modeling 2d multispecies reactive transport in saturated/unsaturated porous media with the...

Modeling 2D Multispecies Reactive Transportin Saturated/Unsaturated Porous Mediawith the Eulerian–Lagrangian Localized Adjoint Method

F. Ramasomanana & A. Younes & M. Fahs

Received: 18 May 2011 /Accepted: 28 September 2011 /Published online: 19 October 2011# Springer Science+Business Media B.V. 2011

Abstract In the present paper, the Eulerian–Lagrangianlocalized adjoint method (ELLAM) formulation devel-oped by Younes et al. (Advances in Water Resources29:1056–1074, 2006) is combined with the sequentialnoniterative approach to accurately simulate 2Dmulticomponent reactive transport in saturated/unsat-urated porous media. The performance and accuracyof the developed model, named ELLAM_REACT, arecompared against those of an existing numericalmodel based on a combination of discontinuousGalerkin and multipoint flux approximation methods(DGMPFA_REACT). Three studied test cases, deal-ing with reactive transport in saturated and unsaturatedporous media and involving chemical reactions withonly aqueous species or both fixed and aqueous species,show the superiority of the ELLAM_REACT modelcompared to the DGMPFA_REACT model.

Keywords Porous media . Reactive transport .

Unsaturated flow . ELLAM . SNIA

1 Introduction

Standard Eulerian methods such as finite elements orfinite differences are not well suited for advection-dominated transport problems since they can generatesolutions with artificial diffusion and/or unphysicaloscillations. The Eulerian–Lagrangian localized adjointmethod (ELLAM), introduced by Celia et al. (1990), isan interesting alternative. It uses space–time testfunctions and treats general boundary conditionsnaturally in its formulation. ELLAM has been usedso far for one-dimensional advection dispersionequations (Celia et al. 1990; Healy and Russell1993; Younes 2004a), one-dimensional reactive trans-port problems (Wang et al. 1995; Vag et al. 1996; Belland Binning 2004; Younes 2004b; Farthing et al.2006; Younes and Fahs 2007; Fahs et al. 2009a),bidimensional transport problems with uniform rect-angular meshes (Binning and Celia 1996; Healy andRussell 1998), and unstructured triangular meshes(Younes and Ackerer 2005).

A new formulation of ELLAM was developed byYounes et al. (2006) to solve 2D advection dispersionequation in saturated porous media. This formulationrequires a limited number of strategic integration points(usually one per element) even for unstructured meshes.The integration points and weights are assigned at thenew time level and then backtracked to the old timelevel without redistributing the weights. In order toavoid excessive numerical diffusion which can occurwhen dealing with several time steps, continuous

Water Air Soil Pollut (2012) 223:1801–1813DOI 10.1007/s11270-011-0985-4

F. Ramasomanana :A. Younes (*) :M. FahsLaboratoire d’Hydrologie et de Géochimie de Strasbourg,Université de Strasbourg/EOST, CNRS, UMR 7517,1 rue Blessig,67084 Strasbourg, Francee-mail: [email protected]

M. FahsLebanese International UniversitySchool of Engineering Beirut,Beirut, Lebanon

characteristics are used. The concentration at the foot ofeach characteristic is obtained by interpolating solelychanges in concentration due to the dispersion process.The scheme was shown to be highly efficient formodeling 2D tracer transport in homogeneous andheterogeneous saturated porous media (Younes et al.2006; Ramasomanana and Younes 2011).

In this work, the ELLAM formulation developedby Younes et al. (2006) is extended to the modeling of2D multicomponent reactive transport in unsaturatedporous media. For these types of problems, accurateand efficient simulators remain a challenge whendealing with sharp concentration fronts.

Multicomponent reactive transport models areoften based on the sequential noniterative approach(SNIA) for coupling transport and chemical process-es. Indeed, SNIA has many advantages: (1) it can beeasily implemented, (2) it leads to a smaller system ofequations than the global approaches (Miller andRabideau 1993), (3) transport and chemical equationscan be solved using different numerical techniquesthat are specifically suited to achieve high accuracyfor each system of equations (Valocchi and Malmstead1992; Carrayrou et al. 2002), and (4) chemicalequations can be solved using smaller time steps thanthe ones used for the transport equations since thecharacteristic times of transport processes are oftenlarger than those of chemical processes (Kaluarachchiand Morshed 1995).

However, SNIA introduces an intrinsic numericalerror do to the splitting (Herzer and Kinzelbach 1989;Barry et al. 1996, 1997; Carrayrou et al. 2004; Fahs etal. 2008); therefore, small time steps are oftenrequired to reduce the operator splitting (OS) error.This can affect the efficiency and the accuracy ofELLAM when combined with SNIA. Indeed, it isknown that Eulerian–Lagrangian methods performwell for problems in which they can successfully usea large time step, however when several time steps areto be used, they suffer from numerical dispersion.Despite the fact that the ELLAM formulation ofYounes et al. (2006) uses a single strategic integrationpoint per element, the total number of particles to betracked may increase significantly after an increasingnumber of small time steps. The reason is that thetotal number of particles is proportional to the numberof elements multiplied by the number of time steps.This has an impact on both memory and CPUrequirement of ELLAM especially in the case of

transient unsaturated flow where the tracking processis more CPU consuming than in the case of steady-state saturated flow.

Given all the above, the aim of this work is todiscuss the applicability and efficiency of the ELLAMin simulating 2D multispecies reactive transport inboth saturated and unsaturated porous media. Theoutline of the paper is the following. In the first part,the mathematical equations governing fluid flow andreactive transport in 2D unsaturated porous media arebriefly recalled. The second part explains the devel-opment of the new numerical model based on thecombination of ELLAM and SNIA to solve thegoverning equations. Finally, three test cases aresimulated in order to analyze the performance of themodel for reactive transport problems in saturated andunsaturated porous media. The efficiency and accuracyof this model are assessed by comparing the resultsto those obtained using an existing numerical modelbased on a combination of discontinuous Galerkin andmultipoint flux approximation methods (Younes andAckerer 2008).

2 The Mathematical Model

The multicomponent reactive transport equationsin the saturated/unsaturated zone can be writtenas:

@ qCið Þ@t

¼ �r � qCið Þ þ r � Di � rCi

� �þ Ri C1;:::;CNc;tð Þ;

i ¼ 1;:::;Nc

ð1Þ

where Ci M � L�3� �

is the concentration of the speciesi, q½ � � the volumetric water content, Nc the numberof species, Ri the rate of the reaction which isgenerally function of all species concentrationsCjðj¼1;:::;NcÞ and Di L2 � T�1½ � the dispersion tensor of

coefficients Dikl defined by:

Dikl ¼ ðaT qj j þ DmÞdkl þ ðaL � aT Þ qkql

qj j ; k; l ¼ 1; 2

ð2Þ

where qk ½L � T�1� and ql½L � T�1� are the componentsof the Darcy’s velocity q½L � T�1�; aL L½ � and aT L½ � the

1802 Water Air Soil Pollut (2012) 223:1801–1813

longitudinal and transverse dispertivities, respective-ly; dij the Kronecker delta function and Dm L2 � T�1½ �the molecular diffusion coefficient.

Equation 1 is solved for the domain Ω ofboundary ∂Ω with initial and boundary conditions

of Dirichlet, Neumann, or total flux boundaryconditions.

SNIA splits the nonlinear system (Eq. 1) into asystem of linear partial differential equations for

advection–dispersion to find the trial solution Cnþ1;»i :

qnþ1Cnþ1;»i ¼ qnCn

i þZtþΔt

t

�r � qCið Þ þ r � Di � rCi

� �� dt; i ¼ 1; :::;Nc ð3Þ

This trial solution is then used as initial conditionfor the solution of a second system formed by thenonlinear reaction equations:

Cnþ1i ¼ Cnþ1;»

i þZtþΔt

t

Ri C1; :::;CNc; tð Þdt; i ¼ 1; :::;Nc

ð4ÞFor transient unsaturated problems, the volumetric

water content and the Darcy’s velocity can vary in bothspace and time. Their distribution inside the domain isobtained from the solution of Richards’ equation:

cðhÞ þ SsSwðqÞ½ � @H@t

þr � q ¼ f ð5Þ

where H[L] is the hydraulic head such that H=h+zwith h[L] the pressure head and z[L] the depth takenpositively upward, Ss½L�1� the specific storage; Sw½��the relative saturation of the aqueous phase Sw ¼ðq qs= Þ; qs½�� the saturated water content; cðhÞ½L�1� thespecific moisture capacity cðhÞ ¼ dq dh=ð Þ; f ½T�1�the source-sink term; q ¼ �KðhÞ � rH the Darcy’swater velocity and KðhÞ½L � T�1� the hydraulicconductivity.

The interdependencies of the pressure head, hy-draulic conductivity and water content are character-ized by their constitutive equations. In this study, weuse the standard models of Mualem (1976) and vanGenuchten (1980):

SeðhÞ ¼ qðhÞ � qrqs � qr

¼1

1þ ahj jnð Þ 1�1 n=ð Þ h < 0

1 h � 0

�

K Seð Þ ¼ KsSe1=2 1� 1� Se

n= n�1ð Þð Þ� � 1�1=nð Þ 2

ð6Þ

where Se½ � � is the effective saturation, qr½�� theresidual water content, Ks½L � T�1� the saturatedconductivity, and α and n the van Genuchten soilparameters.

Note that in the literature, the pressure-head form(Eq. 5) is often avoided since the mixed form of theRichards’ equation presents better conservation prop-erties. However, it was demonstrated (Tocci et al. 1997)that higher order numerical integration is effective forsolving the pressure head form (Eq. 5).

3 The Numerical Model

The coupled system of Eqs. 1, 2, 3, 4, 5, and 6 issolved on an unstructured triangular mesh in asequential way. First, the unsaturated flow system(Eqs. 5 and 6) is solved using the mixed finiteelement (MFE) method for the spatial discretizationand higher order methods for the temporal discretiza-tion via the method of lines. The MFE method givesaccurate and consistent velocity field even for highlyheterogeneous domains (Younes et al. 2010), and itwas also shown to be efficient for solvingRichards’ equation (Bergamaschi and Putti 1999;Farthing et al. 2003; Bause and Knabner 2004a;Belfort et al. 2009; Fahs et al. 2009b). With the MFEformulation, the velocity is defined everywhere in thefield and it is continuous across the interelementboundaries. To perform the tracking, each particleinside an element moves by following the MFEvelocity of that element until it reaches an exit point.Then, this point becomes the entry point of thedownstream element. During the procedure, theresidence time (time of flight) is also stored whenleaving each element. The final position is reached

Water Air Soil Pollut (2012) 223:1801–1813 1803

when the residence time equals the time steplength.

During the simulation of the unsaturated flow, thedistribution of Darcy’s velocity and the water contentwithin the domain are stored at different times and areused eventually when solving the transport equation.

3.1 The ELLAM Formulation for Transportin the Unsaturated Zone

In this section, we briefly summarize the main stagesof the ELLAM for the resolution of the advection–dispersion equation in the context of transport in theunsaturated zone. For simplicity, the transport equa-tion is written in the following form:

@ qCð Þ@t

þr � qCð Þ � r � D � rCð Þ ¼ 0 ð7Þ

The weak formulation of Eq. 7, using space–timetest function ω (x, t), leads to:Z T

0

ZΩ

@ qCð Þ@t

wþr � qCð Þw�r � D � rCð Þw

dx � dt ¼ 0

ð8ÞThe ELLAM selects test functions ω=ω (x, t)

constant along the characteristics:

dwdt

¼ @w@t

þ q

q� rw ¼ 0 ð9Þ

The functions ω (x, t) is discontinuous at the oldtime t n and verifies w x; tð Þ ¼ 0 for t=2 t n; t nþ1½ �. At thenew time t n+1, wi x; t nþ1ð Þ is the piecewise linearfunction in space (chapeau function). Thus, using theGreen’s theorem and expression of Eq. 9, the weakformulation can be written as follows (for details, seeYounes et al. 2006):

ZΩ

qnþ1Cnþ1wnþ1i dxþ

ZT0

ZΩ

ðD � rCÞ � rwidxdt þZT0

Z@Ω

ðqC � D � rCÞwi½ �h@Ωdxdt ¼ZΩ

qnCnwni dx ð10Þ

The concentration C is defined within a finiteelement framework as a combination of piecewiselinear basis functions. Therefore, the first integral inthe left hand side of Eq. 10, represents the standardfinite element mass integral. Using the mass lumpingprocedure, it leads to:ZΩ

qnþ1Cnþ1wnþ1i xð Þdx ¼

XE�i

jEjqnþ1E

3

!Cnþ1i ð11Þ

In this equation, E � i represents all elements E

sharing the node i, qnþ1E is the volumetric water

content of E at the new time tn+1 and |E| the area of E.The second term in Eq. 10 represents the exchange

by dispersion. This term is integrated over time usinga one step backward Euler approximation:

ZT0

ZΩ

D � rCð Þ � rwidxdt ¼ Δt xð ÞZΩ

Deq � rCnþ1� � � rwnþ1

i dx

ð12Þ

Note that the time step size Δt xð Þ in this integral isreduced for elements near the inflow boundary

(Younes et al. 2006). Furthermore, an equivalentdispersion tensor Deq is used instead of Dn+1 in theexpression (Eq. 12). Indeed, as suggested in (Fahset al. 2009a; Ramasomanana and Younes 2011), Dn+1

is not representative of the dispersion experiencedalong the characteristic between tn and tn+1, and it cangenerate errors for large time steps. To evaluate theequivalent dispersion tensorDeq, the dispersion tensorsDEj for all crossed element Ej between tn and tnþ1 are

monitored and stored as well as the duration ΔtEj

required to cross each element Ej. The equivalentdispersion coefficient Deq is then calculated by:

Deq ¼ 1

Δt xð ÞXj

ΔtEjDEj ð13Þ

The equivalent dispersion tensor is hence thearithmetic mean of local dispersion tensors weightedby the mean times spent under the influence of theselocal values. The use of Deq instead of Dn+1 improvesthe results for large time steps (Fahs et al. 2009a;Ramasomanana and Younes 2011).

The third term and the right-hand term in Eq. 10correspond, respectively, to the boundary conditions


and the mass at the old time level. These terms arecalculated numerically. For simplicity, only theintegral of the mass at the old time level is developedhere. This integral can be approximated by:ZΩ

qnCnwni xð Þdx ¼

XE�i

qnEXNE

p¼1

Cn xp; tnð Þwi xp; tnð ÞWp

ð14Þ

where NE is the number of spatial integration points foreach element E and Wp the weight of the numericalintegration point (p). To evaluate wi x p; tnð Þ thelocation in space of xp at the time tn is trackedforward to ex p at tn+1 and the relation wi x p; tnð Þ ¼wi ex p; tnþ1ð Þ is used.

3.2 The Numerical Strategy

To avoid numerical diffusion due to interpolation whensmall time steps are used, the advection–dispersion–reaction system (Eqs. 1, 2, 3, and 4) is solvedaccording to the following:

& Only strategic points are used as numericalintegration points. The location and weights ofthese points are assigned at the new time level.The points are then backtracked to the old timelevel without redistributing the weights. The totalnumber of integration points is proportional to thenumber of time steps multiplied by the number ofthe integration points P per element. If P=1, theintegration point is located at xj, which corre-sponds to the center of the element and Wspace

j

corresponds to its area. If P=4, the triangularelement is subdivided into 4 equal subtriangles(by joining the edge midpoints), xj corresponds tothe center of each subtriangle, and Wspace

j corre-

sponds to the area of the subtriangle.& The solution of the system (Eq. 10) gives the

concentrations at the nodes of the mesh due to the

advection–dispersion processes Cadvþdispi

� �at the

new time level (n+1). This calculation is based onlyon the strategic integration points corresponding tothat time level.

& The concentrations Cadvþdispi are then used as

initial concentrations to solve the chemical Eq. 4at the node level providing thus that the node

concentrations as a result of the advection–

dispersion–reaction processes Cadvþdispþreacti

� �.

& The concentration at the integration points of thesubsequent time steps is updated by interpolatingonly the changes in concentration due to thedispersion and the chemical processes:

Cnp ¼ Cadv

p þ Cdisp þ reactp ð15Þ

where Cadvp is the concentration of the particle (p)

due to the advection process and Cdisp þ reactp the

concentration of (p) due to the chemical anddispersion processes obtained by interpolating the

nodal values Cdisp þ reacti which are calculated as

follows:

Cdisp þ reacti ¼ Cadv þ disp þ react

i � Cadvi ð16Þ

where Cadvi represents the nodal concentration due

to advection, calculated from Eq. 10 with zerodispersion. Note that using the mass lumpingtechnique, this procedure can be done locally atthe node level and there is no linear system to besolved in this case.

Finally, it is relevant to recall that specific time stepsizes are used for the time discretization of the Richards’equation. Therefore, during the tracking process, thevelocity field is updated periodically based on the timediscretization of the flow. As a general rule, specifictime step size is used for the integration (over time) ofthe nonlinear chemical equations.

4 Numerical Experiments

In this section, the performance and the accuracy ofthe developed model, named ELLAM_REACT, arecompared against those of an existing numericalmodel based on a combination of discontinuousGalerkin and multipoint flux approximation methods(DGMPFA_REACT). In this model, the discontinu-ous Galerkin (DG) method is used to discretize theadvection equation and is combined with the multi-point flux approximation (MPFA) method for thediscretization of the dispersion equation (Younes andAckerer 2008). The DG method leads to robust andaccurate numerical results for problems involvingsharp fronts (Shuangzhang and Shahrouz 2005). The


MPFA method is locally conservative and handleswell general irregular grids on anisotropic heteroge-neous domains (Younes and Fontaine 2008). Becauseit is an explicit scheme, the Courant criterion (CFL)has to be fulfilled which may lead to very small timesteps for unstructured meshes.

The SNIA is then used for coupling transport andreaction processes with both ELLAM_REACT andDGMPFA_REACT models. The chemical equationsare solved using the Runge–Kutta method with smallsubtime steps.

In order to assess the accuracy and the efficiencyof both models, three test cases sourced from theliterature are employed. They describe reactive trans-port in saturated and unsaturated porous media withchemical reactions involving mobile chemical speciessolely or both fixed and mobile species.

The first case describes the degradation of PCE insaturated porous media (Clement et al. 1998) andanalyses the performance of the models in the case ofchemical reactions involving only mobile species.The second and third test cases are the m-xylenedegradation in saturated and unsaturated flow regime,respectively. These latter test cases provide the behaviorof the models when chemical reactions involve bothfixed and mobile species.

4.1 The Degradation of the Tetrachloroethene PCE

This test case, inspired fromClement et al. (1998), dealswith the degradation of the tetrachloroethene PCE insaturated porous media. It involves four mobilechemical species (Nc=4), namely the PCE and itsdegradation products: the trichloroethene (TCE), thedichloroethene (DCE), and the vinyl chloride (VC).Under the assumption of first-order sequential kineticbiodegradation and biotransformation of PCE and itsdegradation products, the reaction equations are:

R1 ¼ d

dtPCE½ � ¼ �KPCE PCE½ �

R2 ¼ d

dtTCE½ � ¼ YTCE=PCEKPCE PCE½ � � KTCE TCE½ �

R3 ¼ d

dtDCE½ � ¼ YDCE=TCEKTCE TCE½ � � KDCE DCE½ �

R4 ¼ d

dtVC½ � ¼ YVC=DCEKDCE DCE½ � � KVC VC½ �

ð17Þ

where [PCE], [TCE], [DCE], and [VC] are theconcentrations of the respective contaminants. The first-order degradation ratesKPCE, KTCE, KDCE, and KVC are0.005, 0.003, 0.002, and 0.001 day−1, respectively.The yield coefficients YTCE/PCE, YDCE/TCE, and YVC/DCEare computed from stochiometric relations and corre-spond to 0.79, 0.74, and 0.64, respectively.

The spatial domain is a rectangular confinedaquifer of 500 m length and 300 m wide. Acontinuous source introducing PCE at the rate of2 m3day−1 with a concentration of 200 mg l−1 islocated within the domain at (135,135). Initially, thedomain is assumed free of contaminant. Boundaryconditions for the flow are a constant flux of 1/3 mday−1 at the left boundary and a prescribed head of10 m at the right boundary of the domain. Theporosity is 0.3, the longitudinal and the transversedispersivities are 3 and 1 m, respectively. The overallduration of the simulation is 3 years. A zero concentra-tion is prescribed at the inlet boundary, and freeboundary condition is assumed at the exit boundary.The spatial domain is discretized using an unstructuredtriangular mesh of 13,640 elements with a localrefinement around the source.

Due to the CFL restriction, the simulation of thistest case with DGMPFA_REACT requires a transporttime step of Δt ¼ 1:35 days. Two simulations,ELLAM_REACT1 and ELLAM_REACT2 with asmall transport time step of Δt ¼ 15 days and a largetransport time step of Δt ¼ 182:5 days respectivelywere performed using ELLAM_REACT. For allsimulations, the time step for the chemical equationsis fixed to Δt ¼ 0:01 days regardless of the transporttime step size.

Figure 1 shows the spatial distribution (isolines)of the TCE and DCE at the end of the simulation(after 3 years). The two models ELLAM_REACTand DGMPFA_REACT give similar results. Theresults of ELLAM_REACT1 are very close to theresults of ELLAM_REACT2. This shows that theELLAM_REACT can be used with large time stepswithout altering the accuracy of the results when thereactions involve only aqueous species. Indeed, in thiscase, the mass balance error (Valocchi and Malmstead1992; Kaluarachchi and Morshed 1995), introducedby SNIA, is removed with ELLAM (for details, seeYounes and Fahs 2007).

With respect to the efficiency, DGMPFA_REACTrequires 273 s for the whole simulation, whereas


ELLAM_REACT1 and ELLAM_REACT2 require143 and 32 s, respectively. For this test caseELLAM_REACT2 is highly efficient, it providesaccurate results with eight times less CPU time thanDGMPFA_REACT.

4.2 The m-Xylene Degradation in SaturatedFlow Regime

This test case was developed by Bause and Knabner(2004b). It deals with biochemical degradation of them-xylene in porous media. The microbial activity inthe subsurface depends on the bioavailability of allsubstrates utilized by the microorganisms. The mainsubstrates are the electron donor D, the electronacceptor A, and the primary carbon source X. Theelectron donor and the electron acceptor can bemodeled as mobile species whereas the carbon sourceis considered as fixed species. The chemical reactionsare modeled using Monod kinetics:

R1 ¼ d

dtqCDð Þ ¼ �m

R2 ¼ d

dtqCAð Þ ¼ �aA=Dm

R3 ¼ dCX

dt¼ Y

q1� CX

CX max

� �m� kdCX

ð18Þ

where the Monod term m½L � T�1� is defined by

m ¼ qmmaxCXCD

KD þ CD

KID

KID þ CD� CA

KA þ CA

KIA

KIA þ CA

ð19Þ

In Eq. 18, the constant aA=D denotes the electron

acceptor to donor mass ratio, kd½T�1� the first-orderdecay rate for the biomass, Y the microbial yieldcoefficient per unit electron donor consumed andCX max½M � L�3� the maximum biomass concentration.In Eq. 19 mmax½T�1� denotes the maximum growthrate, Ki¼D;A½M � L�3� the half utilization constant of

the electron donor and acceptor and Ki¼ID;IA ½M � L�3�the Haldane inhibition concentration of the electrondonor and acceptor.

For the simulation, we use the following values forthe parameters: aA=D ¼ 2:16, kd ¼ 0:025 day�1, Y ¼0:52, CX max ¼ 1:0 mg l�1, mmax ¼ 1:13 day�1, KD ¼7:9 � 10�1 mg l�1, KA ¼ 1:0 � 10�1 mg l�1, KID ¼9:17 � 10�1 mg l�1 and KIA ¼ 1.

The spatial domain is a rectangular shape6 � 10 mð Þ including two impermeable regionsdenoted by E1 and E2 (Fig. 2). The domain isdiscretized with an unstructured triangular mesh of12,378 elements (Fig. 2). The fluid flow through thedomain is assumed to be steady state with a conductivityKs ¼ 4:5 � 10�2 m day�1. The flow boundary con-ditions are the following:

H ¼ 0 m for x 2 0; 6½ � � 0f gH ¼ 20 m for x 2 0; 6½ � � 10f gKsrh ¼ 0 m day�1 elsewhere:

8><>:ð20Þ

The electron donor m-xylene is injected at theupper inflow boundary with

DGMPFA_REACT(Δt = 1.35 days) (Δt = 15 days) (Δt = 182.5 days)

ELLAM_REACT1 ELLAM_REACT2

TCE

DCE

Fig. 1 Spatial distribution of the TCE and the DCE after 3 years


CDðt; xÞ ¼ 2:24 mg l�1

CAðt; xÞ ¼ 2:55 mg l�1

(for x 2 1; 5½ � � 10f g

CDðt; xÞ ¼ 0:0 mg l�1

CAðt; xÞ ¼ 2:0 mg l�1

(for x 2 0; 1½ � [ 5; 6½ �ð Þ � 10f g:

ð21ÞThe porosity is 0.33, the dispersivities aL ¼ 0:36 m

and aT ¼ 0:03 m and the molecular diffusion Dm ¼7:4 � 10�5 m2 day�1. The initial conditions for thechemical species are:CDð0; xÞ ¼ 0:0 mg l�1, CAð0; xÞ ¼2:0 mg l�1, and CX ð0; xÞ ¼ 3:0 � 10�2 mg l�1. Theduration of the simulation is T=300 days.

The simulation of this test case with theDGMPFA_REACT model requires a time stepΔt ¼ 0:04 days due to the CFL restriction. Two

simulations ELLAM_REACT1 and ELLAM_-REACT2 with a small transport time step of Δt ¼2 days and a large transport time step of Δt ¼60 days respectively were performed with ELLAM_-REACT. The chemical equations are solved with afixed small time step Δt ¼ 0:01 days for all models.

The results of these simulations are presented inFig. 3. This figure illustrates the spatial distribution ofthe mobile (donor) and fixed (biomass) species after300 days. ELLAM_REACT1 and DGMPFA_REACTgive similar results. This shows that ELLAM_-REACT can give accurate results for the simulationof 2D reactive transport problems involving bothmobile and fixed species. Figure 3 shows also thatresults of ELLAM_REACT2 are different from thoseof DGMPFA_REACT. The difference is due to theOS errors. Indeed, it is well known that OS errorsgenerate numerical diffusion proportional to Δtsmearing sharp concentration fronts (Herzer andKinzelbach 1989; Barry et al. 1997). Contrarily tothe first problem, this error cannot be removed withELLAM. Therefore special attention should be givento the choice of the time step size in order to minimizethis error.

Regarding efficiency, the results of ELLAM_-REACT2 are not accurate and therefore are notconsidered for the comparison. The CPU times withDGMPFA_REACT and ELLAM_REACT1 are 750and 200 s, respectively. Therefore, ELLAM_REACTprovides accurate results using four times less CPUtime than DGMPFA_REACT.

4.3 The m-Xylene Degradation in Variably SaturatedFlow Regime

The previous problem is now simulated consideringan unsaturated porous media as suggested in Bauseand Knabner (2004b). The parameters of the soil are:θr=0.031, θs=0.396, α=0.423, n=2.06, andKs ¼ 4:96 � 10�2 m day�1. The spatial domain isillustrated in Fig. 4.

The initial condition for the hydraulic head is

Hð0; xÞ ¼ 1:0 m for x 2 Ω ¼ 0; 2ð Þ � 0; 3ð Þn R1 [ R2ð Þð22Þ

where R1¼ 0; 1:2½ �� 2:25; 2:50½ � and R2 ¼ 0:8; 2:0½ � �1:5; 1:75½ �.

Fig. 2 The spatial domain and the mesh grid for the second testcase


Dirichlet flow boundary conditions are prescribedas follows:

H ¼ 3:20 m for t 2 7i; 7 iþ 1ð Þ½ �1:0 m for t 2 7 iþ 1ð Þ; 7 iþ 2ð Þ½ � for x 2 0; 1½ � � 3f g; i ¼ 0; 2; 4:::

(

H ¼ 1:0 m for t > 0; x 2 2f g � 0; 1½ �f

ð23Þ

A zero normal flux is prescribed for the rest ofboundaries.

The initial conditions for the species inside Ω are:CDð0; xÞ ¼ 0 mg l�1, CAð0; xÞ ¼ 2:0 mg l�1, a n dCX ð0; xÞ ¼ 3:0 � 10�3 mg l�1. The contaminant m-xylene is injected at the upper inflow boundary with:

CD t; xð Þ ¼ 3:36 mg l�1

CA t; xð Þ ¼ 2:55 mg l�1 for x 2 0; 1½ � � 3f g:

(ð24Þ

The duration of the simulation is 300 days. Thedomain is discretized with 7,500 triangular elements asshown in Fig. 4.

DGMPFA_REACT ELLAM_REACT1 ELLAM_REACT2

Electron Donor

Biomass

(Δt = 0.04 days) (Δt = 2 days) (Δt = 60 days)

Fig. 3 Spatial distribution of electron donor and biomass in the saturated porous media


The problem is first simulated with the DGMPFA_-REACT using a time step of Δt=0.05 days to respectthe CFL criterion. Two simulations are performedwith ELLAM_REACT, the first ELLAM_REACT1uses a small time step ofΔt=0.05 days and the secondELLAM_REACT2 uses a large time step of Δt=

25 days. The flow time step size is fixed to 0.05 days,which corresponds to the transport time step size ofDGMPFA_REACT. The time step for the chemicalequations is fixed to Δt=0.01 days.

Figure 5 shows that DGMPFA_REACT andELLAM_REACT1 give similar results while the resultsof ELLAM_REACT2 are different. ELLAM_REACTcan be used to accurately simulate reactive transportinvolving mobile and fixed species in unsaturated soils,however, as for the saturated case, small time stepsshould be used to reduce OS errors.

Concerning efficiency, ELLAM_REACT1 andDGMPFA_REACT require 520 and 540 s, respec-tively, to perform the simulation. Contrarily to thesaturated case and despite the fact that ELLAM_-REACT1 uses 10 times larger time steps thanDGMPFA_REACT, the two models have comparableperformances. This phenomenon is due to thecomputational effort required by the tracking processwhich becomes more significant for unsaturated flowsince the velocity becomes transient and differentvelocity fields are used for the tracking. Indeed, forthe simulated case, the CPU time required by thetracking is around 75% of the total CPU time of thesimulation.

Note that if we use smaller time steps, theperformances of ELLAM_REACT can be significantlyaltered since the number of integration points to betracked is proportional to the number of elementsmultiplied by the number of time steps. This has animpact on both the memory requirement and thecomputational time of the method. To overcome this

Fig. 4 The spatial domain and the mesh grid for the third testcase

DGMPFA_REACT ELLAM_REACT1 ELLAM_REACT2 ELLAM_REACT1b

Subdivision technique

Biomass

(Δt = 0.5 day) (Δt = 25 days) (Δt = 0.5 day)(Δt = 0.05 day)

Fig. 5 Spatial distribution of biomass in the unsaturated porous media


problem and to improve the efficiency of ELLAM_-REACT, we subdivide the duration of the simulationinto a few number of time periods (denoted by ndt) andwe keep the characteristics continuous only for thosetime steps included in the time period. When startinga new time period, the concentration at the foot ofeach characteristic is calculated by interpolating theconcentration of the nodes. In this way, the number ofintegration points can be divided by the number oftime periods ndt.

Considering the above, the same problem is nowsimulated with ELLAM_REACT1b, which uses thesame step size of Δt=0.5 days as ELLAM_REACT1but the duration of the simulation is subdivided into10 periods (ndt=10). As it can be seen in Fig. 5 thatthe results of ELLAM_REACT1b are comparable tothe results obtained with ELLAM_REACT1. Thisshows that the numerical diffusion caused by theinterpolation is very limited due to a reduced numberof time periods (10) compared to the number of timesteps (400). The efficiency of ELLAM_REACT hasalso significantly improved. The memory requirementfor ELLAM_REACT1b is now lower than forELLAM_REACT1, and the CPU time of ELLAM_-REACT1b is 130 s. Therefore, with the subdivisiontechnique, ELLAM_REACT is four times moreefficient than DGMPFA_REACT even for unsaturatedreactive transport problems involving both fixed andmobile species.

5 Conclusion

The aim of the present work is to assess the accuracy andefficiency of a new 2D model for multicomponentreactive transport in saturated/unsaturated porous media.

The new model, called ELLAM_REACT, com-bines the ELLAM formulation of Younes et al. (2006)for advection–dispersion transport with the SNIA toaccurately solve the multicomponent reactive trans-port. The ELLAM_REACT model uses continuouscharacteristics and only changes due to dispersion,and chemical processes are interpolated. The perfor-mance and the accuracy of ELLAM_REACT havebeen compared against those of an existing numericalmodel based on a combination of discontinuousGalerkin and multipoint flux approximation methods(DGMPFA_REACT). Three test cases have beenconsidered.

In the first test case, reactive transport with chemicalreactions involving only aqueous species was simulated.Results show that ELLAM_REACT is highly efficient,and it maintains its high accuracy when large time stepsare considered.

The second case considered for simulation wasreactive transport in saturated porous media withchemical reactions involving both aqueous and fixedspecies. The new model ELLAM_REACT provesagain to be more efficient than DGMPFA_REACT.However, ELLAM_REACT requires small time stepsto avoid numerical diffusion due to OS errors.

Finally, reactive transport test in unsaturated porousmedia with both fixed and mobile species was simulat-ed. This case scenario shows that the efficiency ofELLAM_REACT can be negatively affected whensmall time steps are used due to a significant increasein the number of particles to be tracked and also due tothe computational effort required for the trackingprocess when dealing with transient unsaturated flow.To overcome this problem, the duration of the simula-tion is subdivided into a few number of time periodswhere characteristics remain continuous only for timesteps included in the time period. By using thistechnique, ELLAM_REACT is found again to be moreefficient than DGMPFA_REACT.

Acknowledgments This work was partly carried out withinthe framework of the GdR MoMas CNRS-2439 sponsored byANDRA, BRGM, CEA, and EDF. Their support is dulyacknowledged.

References

Barry, D. A., Bajracharya, K., & Miller, C. T. (1996).Alternative split-operator approach for solving chemicalreaction/groundwater transports models. Advances inWater Resources, 19, 261–275.

Barry, D. A., Miller, C. T., Culligan, P. J., & Bajracharya, K.(1997). Analysis of split operator methods for nonlinearand multispecies groundwater chemical transport models.Mathematics and Computers in Simulation, 43, 331–341.

Bause, M., & Knabner, P. (2004a). Computation of variablysaturated subsurface flow by adaptative mixed hybridfinite element methods. Advances in Water Resources, 27,565–581.

Bause, M., & Knabner, P. (2004b). Numerical simulation ofcontaminant biodegradation by higher order methods andadaptive time stepping. Computing and Visualization inScience, 7, 61–78.

Belfort, B., Ramasomanana, F., Younes, A., & Lehmann, F.(2009). An efficient lumped mixed hybrid finite element


formulation for variably saturated groundwater flow.Vadose Zone Journal, 8, 352–362.

Bell, L. S. J., & Binning, P. J. (2004). A split operator approachto reactive transport with the forward particle trackingEulerian-Lagrangian localized adjoint method. Advancesin Water Resources, 27, 323–334.

Bergamaschi, L., & Putti, M. (1999). Mixed finite elements andNewton-type linearizations for the solution of Richards’equation. International Journal for Numerical Methods inEngineering, 45, 1025–1046.

Binning, P., & Celia, M. A. (1996). A finite volume Eulerian–Lagrangian localized adjoint method for solution of thecontaminant transport equations in two-dimensional multi-phase flow systems. Water Resources Research, 32, 103–114.

Carrayrou, J., Mosé, R., & Behra, P. (2002). New efficientalgorithm for solving thermodynamic chemistry. AmericanInstitute of Chemical Engineers Journal, 48, 894–904.

Carrayrou, J., Mosé, R., & Behra, P. (2004). Operator-splittingprocedures for reactive transport and comparison of massbalance errors. Journal of Contaminant Hydrology, 68,239–268.

Celia, M. A., Russell, T. F., Herrera, I., & Ewing, R. E. (1990).An Eulerian–Lagrangian localized adjoint method for theadvection–diffusion equation.Advances in Water Resources,13, 187–206.

Clement, T. P., Sun, Y., Hooker, B. S., & Petersen, J. N. (1998).Modeling multispecies reactive transport in ground water.Ground Water Monitoring & Remediation, 18(2), 79–92.

Fahs, M., Carrayrou, J., Younes, A., & Ackerer, P. (2008). Onthe efficiency of the direct substitution approach forreactive transport problems in porous media. Water, Air,and Soil Pollution, 193, 1–4.

Fahs, M., Younes, A., & Delay, F. (2009). On the use of large timesteps with ELLAM for transport with kinetic reactions overheterogeneous domains. American Institute of ChemicalEngineers Journal, 55, 1121–1126.

Fahs, M., Younes, A., & Lehmann, F. (2009). An easy andefficient combination of the mixed finite element method andthe method of lines for the resolution of Richards’ equation.Environmental Modelling & Software, 24, 1122–1126.

Farthing, M. W., Kess, C. E., & Miller, C. T. (2003). Mixedfinite element methods and higher order temporal approx-imations for variably saturated groundwater flow. Advancesin Water Resources, 26, 373–394.

Farthing, M. W., Kess, C. E., Russell, T. F., & Miller, C. T.(2006). An ELLAM approximation for advective–dispersivetransport with nonlinear sorption. Advances in WaterResources, 29, 657–675.

Healy, R. W., & Russell, T. F. (1993). A finite-volumeEulerian–Lagrangian localized adjoint method for solutionof the advection–dispersion equation. Water ResourcesResearch, 29, 2399–2413.

Healy, R. W., & Russell, T. F. (1998). Solution of theadvection–dispersion equation in two dimensions by afinite-volume Eulerian–Lagrangian localized adjoint method.Advances in Water Resources, 21, 11–26.

Herzer, J., & Kinzelbach, W. (1989). Coupling of transport andchemical processes in numerical transport models.Geoderma,44, 115–127.

Kaluarachchi, J. J., &Morshed, J. (1995). Critical assessment of theoperator-splitting technique in solving the advection-dispersion-reaction equation: I. First-order reaction. Advancesin Water Resources, 18, 89–100.

Miller, C. T., & Rabideau, A. J. (1993). Development of split-operator, Petrov–Galerkin methods to simulate transport anddiffusion problems. Water Resources Research, 29, 2227–2240.

Mualem, Y. (1976). A new model for predicting the hydraulicconductivity of unsaturated porous media. Water ResourcesResearch, 12, 513–522.

Ramasomanana, F., & Younes, A. (2011). Efficiency of theEulerian-Lagrangian localized adjoint method for solvingadvection-dispersion equation on highly heterogeneousmedia. International Journal for Numerical Methods inFluids, 66 (in press).

Shuangzhang, T., & Shahrouz, A. (2005). A slope limitingprocedure in discontinuous Galerkin finite element methodfor gas dynamics applications. International Journal ofNumerical Analysis and Modeling, 2, 163–178.

Tocci, M. D., Kelley, C. T., & Miller, C. T. (1997). Accurateand economical solution of the pressure-head form ofRichards’ equation by the method of lines. Advances inWater Resources, 20(1), 1–14.

Vag, J. E., Wang, H., & Dahle, H. K. (1996). Eulerian–Lagrangian localized adjoint methods for systems ofnonlinear advective–diffusive–reactive transport equations.Advances in Water Resources, 19, 297–315.

Valocchi, A. J., & Malmstead, M. (1992). Accuracy of operatorsplitting for advection–dispersion reaction problems. WaterResources Research, 28, 1471–1476.

van Genuchten, M. T. (1980). A closed-form equation forpredicting the hydraulic conductivity of unsaturated soils.Soil Science Society of America Journal, 44, 892–898.

Wang, H., Ewing, R. E., & Celia, M. A. (1995). Eulerian–Lagrangian localized adjoint method for reactive transportwith biodegradation. Numerical Methods Partial forDifferential Equations, 11, 229–254.

Younes, A. (2004a). An accurate moving grid Eulerian–Lagrangian localized adjoint method for solving the one-dimensional variable-coefficient ADE. International Jour-nal for Numerical Methods in Fluids, 45, 157–178.

Younes, A. (2004b). Modélisation du transport réactif enmilieux poreux non saturés avec une méthode ELLAMen maillage variable. Comptes Rendus de l’Académie desSciences, Géosciences, 336, 547–552.

Younes, A., & Ackerer, P. (2005). Solving the advection–diffusion equation with the Eulerian–Lagrangian localizedadjoint method on unstructured meshes and non uniformtime stepping. Journal of Computational Physics, 208,384–402.

Younes, A., & Ackerer, P. (2008). Solving the advection–dispersion equation with discontinuous Galerkin andmultipoint flux approximation methods on unstructuredmeshes. International Journal for Numerical Methods inFluids, 58, 687–708.

Younes, A., & Fahs, M. (2007). Direct and split operatorapproaches with ELLAM for reactive transport equations.American Institute of Chemical Engineers Journal, 53,2161–2169.


Younes, A., & Fontaine, V. (2008). Hybrid and multi-pointformulations of the lowest-order mixed methods forDarcy’s flow on triangles. International Journal forNumerical Methods in Fluids, 58, 1041–1062.

Younes, A., Ackerer, P., & Lehmann, F. (2006). A newefficient Eulerian–Lagrangian localized adjoint method

for solving the advection–dispersion equation onunstructured meshes. Advances in Water Resources, 29,1056–1074.

Younes, A., Ackerer, P., & Delay, F. (2010). Mixed finiteelements for solving 2-D diffusion-type equations.Reviews of Geophysics, 48.


modeling 2d multispecies reactive transport in saturated/unsaturated porous media with the...

Documents