile phase component in this problem There are about 10 reaction parameters that are need to be set by the user The reaction package for this problem is 6 Press the generate input template button and input the reaction parameters as shown in Figure 35 The parameters set here are for high flow experimental dataset from the paper There is a constant supply of nitrate and acetate throughout the length of the experiment from the inlet and there was an initial concentration of aqueous and solid biomass which continue to grow The
16. results from the RT1D simulations are shown in Figure 36 a and b 42 Effluent NO mg L A o oO o Advection Dispersion Parameters Kinetics Type Problem 50 1 amp 2 Geochemistry Type Problem 50 3 amp 4 Figure 34 Advection dispersion parameters for the example problem 6 Figure 35 Kinetic parameters for the example problem 6 10 Nitrate RT1D 6 Nitrate Model T A Nitrate Data 1 c O OG Bm 0 1 3 E 2 CO T 0 01 i O Aqueous Biomass RT1D 2 Aqueous Biomass Model LU A Aqueous Biomass Data 0 001 5 10 15 20 25 0 5 10 15 20 Time days Time days o 43 25 Figure 36 a Nitrate concentration for the high flow experiment b Aqueous biomass concentration comparison for the high flow experiment 4 7 EXAMPLE 7 REACTION PACKAGE 7 This is the benchmark problem 3 from Torlapati et al 2012 This reaction package describes the biodegradation of carbon tetrachloride in the presence of nitrate and acetate The code form of the reaction package can be written as follows 1 Input the advection dispersion parameters for the example problem as shown in Figure 37 There are a total of 4 mobile components and 2 immobile components for this problem 2 The reaction package for this problem is 7 and the user needs to input 16 reaction terms Press the generate input template button 4 Input the retardation factor initial condition and boundary c
17. strength moles L C4 in Westall 1979a CAPI Inner capacitance F m C5 in Westall 1979a CAP2 Outer Capacitance F m C6 in Westall 1979a Sorbed Species ID When generating isotherms the program requires Species index numbers of the sorbed species in the tableau The number of sorbed species displayed here is based on the Sorbed concentrations set before pressing the Generate Input Template button Please input all the ID s of the sorbed species here This is used to calculate the amount of total sorbed species in moles L The program reads the concentrations of all the sorbed species entered here and displays the total sum of their concentrations as total sorbed concentration In case of a speciation problem where the sorbed concentrations are not required this should be left blank Component of the aqueous species This is only required when generating isotherms Enter the component number of the aqueous species The aqueous concentration of the component is calculated at the end of the simulation by subtracting the sorbed concentration from the total concentration of the component in moles L pen pa 1 2 6 GEOCHEMISTRY EQUILIBRIUM PARAMETERS WITH TRANSPORT The input template is similar to the above except we need a few additional parameters This section discusses only the additional parameters to avoid any repetition 1 11 111 IV Boundary The total concentrations of all the components at
18. 0 05 cm day dx 0 1cm 3 2 2 2 FULL Y IMPLICIT ADVECTION DISPERSION SCHEME Simulations were performed with a time step of 0 1 0 05 0 01 and 0 001 for both high and low Peclet numbers using the fully implicit advection dispersion scheme to generate Courant numbers of 1 0 5 0 1 and 0 01 respectively Figure 9 and 10 show the results for low and high Peclet number simulations respectively 1 2 A Pe 0 5 Implicit CR 0 01 E CR 0 10 0 8 CR 0 50 0 6 CR 1 00 Analytical E 0 4 yt O 0 0 0 10 20 30 40 50 Length cm Figure 9 RT1D results for low Peclet number simulations with varying Courant number using the fully implicit advection dispersion scheme v 1cm day D 0 2 cm day dx 0 1cm 20 N Pe 2 Implicit CR 0 01 1 0 CR 0 10 os CR 0 50 O E Zos CR 1 00 Analytical o 0 4 O am 8 0 2 0 0 0 10 20 30 40 50 Length cm Figure 10 RT1D results for high Peclet number simulations with varying Courant number using the fully implicit advection dispersion scheme v 1cm day D 0 05 cm day dx 0 1cm It was observed from the simulations that the fully implicit scheme performed a lot better than the explicit scheme However there was some numerical dispersion when the Peclet number was high 3 2 2 3 TVD ADVECTION AND IMPLICIT DISPERSION SCHEME Simulations were performed with a time step of 0 1 0 05 0 01 and 0 001 for both
19. 10 15 20 25 30 Length cm Figure 16 RT1D results for high Peclet number simulations with varying Courant number using the fully implicit advection dispersion scheme v 0 4 cm day D 0 08 cm day dx 0 4 cm k 7 5E 2 day T 50 days 3 2 3 3 TVD ADVECTION AND IMPLICTT DISPERSION Simulations were performed with a time step of 1 0 5 0 1 and 0 01 for high Peclet number simulations and 26 5 13 25 2 65 and 0 263 days for low Peclet number simulations to generate Courant numbers of 1 0 5 0 1 and 0 01 respectively Figure 17 and 18 show the results for low and high Peclet number simulations respectively 100 m Pe 0 5 TVD 7790 5 80 CR 0 10 CR 0 50 g CR 1 00 qu i 40 m Analytical w O 6 20 O 0 900 1000 1500 2000 2500 3000 Length cm 25 Figure 17 RT1D results for low Peclet number simulations with varying Courant number using the fully implicit advection dispersion scheme v 1 cm day D 10 cm day dx 5 cm k 7 0E 4 day T 3000 days 1 0 Z Pe 2 TVD CR 0 01 i d 0 8 CR 0 10 CR 0 50 2 e CR 1 00 CO i 0 4 Analytical vw O 6 0 2 O 0 0 0 5 10 15 20 25 30 Length cm Figure 18 RT1D results for high Peclet number simulations with varying Courant number using the fully implicit advection dispersion scheme v 0 4 cm day D 0 08 cm day dx 0 4 cm k 7 5E 2 day T 50 days There was some numerical dispersion for very low courant numbers in the high Pecl
20. C OC 0 Es 22 R Fa ENE ae D ae 2 Ara ea C 1 HE32 Ye Ko C k 3C oC a OC o Cor 23 4 at ER Ae 04 02 24 02 04 03 1 24 03 Where R is the retardation factor V the seepage velocity LT D is the hydrodynamic dispersion coefficient LTS Y is the yield coefficient F is the fraction k T is the first order decay constant The above reaction equations could be written in code format as follows dydt 1 RC 1 Conc 1 R 1 dydt 2 RC 11 RC 5 RC 1 Conc 1 RC 2 Conc 2 RC 12 RC 6 RC 3 Conc 3 R 2 dydt 3 RC 13 RC 7 RC 1 Conc 1 RC 14 RC 8 REO Conc 2 RC 3 Conc 3 R 3 dydt 4 RC 14 RC 9 RC 2 Conc 2 RC 16 RC 10 RC 3 Conc 3 RC 4 Conc 4 R 4 where dydt 1 is the expression for ODE for component 1 1 could be either 1 2 3 or 4 RQ is the retardation factor for the component 1 RC 1 is the user set reaction parameters Conc 1 is the concentration of the component 1 3 3 3 FOUR COMPONENT DECAY CHAIN RXNTYPE 3 2 Bauer et al 2001 presented analytical solutions for the transport of a decay chain for in homogenous porous media The reaction equations presented in the paper were used as a built in reaction package for the RT1D The reaction equations are as follows 2 R Say Xe Ec 24 Ot OX OX OC OC 8C 25 R a gi Ape TE k R C k R C OC OC OC 26 R Pa red r k R C k R C 2 27 R
21. C is the concentration of mobile phase mg L S is the concentration of the immobile phase mg mg k is the aqueous phase decay constant 1 day Ka L mg is the linear sorption constant is the porosity p is the bulk density of the soil mg L The code form of this reaction package could be written as follows 29 dydt 1 RC 2 Conc 1 Conc 2 RC 3 RC 5 Conc 1 dydt 2 RC 4 RC 1 RECO Conc 1 Conc 2 RC 3 3 3 6 DENITRIFICATION RXNTYPE 6 Clement et al 1997 studied the effects of denitrifying conditions on the growth and transport of bacteria in a porous media column under two substrate loading conditions A numerical model was developed to generate the breakthrough profiles of bacterial cells and substrates A first order attachment and detachment model was used to describe the exchange processes between mobile and immobile phase bacterial cells This reaction package considered three mobile components namely nitrate acetate and aqueous phase bacteria and one immobile component namely immobile bacteria The reaction package used in the problem is given below 2 Sea V ES Dax BP 34 X X n OC OC QC Xp 35 2 36 V gt D TRA KX F Bak X X n 37 dX _ LX K X4 nK X 37 dt p Where Cy Ca Xa and X are concentrations mg L of nitrate acetate aqueous phase bacteria and immobile phase bacteria mg mg respectively The parameters Kat day and Kade da
22. Dispersion Parameters Kinetics Type Problem SO 1 amp 2 Geochemistry Type Problem SO 3 amp 4 Figure 52 Parameters to generate the input template sheet for example problem 12 96 Input the tableau the total and guess concentration for each component as shown in Figure 53 Since we are not generating isotherms enter 0 for both Component of Aqueous species and Sorbed species ID This tells the program not to calculate the aqueous phase and solid phase concentrations The speciation for this example problem is presented in Table 4 Species Name Conci E BET Cant EOS CaHco3 458606 O Cacoataq EO pHacos o 205608 O 265041 0 11 4341 000 E E 0 II OH E AAA Cae 0 001 3163110315 3 363665504 Hy 1041270197 Table 4 RT1D simulation results for example problem 12 57 Figure 53 Geochemistry parameters for example problem 12 98 REFERENCES Bauer P Attinger S Kinzelbach W 2001 Transport of a decay chain in homogenous porous media analytical solutions Journal of Contaminant Hydrology 49 217 239 Cederberg G A Street R L Leckie J O 1985 A groundwater mass transport and equilibrium chemistry model for multicomponent systems Water Resources Research 21 1095 1104 Chapra S C Canale R P 1998 Numerical Methods for Engineers with Programming and Software Applications Third edn WCB McGraw Hill Clement T P Sun Y
23. Figure 30 39 Figure 29 Monod parameters for example problem 4 TCE RT1D 0 08 DCE RT1D nd VC RT1D 9 0 06 Ethene RT1D E hee TCE Analytical 5 DCE Analytical 3 0 02 x VC Analytical e Ethene Analytical 0 00 Time h Figure 30 Results for example problem 4 4 5 EXAMPLE 5 REACTION PACKAGE 5 This is the first benchmark problem presented in Torlapati and Clement 2012 1 Input the advection dispersion parameters as shown in Figure 31 2 The reaction package for this problem is 5 and the simulation option is 2 because this 1s coupled with transport 40 Advection Dispersion Parameters Kinetics Type Problem SO 1 amp 2 Geochemistry Type Problem 50 3 amp 4 YP ry Typ Figure 31 Advection dispersion parameters for example problem 5 Figure 32 Reaction parameters for example problem 5 3 There is one mobile component and a corresponding immobile phase component and there are a total of 5 reaction parameters that can be set by the user Press the generate input template button once the parameters have been set 4 Input the reaction parameters and the initial and boundary conditions as shown in Figure 18 The concentration of both components before the beginning of the experiment is zero and a concentration pulse of 1 mg L was supplied near the inlet for the entire duration of the experiment 5 Press the solve button to generate the results in Sheet 2 and 3 The
24. Hooker B S Petersen J N 1998 Modeling multispecies reactive transport in ground water Ground Water Monitoring amp Remediation 18 79 92 Dzombak D A Morel F M M 1990 Surface complexation modeling Hydrous ferric oxide Wiley New York Jeppu G P Clement T P Barnett M O Lee K K 2012 A modified batch reactor system to study equilibrium reactive transport problems Journal of Contaminant Hydrology 129 130 2 9 Parkhurst D L Appelo C A J 1999 User s guide to PHREEQC version 2 A computer program for speciation batch reaction one dimensional transport reaction path and inverse geochemical calculations U S Geological Survey Water Resources Investigations Report p 312 pp 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 biodegradation in intermittently fed aquifer columns Water Resources Research 38 1 13 99 Quezada C R Clement T P Lee K K 2004 Generalized solution to multi dimensional multi species transport equations coupled with a first order reaction network involving distinct retardation factors Advances in Water Resources 27 507 520 Schaefer C E Condee C W Vainberg S Steffan R J 2009 Bioaugmentation for chlorinated ethenes using Dehalococcoides sp Comparison between batch and column experiments Chemosphere 75 141 148 Torlapati J Clement T P
25. ODEs in reaction package xi Adaptive Runge Kutta Fehlberg solver for solving ODE s in the reaction package xii MICROQL II with surface complexation reactions for geochemistry package xili Subroutine to couple the advection solvers with MICROQL package 3 2 TRANSPORT MODULES The explicit finite difference schemes used in the advection modules of RT1D program are constrained by certain stability condition criteria To ensure that the results generated by the numerical models are stable and error free researchers have used Courant number Vdx R dt and Peclet number Vdx D to determine the grid size and time step for the simulations In the following sections we have tested the stability of our numerical models for different Courant and Peclet numbers For explicit schemes the Courant number should be less than 1 to obtain oscillation free results If the Courant number exceeds 1 numerical oscillations are observed near the advective front In addition to obtain good quality solution the Peclet number should be set below 2 3 2 1 PURE ADVECTION In this section we tested the Explicit and TVD schemes for different Courant numbers by setting the value of dispersion to 0 Simulations were performed for a one dimensional column of 50 cm length The pore velocity was about cm day and the simulations were performed for duration of 20 days The grid size was set to 1 cm and the time step was varied to generate Courant numbers of 1 0 5 0 1 and
26. RET 27 28 29 30 al 32 33 34 35 36 37 38 39 20 41 42 43 44 45 46 47 dE 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 65 66 67 68 69 70 71 72 73 74 75 76 TI TE 79 20 Figure 41 Geochemistry input details for example problem 8 48 1 00E 01 1 00E 02 Aqueous Cd RT1D Chloride RT1D Sorbed Cd RT1D 1 00E 03 Chloride data 5 a Sorbed Cd data Aqueous Cd data 1 00E 04 c O O 1 00E 05 1 00E 06 Distance cm Figure 42 RT1D simulation results for the example problem 8 4 9 EXAMPLE 9 SEQUENTIAL BATCH REACTOR This is the benchmark problem in Torlapati and Clement 2012 We simulate a sequential batch reactor set up using the transport coupled with adsorption of arsenic described by the surface complexation 1 There are a total of 3 sequential batch reactors for this problem and we will consider an equivalent 3cm length column with 4 nodes where one extra node is designated as the boundary node to supply the concentration to the other nodes 2 We will consider 1cm day velocity and dispersion effects are ignored We will grid size of lcm and time step of day to ensure that the courant number is 1 3 We will use explicit advection scheme shortcode 0 to simulate one node advection with courant number 1 Set the transport parameters as shown in Figure 43 Kinetics Type Problem 50 1 amp
27. ROBLEM The geochemical equilibrium reaction problem is formulated in the form of a tableau that represents the interactions between all the components and species involved in the chemical system As defined by Westall 1976 a species is a chemical entity of interest present in the system whereas a component is a basic building block used for forming various chemical species in the system The stoichiometric relationship between the components and the species can be represented in the form of a matrix known as the tableau The chemical speciation problem defined by the tableau is solved using an EXCEL VBA version of MICROQL code The details of the numerical solution schemes employed by MICROQL are discussed in Westall 1979b Within RTID the transport equations that involve geochemical or equilibrium reactions are solved using an approach proposed by Cederberg et al 1985 which is slightly different from the approach used for the solving the transport equations involving kinetic reactions As discussed in Cederberg et al 1985 first the aqueous concentration of a component of interest will be transported using the transport module The aqueous concentration of a component at a particular node is calculated by subtracting the concentrations of all the sorbed species associated with that component from its total concentration After the transport step the updated advection dispersed aqueous component concentration is added back to the sorbed
28. USER MANUAL FOR RTID Multi Component Reactive Transport Model for 1 D Systems RT1D v1 0 Beta By Jagadish Torlapati T Prabhakar Clement Professor Department of Civil Engineering Email clement auburn edu Auburn University Auburn AL 36849 ACKNOWLEDGMENTS RT1Dv1 0 beta is an EXCEL based VB code this first beta version was coded by Dr Torlapati as part of his PhD dissertation research completed at Auburn University RTID uses several numerical solvers and routines that were originally available within the FORTRAN tool RT3D and the associated research software tools Therefore we first like to thank the RT3D development team members who over the past two decades have contributed several ideas in this area while working with Professor Clement at Battelle Pacific Northwest National Laboratory and at University of Western Australia We also like to thank the researchers at Auburn University who have developed various types of reactive transport models while working in Professor Clement s research group Dr Tirtha Gautham Anjani Kumar and Dr Gautham Jeppu developed various types of geochemical reaction models and coupled them with transport routines their FORTRAN models provided a template for developing the geochemistry module in RT1D Jared McLaughlin investigated various types of TVD solvers during his tenure at Auburn University We like to thank Dr Charles E Schaefer of Shaw Environmental and Infrastructure NJ
29. alues 0 075 and 0 When k is set to 0 the plume acts like a tracer 9 Press the solve button to display the spatial variation of concentration in Sheet 2 and the breakthrough concentrations at the outlet are displayed in Sheet 3 The results for this example for both k values are presented in Figure 21 1 2 RT1D k 0 075 3 1 0 Analytical k 0 075 D RT1D Tracer Eos N 4 Analytical Tracer 0 6 0 4 O O 0 2 0 0 0 5 10 15 20 25 30 Length cm Figure 21 Results for two different k values for example problem 1 4 2 EXAMPLE 2 REACTION PACKAGE 2 In this example problem we will learn to use the reaction package 2 which simulates the four species first order sequential decay as described in section 3 3 2 1 Input the advection dispersion and reaction parameters as shown in Figure 22 This information can be obtained from Quezada et al 2004 2 This is a four species problem so we will set the number of mobile components in cell B16 as 4 and the reaction package we will use for this problem is 2 3 The ODE solver can be either 0 or 1 and this shouldn t change the results by much 4 There are a total of 16 reaction parameters for this problem and the values for these reaction parameters are available in Figure 23 34 6 PY a o bB _ c np IN Advection Dispersion Parameters WO OO psa em Gr The retardation factors for the components are 1 2 3 and 4 respectively and the i
30. concentrations of the respective component to compute the total component concentration at that 15 node This total component concentration is then transferred to MICROQL to solve the geochemical speciation problem The equilibrated species concentrations are used to update the values of aqueous component concentrations for the next time transport step Further details of this transport algorithm are discussed in Cederberg et al 1985 16 3 PROGRAM DETAILS 3 1 LIST OF SUBROUTINES 1 An input generator based on the number of species and the type of reaction a Subroutine to generate input template for kinetic transport amp batch reactions b Subroutine to generate input template for the geochemical equilibrium problems c Subroutine to generate input template for the geochemical equilibrium problems coupled with transport ii RTID Main program that calls the required subroutines based on the advection solver and reaction type 11 Subroutine to set the boundary condition based on the pulse time iv Subroutine to set the initial condition v The backward difference explicit advection solver vi TVD solver with van leer flux limiter for advection vil Implicit finite difference solver for dispersion vill Fully implicit finite difference solver for advection dispersion ix Tridiagonal solver for solving the tridiagonal matrix generated by the implicit finite difference method for dispersion x Fourth order Runge Kutta solver for solving
31. dule using the implicit finite difference method In addition to these explicit implicit solvers there is also a fully implicit option that solves the advection dispersion terms together using a fully implicit approach 2 1 1 EXPLICIT ADVECTION SCHEME The advection part of the transport equation can be solved using the explicit backward difference approximation as shown below cc VCC p i i 7 At R Ax where C is the concentration of the component at the current time step at the current node C is the concentration of the mobile component at the previous time step at the current node and C is the concentration of the mobile component at the previous time step at the preceding node After further simplification we can solve for the concentration of mobile component at the current node CE as shown below 12 cua aa 8 1 where Cr ES 1s known as the grid Courant number X 2 1 2 EXPLICIT TVD SCHEME Numerical dispersion is a major concern while solving the advection dominated problems RT1D includes a robust total variation diminishing TVD scheme that minimizes numerical dispersion errors Details of this scheme are given below Using the Taylor series expansion the standard Lax Wendroff LW scheme for the advection term can be written as Leveque 2002 cr c E C C i e C 9 The above equation can be rearranged and written in a flux balance format as C C as _ Bap F 10 AL Ax Where E VC
32. e concentration pulse has been input should also be provided by the user For a continuous pulse the pulse time should be equal to the total simulation time The user should also provide the type of advection dispersion solver to be used for the transport module The simulation option SO describes the different types of problems that RT1D can solve The options are batch kinetics 1 kinetic reaction coupled reactive transport 2 batch geochemistry 3 and geochemistry coupled reactive transport 4 Step 2 Simulation option parameters Depending on the simulation option chosen within Section 1 the kinetic or the geochemistry parameters should be input in Section 2 For a kinetic type problem the user should input total number of mobile and immobile components the reaction package number and type of ODE solver module used to solve the reaction package The user can also set the total number of user defined reaction parameters kinetic parameters that will be used within the reaction package For a geochemistry problem the user should input the number of components species components whose concentrations are fixed example fixed pH number of aqueous component concentrations that need to be tracked number of sorbed concentrations that are required for correcting the aqueous concentrations during transport and the type of surface complexation model This information is input in the form of standard reaction tableau Westall 1976 S
33. et number simulations but this was negligible and the results from RTID matched the analytical solutions well 3 3 LIST OF KINETIC REACTION PACKAGES In the following section we will present a discussion about the built in reaction packages in the RTID program Each reaction package and its code form will be presented 3 3 1 FIRST ORDER SEQUENTIAL DEGRADATION RXNTYPE 1 A generalized reactive transport equation for a single component first order decay is present below OC OC _ aC R V D KC 19 Ot OX Ox 26 Where C is concentration of the mobile component k 1s the first order decay constant T The above equation simulates a tracer when the decay constant is set to zero This reaction package could be written in the code form as follows dydt 1 1 R 1 RC 1 Conc 1 Where dydt 1 is the ordinary differential equation for the mobile component 1 R 1 is the retardation factor for component and RC 1 is the user set reaction parameter k Conc 1 is the aqueous concentration of the mobile component 3 3 2 FOUR SPECIES COUPLED SEQUENTIAL DEGRADATION RXNTYPE 2 Quezada et al 2004 presented the analytical solutions for a four species coupled sequential first order degradation reactions The reaction equations simulate the transport and coupled decay of four mobile components The governing equations are as follows VG ASK X ac a gt c 21 2 at ae Ox 02 01 A c2 cl 02103 1 c2 c3 O
34. for his support in developing some of the reaction packages discussed in this manual This code development effort was in part funded by the Office of Science BER U S Department of Energy Grant No DE FG02 06ER64213 Auburn University l TABLE OF CONTENTS INTRODUCTION 1 1 Understanding the Spreadsheet 1 2 Input Spreadsheet Label Details 1 2 1 Transport Module 1 2 2 Kinetic Reaction Module 1 2 3 Geochemistry Equilibrium Module 1 2 4 Kinetic Reaction Parameters 1 2 5 Geochemistry Equilibrium Parameters Without Transport 1 2 6 Geochemistry Equilibrium Parameters With Transport MODEL DEVELOPMENT AND NUMERICAL SOLUTION 2 1 Transport Module 2 1 1 Explicit Advection Scheme 2 1 2 Explicit TVD Scheme 2 1 3 Implicit Finite Difference Method for Dispersion Scheme 2 1 4 Fully Implicit Numerical Scheme for Advection Dispersion 2 2 Reaction Module 2 2 1 Kinetic Type Problem 2 2 2 Geochemistry Equilibrium Type Problem PROGRAM DETAILS 3 1 List of Subroutines 3 2 Transport Module Comparison with Analytical Schemes 3 2 1 Pure Advection Modules 3 2 2 Advection Dispersion Modules 3 2 2 1 Explicit Advection and Implicit Dispersion Scheme 3 2 2 2 Fully Implicit Advection Dispersion Scheme 3 2 2 3 TVD Advection and Implicit Dispersion Scheme 3 3 Reactive Transport Module Comparison with Analytical Schemes 3 3 1 Explicit Advection and Implicit Dispersion 3 3 2 Fully Implicit Advection Dispersion Scheme 3 3 3 TVD Advection and Implicit Dispersion Sc
35. form OC VC DOC Pp ot R ox R x R where i 1 2 3 m 1 1 e p 2 ot where j m 1 m 2 m 3 n where V is the velocity m day D is the hydrodynamic dispersion coefficient m 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 mobile components n 1s the total number of components note that a total of n m immobile components are numbered sequentially after numbering all the mobile components R is the linear retardation factor of the it mobile component R 3 where p is the bulk density mg L is the porosity Kg is P the linear sorption constant L mg and B and Bj are the reactions involving mobile and immobile components respectively The expressions used for B and pj terms would vary depending on the type of reactions involved in the system Note the immobile component equations do not have advection dispersion terms but will have reaction terms that will be coupled to some of the mobile component reaction terms Also the mobile component reaction terms themselves could be coupled to each other The coupled set of reactive transport equations represented by equations 1 and 2 are solved using the operator split approach Clement et al 1998 Torlapati and Clement 2012 Using this approach the governing set of transport equations can be written as
36. gonal matrix solver to solve for all the unknown concentrations at the new time level i 1 2 1 4 FULLY IMPLICIT NUMERICAL SOLUTION FOR ADVECTION DISPERSION In this option we solve the advection and dispersion together implicitly We use a central difference approximation for the advection term and the numerically discretized form for the advection dispersion equation is as follows emne Vv Con D CO At R 2Ax R Ax The above equation can be further simplified as follows 16 1 1 14 C R P a 1 C 2 RP 1 0 CC 17 i y Ax VAX _ QU ZA where DAt and DD Expanding the equation 17 for all the nodes would yield a tridiagonal matrix similar to 15 For this problem the values of a b c and d are given asa 1 2 R Y 1 aand C R P respectively 2 2 REACTION MODULE 2 2 1 KINETIC TYPE PROBLEM For a multi species the reaction part shown in equation 5 amp 6 simplifies into a set of ordinary differential equations ODE These ODEs are referred to as a reaction package The set of ODEs described within a kinetic reaction package can be solved using two different ODE solvers a standard 4th order Runge Kutta RK solver or a more robust Runge Kutta Fehlberg RKF solver Chapra and Canale 1998 The RK solver uses a constant reaction time step whereas the RKF solver will automatically subdivide the reaction time step into sub steps to minimize the local error 2 2 2 GEOCHEMISTRY EQUILIBRIUM TYPE P
37. heme 3 3 List of Kinetic Reaction Packages 3 3 1 First Order Sequential Degradation RXNTYPE 1 3 3 2 Four Species Coupled Sequential Degradation RXNTYPE 2 3 3 3 Four Component Decay Chain RXNTYPE 3 3 3 4 Modified Monod Kinetics of TCE Bioaugmentation RXNTYPE 4 3 3 5 Rate limited Sorption RXNTYPE 5 3 3 6 Denitrification RXNTYPE 6 3 3 7 Biodegradation of Carbon Tetrachloride RXNTYPE 7 3 3 8 User defined Reaction Package RXNTYPE 10 EXAMPLE PROBLEMS 4 1 Example 1 Reaction Package 1 4 2 Example 2 Reaction Package 2 4 3 Example 3 Reaction Package 3 4 4 Example 4 Reaction Package 4 4 5 Example 5 Reaction Package 5 4 6 Example 6 Reaction Package 6 4 7 Example 7 Reaction Package 7 4 8 Example 8 Surface Complexation 4 9 Example 9 Sequential Batch Reactor 4 10 Example 10 Ion Exchange 4 11 Example 11 Arsenic Adsorption Isotherm 4 12 Example 12 Calcium Carbonate Speciation 1 INTRODUCTION Reactive transport problems in porous media systems could be mediated by either kinetic or equilibrium processes Kinetic models are used to describe relatively slow chemical reactions whereas equilibrium reactions are used to describe fast chemical reactions Several reaction transport codes are available in the literature but they can only handle kinetic type reactions e g RT3D or can handle equilibrium type reactions Cederberg et al 1985 Parkhurst and Appelo 1999 RTID was designed to provide a unified platform for s
38. high and low Peclet numbers using the TVD advection and implicit dispersion scheme to generate Courant numbers of 1 0 5 0 1 and 0 01 respectively Figure 11 and 12 show the results for low and high Peclet number simulations respectively It was observed from the results that the TVD scheme was consistent with the analytical results for all the Courant and Peclet numbers It is highly recommended that the users of the RT1D use TVD scheme for accurate results 21 Pe 0 5 TVD CR 0 01 CR 0 10 CR 0 50 CR 1 00 Analytical 0 10 20 30 40 50 60 Length cm Figure 11 RT1D results for low Peclet number simulations with varying Courant number using the TVD advection and implict dispersion scheme v 1cm day D 0 2 cm day dx 0 1cm 1 2 46 Pe 2 TVD cR 0 01 E CR 0 10 0 8 CR 0 50 0 6 CR 1 00 Analytical 0 4 yt O O 8 0 2 0 0 0 10 20 30 40 50 Length cm Figure 12 RT1D results for high Peclet number simulations with varying Courant number using the TVD advection and implict scheme v 1cm day D 0 05 cm day dx 0 1cm 3 2 3 ADVECTION DISPERSION AND REACTION MODULES In this section we present the results to test the program s stability under varying Peclet and Courant numbers in the presence of a first order decay constant for a single component The simulations were compared with analytical solutions For the high Peclet number simulations the data from the f
39. iation of the component concentrations after the completion of the simulation time Sheet 3 provides the breakthrough component concentrations at the outlet after each time step By default the breakthrough component concentrations are printed at the last node end of the column The program can be modified to print the breakthrough component concentrations at any node in the column WARNING The spreadsheet erases the data below the buttons after cells A27 when either button 3 or 6 are pressed The output data in Sheets 2 amp 3 is cleared every time the button 5 is pressed The user is advised to use a different a spreadsheet for post processing the results and for completing other intermediate calculations 1 2 INPUT SPREADSHEET LABEL DETAILS 1 2 1 TRANSPORT MODULE 1 Length L Length of the column for the transport module 11 Total time T Total simulation time 111 IV VI VII Vill 1X Pulse time T Total time for which the concentration is pumped through the column during the experiment at the boundary This is less than or equal to the Total time set in 11 delx L Grid size This is used to calculate the total number of nodes using the formula length delx 1 delt T Time step The total number of iterations is calculated using the formula total time delt Velocity LT This is the pore velocity of the liquid flowing through the column This is calculated fro
40. if there are 6 mobile species and 2 immobile species we will count the immobile species as 7 amp 8 Please look at the example problem for further explanation Reaction package This is a short code for the type of reaction kinetics The different kinds of reaction kinetics that have been pre programmed into the RT1D model We will be adding more packages as we continue developing the model The users can program their reaction kinetics by selecting Option 10 1 First order sequential degradation 2 Four component coupled sequential degradation Quezada et al 2004 7 3 Four component decay chain Bauer et al 2001 4 Modified Monod kinetics of TCE bioaugmentation 5 Rate limited sorption Benchmark problem 1 Torlapati and Clement 2012 6 Denitrification Benchmark problem 2 Torlapati and Clement 2012 7 Biodegradation of Carbon Tetrachloride Benchmark problem 3 Torlapati and Clement 2012 8 Open 9 Open 10 User defined reaction package 1v ODE Solver type Short code for the type of reaction solver O Adaptive Runge Kutta Fehlberg solver 1 Fourth order Runge Kutta solver v f Reaction parameters Set the number of user defined reaction terms needed 1 2 3 GEOCHEMISTRY EQUILIBRIUM MODULE These variables become active only when you re using the geochemistry module of the code This is when you enter either 3 or 4 for the reaction type 1 Components
41. imulating transport problems involving both geochemical and kinetic reactions However it is important to note that the current version can either simulate a set of pure kinetic reactions or a set of pure geochemical reactions but one cannot mix both types of reactions The capabilities of RT1D model which are designated as simulation options are summarized in Figure 1 The model currently supports four different simulation options involving the use of two types of reaction modules a kinetic module and an equilibrium module which can simulate kinetic and equilibrium reactions respectively There are four types of simulation options available within RT1D The first simulation option can be used to solve batch kinetic problems The second option can be used to solve one dimensional reactive transport problems involving kinetic reactions The third option can solve batch equilibrium problems The fourth option can solve one dimensional reactive transport problems involving geochemical equilibrium reactions If the kinetic module is selected the user should also provide a problem dependent reaction package The kinetic module supports several standard reaction models that are already coded within a set of pre programmed reaction packages In addition to these preprogrammed packages RT1D also supports a user defined reaction package via which the user can input any type of kinetics The geochemical module on the other hand does not require a
42. irst component of Bauer ef al 2001 was used and for low Peclet number 22 simulations the date from the first component of Quezada et al 2004 was used Simulations were performed for different Courant numbers of 0 01 0 1 0 5 and 1 0 The parameters for these simulations are presented in Table 1 igh Low Velocity em day 04 gt 1 Dispersion Coefficient em day 0 08 10 E A Table 1 Parameters for low and high Peclet aber simulations 3 2 3 1 EXPLICIT ADVECTION AND IMPLICIT DISPERSION Simulations were performed with a time step of 1 0 5 0 1 and 0 01 for high Peclet number simulations and 26 5 13 25 2 65 and 0 265 days for low Peclet number simulations to generate Courant numbers of 1 0 5 0 1 and 0 01 respectively Figure 13 and 14 show the results for low and high Peclet number simulations respectively It was observed from the figures that explicit scheme performed well with both high and low Peclet numbers There was some numerical dispersion in the presence of high Peclet number However this is negligible 100 _ Pe 0 5 Explicit cr 0 01 gt 80 CR 0 10 60 CR 0 50 CR 1 00 40 Analytical 20 O 0 900 1000 1500 2000 2500 3000 Length cm Figure 13 RT1D results for low Peclet number simulations with varying Courant number using the explicit advection and implict dispersion scheme v 1 cm day D 10 cm day dx 5 cm k 7 0E 4 day T 3000 days 23
43. ispersion parameters 1f the simulation option is set to 1 2 The number of components in this example problem is five and this is a batch kinetics problem so we will consider all the components as immobile components 3 The reaction package number for this problem is 4 and there are a total of 10 reaction parameters that need to be input by the user 38 Advection Dispersion Parameters Kinetics Type Problem 50 1 amp 2 Geochemistry Type Problem 50 3 amp 4 Figure 28 Advection dispersion parameters for the example problem 4 The Monod parameters that need to be input are shown in Figure 29 Schaefer et al 2009 used retardation factor that was calculated due to the presence of air gap in the batch experiments This is not similar to the retardation caused due to the linear sorption in transport experiments However for convenience the R values used in paper are input in the R column for the components This is a batch experiment so there is no need to set a boundary condition The program ignores the boundary condition even if it is set by the user The initial concentrations of the components should be input in the Initial column Input all the Monod parameters as shown in Figure 29 The reaction parameter labels are set for the user s convenience and are ignored by the program Press the solve button and the results for the batch simulations are presented in Sheet 3 and the results for this simulation are shown in
44. lexes of As V that must be kept track of to calculate the equilibrium aqueous concentration So we will set the number of sorbed concentrations to 3 3 The pH is fixed for this problem and hence the number of fixed component concentrations is 1 Now we re ready to generate the input template sheet 4 Input the geochemistry tableau the initial aqueous concentration of arsenic and the surface complexation parameters as shown in Figure 50 93 5 The 3 sorbed species of As V are 8 9 and 10 in the species tableau These will allow the program to calculate the sorbed concentration of As V and the equilibrium As V concentration in moles L The program sums up the concentrations of species 8 9 and 10 to get the sorbed species concentration and subtracts this from the total aqueous concentration of As V to get the aqueous concentration of As V 6 The concentration of each species is presented in Table 3 The program also calculates the total aqueous and sorbed concentrations of As V in moles L This process can be repeated for different values of initial As V concentration and the generated data can be plotted go obtain isotherms 7 The values of sorbed and aqueous arsenic concentration were consistent with the values observed in the paper and the comparison between the published data and the RTID models simulations are presented in Figure 51 4 nn q B_ e _ _b E Advection Dispersion Parameters Kinetics Type Problem 50 1 amp
45. m automatically sets the unused options to N A The Section 3 in Figure 2 is a Generate Input Template button that will automatically generate a spreadsheet input table depending on the simulation option and the reaction type parameters set in Section 2 It can be observed from this figure that the input for kinetic type reaction has been set to N A by the program after pressing the button in Section 3 An example for a kinetic type input sheet 1s shown in Figure 3 and a geochemistry type input sheet is shown in Figure 4 lees PI B C D E 1 Advection Dispersion Parameters _2 _3 Ea 6 7 8 9 10 11 12 13 Kinetics Type Problem 50 1 amp 2 Geochemistry Type Problem SO 3 amp 4 23 TEI 3 Generate Input _24 Template Solve Clear A Figure 2 Spreadsheet layout for RT1D parameters to generate reaction specific input template In the following section we will briefly discuss various steps involved in running a simulation using RT1D The parameters and the required short codes for each module and options are made available in a textbox in the spreadsheet for the convenience of the user Step 1 Transport module To perform simulations that involve the use of a transport module the advection dispersion parameters along with the length of the column need to be input in their respectively columns in Section 1 The total simulation time and the amount of time for which th
46. m the flowrate Q as follows Q cross sectional area porosity Please make sure that units are same as the length total time Dispersion coefficient LTS This is the hydrodynamic dispersion coefficient calculated by multiplying the value of dispersivity with the velocity Please provide the hydrodynamic dispersion coefficient value and not the dispersivity value The code does not multiply with the velocity Adv Disp type This takes a short code for the type of Advection Dispersion solver The choices are 0 1 and 2 for explicit advection method fully implicit advection dispersion or TVD advection respectively For choices O and 2 dispersion is solved by fully implicit dispersion module Simulation option Type of problem 1 Batch Kinetics 2 Reactive Transport 3 Batch Geochemistry 4 Geochemistry coupled with Transport 1 2 2 KINETIC REACTION MODULE 1 11 111 Mobile components This is the number of aqueous components inside the system The mobile components will have an advection dispersion and reaction term There should be at least 1 mobile species for the program to work Immobile components This is the number of solid phase components in the system that do not undergo advection dispersion They only undergo reaction The immobile components will be placed at the bottom after all the mobile components In the reaction package the immobile species number will start after the mobile species That is
47. nitial concentration for all the components was zero The inlet was supplied with a constant concentration of 1 mg L of Component 1 for the entire duration of the simulation The results from this simulation are shown in Figure 24 Kinetics Type Problem 50 1 amp 2 Geochemistry Type Problem 50 3 amp 4 Figure 22 The advection dispersion parameters for the example problem 2 27 28 30 31 32 alo 37 38 39 40 41 42 a a Ed 47 49 Figure 23 Complete list of reaction parameters for the example problem 2 35 O Comp 1 RT1D 20 8 Comp 2 RT1D E Comp 3 RT1D 6 0 6 Comp 4 RT1D Comp 1 Analytical 0 4 Comp 2 Analytical 0 2 Comp 3 Analytical O Comp 4 Analytical o o Length cm Figure 24 Results for the example problem 2 4 3 EXAMPLE 3 REACTION PACKAGE 3 In this example we will learn to use the reaction package 3 which describes the transport of a decay chain of a four component system l Input the advection dispersion parameters for the problem as shown in Figure 25 2 There are a total of 4 mobile components and the reaction package for this example 1s built into the RT1D model as option 3 There are a total of 4 first order decay constants for each mobile component and the ODE solver type can be either 0 or 1 Press the Generate Input Template button to generate a table to input all the reaction parameters The column is s
48. njection of advanced municipal effluent into the aquifer The principal chemical mechanism involved is the heterovalent ion exchange of Na Mg2 and Ca2 Input the advection dispersion and the geochemistry parameters as shown in Figure 46 Myo BE 1 Advection Dispersion Parameters Kinetics Type Problem SO 1 amp 2 Geochemistry Type Problem 50 3 amp 4 YP ry Typ Figure 46 Transport parameters for example problem 10 91 2 There is no surface complexation in this model and the adsorption occurs by ion exchange so we set the SCM type to 0 3 The initial boundary guess concentrations for the components and the tableau are given in Figure 47 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 2 1 5 65 3 1 6 Figure 47 Geochemistry parameters for example problem 10 pa 4 The sorbed species Na X Mg X and Ca X are composed of species 4 3 and 6 respectively These sorbed concentrations are used to correct the aqueous concentrations of Na Mg2 and Ca2 components respectively 5 The results from these simulations are shown in Figure 48 a b and c 92 1 00E 04 1 00E 03 Na gt Mg2 x Na Anjani a a E x Mg2 anjani E 1 00E 03 3 1 00E 02 i O S Q 1 00E 02 1 00E 01 MBE m 1 00E 02 1 00E 03 1 00E 04 1 00E 05
49. ondition as shown in the Figure 38 5 Due to the presence of a slug injection zone as described in Phanikumar et al 2002 the initial condition for an llem section of the column is different from the rest of the column a To add this initial condition to the program would require editing the initial condition subroutine called setinit b Find this subroutine and uncomment the lines of under the this is needed for initial condition in Phanikumar er al 2002 c Copy paste the information available in Table 2 in Sheet 4 in the spreadsheet Create a Sheet 4 if it is not available The uncommented code segment looks for the initial condition for the slug injection zone in the Sheet 4 6 Press the solve button and simulation results are available in Sheets 2 amp 3 The results for this example problem are shown in Figure 39 U Advection Dispersion Parameters Kinetics Type Problem SO 1 2 Geochemistry Type Problem SO 3 4 YP ry yp 44 Figure 37 Advection dispersion parameters for example problem 7 OJO 160 42 1 8 0 000000028 0 pt ot 650 42 8 0 000000028 0 gt 2loa 1650 42 118 0 000000028 0 pot 160 42 1 8 0000000028 0 4 01 so af E DI 1650 42 118 0 000000028 0 1 1650 42 118 000008 0 2 42 118 o 000000028 0 42 8 0000000028 0 42 11 8 o 000000028 0 1 1650 42 118 o 000000028 0 rt seso ae s 000000002 o Table 2 Initial c
50. ondition for the slug injection zone for example problem 7 af 28 29 30 31 e e lelg 37 38 39 40 41 42 SG BG 47 49 50 51 Figure 38 Kinetic parameters for example problem 7 45 0 18 0 16 0 14 20 12 S 0 10 m E 0 08 Ja E CT RT1D 0 06 CT Model O 0 04 a CT Data 0 02 0 00 0 50 100 150 200 a Distance cm Figure 39 Carbon tetrachloride concentration after 4 days 4 8 EXAMPLE 8 SURFACE COMPLEXATION This is benchmark problem 4 for Torlapati and Clement 2012 Cederberg et al 1985 developed a mass transport model TRANQL for multicomponent solution system They presented a finite element solution for cadmium chloride and bromide transport in a one dimensional column where complexation and sorption were considered The reaction involves formation of aqueous and surfaces complexes and sorption of free cadmium described by constant capacitance model 1 For this problem we need to set up the parameters in the geochemistry type problem The simulation option will be 4 because this problem couples transport with geochemistry equilibrium 2 Input the advection dispersion parameters for this problem as shown in Figure 40 and set the number of components species aqueous components of interest and the sorbed components that are required to correct the concentrations of the aqueous components The pH is fixed at 7 for this problem and hence the number
51. owing four mobile components carbon tetrachloride acetate nitrate and mobile phase bacteria The two immobile phase components are sorbed carbon tetrachloride and immobile phase bacteria The reaction package for this problem is as follows note that the advection dispersion terms are omitted here for brevity fK dC ore es Kk Con Any X 4 14 K Cor Ser Si p j dt p dC Mina Ma M a dt 7 Y Ay A 39 dC Unax MaM b n t a Y Ay A py E 1 M yM ATAN 40 m Hima MaM bc M Ri Ka Ay Ka d M Xm ov gt K 14 K Co Ser 42 anu Hima MM Dic d i M o Ka 1 g MIX o K Ay 9 Where f is the fraction of equilibrium sites bxc 1s the microbial decay rate day Kart 1s the attachment coefficient day Kae 1s the detachment coefficient day k 1s the CT reaction rate day 1 y is the nitrate reaction rate day x is the kinetic desorption rate day Umax is the maximum specific growth rate day Ya Yn and Y py are the yield rates of acetate nitrate and biomass respectively Cer Ca Cn and Scr are the aqueous concentrations of carbon tetrachloride acetate nitrate and the sorbed concentration of carbon tetrachloride respectively Xy and Xi are the concentrations of mobile and immobile bacteria respectively Also M and M are the Monod terms for acetate and nitrate reactions respectively and given by the expressions a and M Ca where K and K are the half sat
52. ration of the species in the column It is possible that there is a residual component concentration present in the column before the simulation time has begun This option sets a constant initial concentration as specified in the spreadsheet across the all the nodes in the column It is possible to set a variable initial concentration by modifying the setinit subroutine in the code Boundary This is the inlet concentration of the species at time 0 This boundary concentration will be supplied to the column until the end of the pulse time Reaction terms Generates a table for entering the reaction terms and labels based on the total reaction terms 1 2 5 GEOCHEMISTRY EQUILIBRIUM PARAMETERS WITHOUT TRANSPORT The inputs are different when the geochemistry package is used with advection or without advection The input template also changes based on the type of surface complexation reaction chosen in the geochemistry input sheet 1 11 111 Mobile species This presents with 4 columns of different parameters for the mobile species a Index is the serial number of the component name This is automatically populated and should not be changed b Comp Name A default component name is generated automatically It is advised that this component name be changed to something more suitable The components with fixed concentration are input after all the variable components have been entered into the spreadsheet cells c Total concen
53. simulation the user should ensure that the following list of tasks are completed a Kinetic problem Simulation options 1 2 1 11 111 IV Select an existing reaction package from the several built in packages using the short code Program your own kinetic reaction package by opening the code editor Input the reaction parameters as necessary Click the solve button shown in Section 5 to perform the simulation b Geochemistry equilibrium problem Simulation options 3 4 1 11 111 IV V Input the total component concentrations at the boundary and the initial concentrations at the nodes if any Input the guess concentrations this is used by the solver for the starting solution to converge to the correct solution A good starting point would be the total concentration Please do not input 0 for the guess values as the logarithm is calculated for these guess concentrations as a part of the solution process Input the tableau and the surface complexation parameters 1f any Input the initial sorbed concentration present in the column and the species that combine to form the sorbed concentration This sorbed concentration is used to calculate the aqueous species concentrations at each time step Click the solve button shown in Section 5 to perform the simulation Step 5 Viewing the solutions The results are presented in Sheets 2 amp 3 of the model spreadsheet Sheet 2 provides the spatial var
54. tep 3 Generate the input template The Generate Input Template button shown in Section 3 of Figure 2 will generate a table to enter the parameters specific to the kind of simulation option selected by the user Example input template for a kinetic type and geochemistry equilibrium type problem are shown in Figures 3 and 4 respectively The Section 4 shown in these figures changes depending on the parameters set in Section 2 of the spreadsheet For the simulation options and 2 kinetic the button generates a template to input retardation factor initial and boundary conditions based on the number of mobile and immobile species entered in Section 2 The input template also provides an area for the user to input reaction parameters specific to that reaction package For the simulation options 3 and 4 geochemistry the button 3 will generate an input layout similar to Figure 3 to enter the tableau surface complexation parameters initial and boundary conditions for the aqueous component concentrations initial condition for the sorbed concentrations the composition of the sorbed concentrations and the surface complexation parameters Figure 3 Example input template for a kinetic problem es EO isla BE a zsaszasaaszans aula 2 2lal8 a a al2 ajal 8 8 29 85 e 3 2 A EEE Figure 4 Example input template for a geochemical equilibrium problem Step 4 Solve Prior to running a
55. the inlet have to be entered here The initial guess value for the concentrations is also entered here Initial The total residual concentration of the existing in the column before the beginning of the simulation The tableau information is similar to the batch geochemistry problem Sorbed phase concentrations The initial concentration of the sorbed species 1s input in the cells The number of sorbed concentrations is dependent on the input parameter set in Section 2 The user needs to enter the sorbed species index and the initial concentration for the sorbed phase Sorbed phase species composition In this section we define the composition of sorbed phase concentrations We have to provide the index of the sorbed phase concentration and the number of species it is composed of and the species index of all the species in the same row 10 2 MODEL DETAILS AND NUMERICAL SOLUTION The multi component one dimensional reactive transport model developed in this study designated as RT1D solves a coupled set of advection dispersion reaction equations for a total of n components The model simulates the transport of m mobile components that are either fully or partially coupled to a set of n m immobile components The reactions between these components could be mediated by biological geochemical kinetic reactions or geochemical equilibrium reactions The governing set of equations solved by the model can be written in a general
56. tration If the total concentration of the component is known please enter the value here d Guess concentration If the values are unknown please enter a guess value so the program has a starting value for the iteration process The iterative process converges faster if suitable starting concentrations are chosen A good guess for the guess concentration is the initial concentration itself unless it is zero Since the logarithm of the guess concentrations are calculated during the solution procedure please do not use 0 s as guess concentrations If the solution does not converge try a different guess value or check the problem inputs Tableau This is the tableau where you fill in the stoichiometric matrix Make sure the components are in the same order as the Total concentrations in the column Also enter the Log K values in the end for each species for the mass action matrix Additional parameters are required based on the surface complexation model chosen a LSIGO Index for PSIO in the component list LO in Westall 1979a b LSIGI Index for PSII in the component list L1 in Westall 1979a c LSIG2 Index for PSD in the component list L2 in Westall 1979a IV LSIG3 Index for SOH in the component list L3 in Westall 1979a SSD Surface site density sites m C1 in Westall 1979a SURFA Specific surface area m g C2 in Westall 1979a CONCS Concentration of sorbing solid g L C3 in Westall 1979a XMU Ionic
57. upplied with 100 mg L of Component 1 for the entire duration of the simulation at the inlet and there was no initial concentration before the beginning of the simulation so set the initial condition for all the components as O as shown in Figure 26 Input the retardation factors and the first order decay constant for each component as shown in Figure 25 Press the solve button to generate the solutions The results for this simulation are shown in Figure 27 36 Advection Dispersion Parameters 1 1 2 26 5 3 4 ES 6 7 8 0 037735849 9 0 5 10 11 12 14 15 KineticsTypeProblem SO 18 2 _ 16 17 18 19 20 Figure 25 Advection dispersion parameters for example problem 3 Figure 26 Reaction parameters for example problem 3 3 100 Comp T RTTD Comp 2 RT1D 80 n Comp 3 RT1D Comp 4 RT1D Comp 1 Analytical o 60 Comp 2 Analytical Comp 3 Analytical 40 Comp 4 Analytical O 20 O O 0 500 1000 1500 2000 2500 3000 Length cm Figure 27 Reaction parameters for example problem 3 4 4 EXAMPLE 4 REACTION PACKAGE 4 This is the reaction package that describes the bioaugmentation of TCE using Dehalococcoides Sp in a batch mode We will run this example with the simulation option 1 to exclude the transport part of the code 1 Input the parameters for the example problem as shown in Figure 28 The model automatically ignores advection d
58. uration coefficients of K C K C acetate and nitrate utilization reactions respectively The kinetic equations 38 to 40 describe biodegradation of carbon tetrachloride utilization of an electron donor acetate and an electron acceptor nitrate Equation 41 describes the growth decay and attachment of the mobile phase bacteria equation 42 describes the sorption of carbon tetrachloride using a two site sorption model and equation 43 describes the growth decay and detachment of immobile phase bacteria The code format for this reaction package could be written as follows 31 Ma Conc 2 RC 6 Conc 2 Mn Conc 3 RC 10 Conc 3 dydt 1 RC 14 Conc 1 Conc 4 Conc 6 RC 4 RC 7 RC 15 1 RC 5 REO Conc 1 Conc 5 RC dydt 2 RC 11 Ma Mn RC 8 Conc 4 Conc 6 dydt 3 RC 11 Ma Mn RC 12 Conc 4 Conc 6 RC 1 RC 16 1 Ma RC 3 Mn Conc 6 Conc 4 dydt 4 RC 11 Ma Mn RC 1 1 Ma RC 9 Conc 4 RC 13 1 Ma Conc 6 dydt 5 RC 7 1 RC 5 RC 2 Conc 1 Conc 5 dydt 6 RC 11 Ma Mn RC 1 1 Ma RC 13 1 Ma Conc 6 RC 9 Conc 4 3 3 8 USER DEFINED REACTION PACKAGE R NI YPE 10 RTID is best utilized when the user supplies their reaction packages To access this mode the user is required to open the code and search for the term RXN PACKAGE
59. vely i can be either TCE DCE and VC qi is the maximum biomass utilization rate K is the half velocity coefficient of the compound I is the competition coefficient R is the retardation due to the presence of air gap For further details about the model and the experimental methods refer to Schaefer et al 2009 The code form of this reaction package 1s given below mTCE RC 4 Conc 1 Conc 1 RC 1 mDCE RC 5 Conc 2 Conc 2 RC 2 1 Conc 1 RC 7 mVC RC 6 Conc 3 Conc 3 RC 3 1 Conc 1 RC 7 Conc 2 RC 8 dydt 1 1 R 1 Conc 5 mTCE dydt 2 Conc 5 mDCE R 2 mTCE R 1 dydt 3 Conc 5 mVC R 3 mDCE R 2 dydt 4 Conc 5 m VC R 3 dydt 5 RC 10 Conc 5 mTCE R 1 mDCE R 2 mVC R 3 To simpify the equations we have defined three different variables mTCE mDCE and mVC to define the Monod terms for each component 3 3 5 RATE LIMITED SORPTION RXNTYPE 5 In case of non equilibrium conditions the sorption is controlled by a mass transfer coefficient 6 When the value is really low the plume acts like tracer because there is no sorption and when the value is really high the plume acts like a retarded plume due to the linear sorption This kind of rate limited sorption kinetics can be modeled using the following reaction package 2 CoN D MO Oa 32 Ot Ox OX K as ofo S 33 dt p K Where
60. y are the attachment and detachment coefficients of mobile and immobile phase bacteria respectively n is the porosity of soil and p is the bulk density of the soil mg L The rate expression Ty is the nitrate utilization rate described by the Monod kinetics C C i l A ty x Where Qmax is the maximum nitrate utilization rate mg nitrate KFC Kick mg biomass day Ky 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 r and biomass growth rate 1 are given by the expressions r Yaniy and ry Y my Ky where Yan and Yx n are the yield coefficients for acetate and biomass respectively and Kg is the cell decay rate coefficient day The code format for this reaction package could be written as follows tN RC 3 Conc 1 RC 1 Conc 1 Conc 2 RC 2 Conc 2 rA RC 6 rN rX RC 7 rN RC 5 dydt 1 rN Conc 3 rN Conc 4 RC 9 RC 10 dydt 2 rA Conc 3 rA Conc 4 RC 9 RC 10 30 dydt 3 rX Conc 3 RC 4 Conc 3 RC 8 Conc 4 RC 9 RC 10 dydt 4 rX Conc 4 RC 8 Conc 4 RC 10 RC 4 Conc 3 RC 9 3 3 7 BIODEGRADATION OF CARBON TETRACHLORIDE RXNTYPE 7 Phanikumar et al 2004 studied the biodegradation of carbon tetrachloride CT in the presence of bacteria The transport problem considered the foll
61. y were used to perform the simulations The results from these simulations were compared against the analytical solutions presented in van Genuchten and Alves 1982 3 2 2 1 EXPLICIT ADVECTION AND IMPLICIT DISPERSION Simulations were performed with a time step of 0 1 0 05 0 01 and 0 001 for both high and low Peclet numbers to generate Courant numbers of 1 0 5 0 1 and 0 01 respectively Figure 7 and 8 show the results for low and high Peclet number simulations respectively 1 2 310 Pe 0 5 Explicit CR 0 01 gt CR 0 10 0 8 CR 0 50 0 6 CR 1 00 Analytical 0 4 yt O 6 8 0 2 0 0 0 10 20 30 40 50 Length cm Figure 7 RT1D results for low Peclet number simulations with varying Courant number using the explicit advection and implicit dispersion scheme v 1cm day D 0 2 cm day dx 0 1cm It was observed from the figures that the explicit scheme showed numerical dispersion in the presence of low Peclet number and this numerical dispersion decreased with increase in the Peclet number and the simulation results were closer to the analytical solutions 19 N Pe 2 Explicit CR 0 01 1 0 CR 0 10 E 08 CR 0 50 CR 1 00 E 0 6 5 Analytical o 0 4 O O 3 0 2 0 0 0 10 20 30 40 50 Length cm Figure 8 RT1D results for high Peclet number simulations with varying Courant number using the Explicit advection and Implicit dispersion scheme v 1cm day D

USER MANUAL FOR RT1D – Multi-Component

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USER MANUAL FOR RT1D &ndash; Multi-Component

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USER MANUAL FOR RT1D – Multi-Component