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Geochimica et Cosmochimica Acta 141 (2014) 656–669

Multicomponent ionic dispersion during transportof electrolytes in heterogeneous porous media: Experiments

and model-based interpretation

Muhammad Muniruzzaman a, Christina M. Haberer a, Peter Grathwohl a,Massimo Rolle a,b,⇑

a Center for Applied Geosciences, University of Tubingen, Holderlinstr. 12, D-72074 Tubingen, Germanyb Department of Civil and Environmental Engineering, Stanford University, 473 Via Ortega, 94305 Stanford, CA, USA

Received 17 February 2014; accepted in revised form 17 June 2014; available online 30 June 2014

Abstract

This study investigates the effects of Coulombic interactions during transport of electrolytes in heterogeneous porousmedia under steady-state flow and transport conditions. We performed flow-through experiments in a quasi two-dimensionalsetup using dilute solutions of strong 1:1 and 1:2 electrolytes to study the influence of electrochemical cross-coupling on masstransfer of charged species in saturated porous media. The experiments were carried out under advection-dominatedconditions (seepage velocity: 1 and 1.5 m/day) in two well-defined heterogeneous domains where flow diverging around alow-permeability inclusion and flow focusing in high-permeability zones occurred. To quantitatively interpret the outcomesof our laboratory experiments in the spatially variable flow fields we developed a two-dimensional numerical model based on amulticomponent formulation and on charge conservation. The results of the multicomponent transport simulations werecompared with the high-resolution concentration measurements of the ionic species at the outlet of the flow-through domain.The excellent agreement between the measured concentrations and the results of purely forward numerical simulationsdemonstrates the capability of the proposed two-dimensional multicomponent approach to describe transport of charged spe-cies and to accurately capture the Coulombic interactions between the ions, which are clearly observed in the flow-throughexperiments. Furthermore, the model allowed us to directly quantify and visualize the ionic interactions by mapping theCoulombic cross-coupling between the dispersive fluxes of the charged species in the heterogeneous domains.� 2014 Elsevier Ltd. All rights reserved.

1. INTRODUCTION

Aqueous diffusion of charged species is significantlyaffected by the electrochemical migration term resultingfrom the interactions among the dissolved species and/orwith charged surfaces (e.g., Vinograd and McBain, 1941;Ben-Yaakov, 1972; Lasaga, 1979; Felmy and Weare,1991; Van Cappellen and Gaillard, 1996; Liu, 2007;

http://dx.doi.org/10.1016/j.gca.2014.06.020

0016-7037/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +1 49707129171.E-mail address: massimo.rolle@uni-tuebingen.de (M. Rolle).

Appelo and Wersin, 2007; Steefel and Maher, 2009). Insuch systems, the diffusive flux of a species is not just func-tion of its properties and concentration gradient but it islinked to the presence and to the concentration gradientsof other charged species in the system (Lasaga, 1979). Thus,a simple expression of Fick’s law does not allow providing arigorous and accurate description. Hence, a multicompo-nent formulation is required to model diffusion (e.g., Ben-Yaakov, 1972; Lasaga, 1979; Felmy and Weare, 1991;Boudreau et al., 2004; Liu et al., 2004; Liu, 2007; Appeloand Wersin, 2007; Li et al., 2008; Steefel and Maher,2009). In several earlier studies, multicomponent diffusion

M. Muniruzzaman et al. / Geochimica et Cosmochimica Acta 141 (2014) 656–669 657

models, based on local charge balance approaches, havebeen developed to interpret conservative and reactive trans-port of charged species in both laboratory (e.g., Vinogradand McBain, 1941; Ben-Yaakov, 1972; Felmy and Weare,1991; Liu et al., 2011) and field conditions (e.g.,Giambalvo et al., 2002; Appelo and Wersin, 2007; Appeloet al., 2008, 2010).

In flow-through systems, the exchange of mass betweendifferent streamlines occurs through diffusion and local dis-persion. The quantification of such processes is of criticalimportance to accurately describe conservative and reactivesolute transport in the subsurface (e.g., Kitanidis, 1994;Wang and Van Cappellen, 1996; Boudreau, 1997;Thullner et al., 2005; Zhang et al., 2010; Haberer et al.,2011; Molins et al., 2012). In particular, the importanceof diffusion in saturated porous media and its role as con-trolling transport mechanism not only at the small poreand Darcy scales but also at larger field scales is increas-ingly recognized in studies of subsurface solute transport(e.g., Carrera et al., 1998; LaBolle and Fogg, 2001;Chiogna et al., 2011; Hadley and Newell, 2014; Rolleet al., 2013b). Focusing on multicomponent transport ofcharged species, in a recent contribution (Rolle et al.,2013a) we proved the existence and relevance of Coulombiceffects in homogeneous saturated porous media underadvection-dominated flow regimes. In that work we per-formed quasi 2-D steady-state flow-through experiments,at high groundwater flow velocities (1.5 and 6 m/day) inhomogeneously packed flow-through systems, along with1-D transient numerical modeling. We showed that electro-static cross-coupling of dispersive-fluxes of ionic species inporous media substantially affects the lateral displacementof charged species not only under purely diffusive but alsounder flow-through conditions.

In this study, we extend the investigation of multicom-ponent ionic transport of electrolytes to heterogeneousporous formations. We perform flow-through bench-scalelaboratory experiments in advection-dominated flowregimes and under steady-state flow and transport condi-tions. We work in two different heterogeneous saturatedporous media containing well-defined coarse and finematerial inclusions, respectively. The aim of the study isto investigate transport of charged species and to identifyCoulombic cross-coupling effects among the different ionsin solution and their impact on lateral spreading and mix-ing in spatially variable flow fields. To quantitatively inter-pret the experimental results we present a two-dimensionalnumerical model based on a multicomponent formulationof ionic transport and on the cross-coupling of dispersivefluxes.

2. THEORETICAL BACKGROUND

Multicomponent diffusive fluxes of ionic species in dilutesolutions can be described by the Nernst-Plank equation(Bard and Faulkner, 2001; Cussler, 2009), which in absenceof strong gradients of ionic strength (Lasaga, 1979) readsas:

J i ¼ �DirCi � DiziFRT

CirU i ¼ 1; 2; 3; . . . ;N ð1Þ

where Di is the self-diffusion coefficient, Ci is the concentra-tion of ionic species i, zi is the charge, F is the Faraday’sconstant, R is the ideal gas constant, T is the temperature,U is the electrostatic potential, and N is the number of thespecies. The multicomponent formulation is derived usingelectroneutrality, which is associated with two distinctphysical constraints (Boudreau et al., 2004): (i) vanishingcharge in every point of the solution (i.e.,

PNi¼1ziCi ¼ 0)

and no electrical current (i.e.,PN

i¼1ziJ i ¼ 0). Substitutingthe expression for the diffusive flux into the solute massbalance equation yields the governing equation for multi-component diffusion problems (e.g., Ben-Yaakov, 1972;Lasaga, 1979; van Cappellen and Gaillard, 1996;Boudreau et al., 2004; Liu et al., 2004, 2011; Appelo andWersin, 2007). Under flow-through conditions, the advec-tive component should also be considered and mass transferbetween different streamlines occurs through local disper-sion. Therefore, in multicomponent flow-through problems,the diffusion coefficient of a charged species (Di) needs to bereplaced by its hydrodynamic dispersion coefficient. Thelatter depends on the interaction between the basic trans-port mechanisms of advection and diffusion and it is com-monly parameterized as the additive contribution of avelocity-independent pore diffusion term and a mechanicaldispersion term, accounting for the variability of the veloc-ity in the pore space.

Following an analogous procedure as in the case ofpurely diffusive problems (e.g., Boudreau et al., 2004), weuse the zero charge constraint and we substitute thedispersive flux into the classical advection–dispersionequation. This yields the governing transport equation ofcharged species for multicomponent flow-throughproblems:

@Ci

@tþ v � rCi �r �

XN

j¼1

DijrCj

!¼ 0 ð2Þ

in which t is time, v is the velocity vector, and Dij is the dis-persion tensor. The entries of Dij in a two-dimensionaldomain, considering a reference system oriented along theprincipal directions, can be expressed as:

Dij ¼DL

ij 0

0 DTij

" #

DLij ¼ dijDL

i �zizjDL

i DLj CiPN

k¼1ðz2kDL

k CkÞ

DTij ¼ dijDT

i �zizjDT

i DTj CiPN

k¼1 z2kDT

k Ck

� �ð3Þ

where DLij and DT

ij are the matrices of longitudinal andtransverse cross-coupled dispersion coefficients, respec-tively, with DL

i being the longitudinal component and DTi

being the transverse component of the dispersion coefficientfor species i in its liberated state (i.e., “self-dispersion”). dij

is the Kronecker delta which has a value of 1 (if i = j) or 0(if i – j).

658 M. Muniruzzaman et al. / Geochimica et Cosmochimica Acta 141 (2014) 656–669

3. TWO-DIMENSIONAL MULTICOMPONENT

TRANSPORT MODEL

The governing flow equations in a two-dimensionaldomain under steady-steady conditions can be describedas (Cirpka et al., 1999a):

r � ðKrhÞ ¼ 0

r � 1

Krw

� �¼ 0

ð4Þ

in which, h and w are the hydraulic head and the stream-function, and K is the tensor of hydraulic conductivity.We solve Eq. (4) numerically by bilinear finite elementson a rectangular coordinate system. For the hydraulic headproblem, constant head boundaries (Dirichlet) are appliedboth at the inlet and at the outlet, and no-flow boundary(Neumann) conditions are considered at the top and bot-tom of the 2-D domain. The steady-state stream-functionequation is solved applying fixed values at the top and bot-tom boundaries. The difference between these values equalsthe volumetric flux through the domain.

In order to solve the multicomponent transport problemfor different ionic species, we construct streamline-orientedgrids based on the results of the flow simulations and solvethe advective and dispersive terms with the finite volumescheme described by Cirpka et al. (1999b), which minimizesartificial mixing. We restrict our analysis to steady-statetransport problems for which the governing multicompo-nent equations read as:

v � rCi �r �XN

j¼1

DijrCj

!¼ 0 ð5Þ

This system of equations is solved on the generated stream-line-oriented grid using the direct solver UMFPACKimplemented in MATLAB (Davis and Duff, 1997). We con-sider a constant concentration boundary at the inflow andzero dispersive-flux conditions at the remaining boundaries.The simulation domain is 100 cm � 12 cm. It represents thesaturated zone of our 2-D experimental setup and was dis-cretized into 100 � 240 cells (Dx = 1 cm, Dz = 0.5 mm).

Table 1Algorithm for the solution of the steady-state multicomponenttransport problem by Picard iteration.

whilenormðCjþ1

i � Cji Þ > e; i = 1, 2, 3,. . ., N

Step 1:

DL;jij ¼ dijDL

i �zizjDL

i DLj Cj

iPN

k¼1ðz2

k DLk Cj

k Þ

DT ;jij ¼ dijDT

i �zizjDT

i DTj Cj

iPN

k¼1ðz2

k DTk Cj

k Þ

Step 2: Calculate Mmobji

Step 3: Solve for the concentration:

Cji ¼ ðMmobj

i Þ�1bi

Step 4: Next iteration: j = j + 1end

The electrostatic coupling between the dispersive fluxesof different ionic species, introduces non-linearities in themulticomponent transport equations. Thus, we implementan iterative scheme to linearize the set of non-linear trans-port equations within a Picard loop, applying the directmatrix solver in each iteration. The computational stepsare summarized in Table 1. The iteration starts with aninitial guess neglecting the cross-coupling term (Eq. (3))between different species and considering the transportidentical to uncharged compounds (i.e., off-diagonal ele-ments of Dij as zero). In step 1, the matrix of the cross-cou-pling dispersion coefficients is determined for every cell ofthe heterogeneous domain, discretized on the generatedstreamline-oriented grid. Successively (step 2), we calculatethe mobility matrix (Mmobi) describing the divergence ofadvective–dispersive fluxes for each compound, using thefinite volume method (Cirpka et al., 1999b). The parame-ters determined in step 1 and step 2 are used to calculatethe concentration vector for a charged species i in step 3,where bi denotes the right-hand side vector resulting fromthe inlet boundary conditions. The iterative process contin-ues until the norm of the differences between the concentra-tion vectors in two consecutive iterations converges to auser-defined threshold value, e (=1 � 10�15 in oursimulations).

The hydrodynamic dispersion coefficients for each ionwere parameterized following the non-linear compound-specific relationship developed to interpret the multitracerexperiments of Chiogna et al. (2010) and Rolle et al.(2012) and inspired by the earlier statistical model of Bearand Bachmat (1967):

Di ¼ DPi þ Daq

iPe2

Peþ 2þ 4d2

� �b

ð6Þ

in which DPi is the pore diffusion coefficient, defined by the

ratio of aqueous diffusion coefficient ðDaqi ) and the tortuos-

ity of the porous medium and approximated as DPi � hDaq

i ,where h is the porosity. Pe is the grain Peclet number:Pe ¼ vd=Daq

i , where v is the seepage velocity and d is thegrain size diameter of the porous material. d expressesthe ratio between the length of a pore channel and itshydraulic radius, and b is an empirical exponent takinginto account the incomplete mixing at the pore scale. Val-ues of d = 6.2 and b = 0.5 were determined for the trans-verse dispersion coefficient with an excellent agreementbetween laboratory experiments and numerical pore-scalesimulations (Rolle et al., 2012; Hochstetler et al., 2013).As done in a recent study (Rolle et al., 2013b), we usethe same parameterization for the local longitudinal dis-persion coefficient (with same d and b = 0.89, derivedfrom transient pore-scale simulations) even though forsteady-state transport problems, as considered in thisstudy, DL

i is not a sensitive parameter.In heterogeneous porous media the average grain size

diameter, d, used to calculate the mechanical dispersionterm is spatially variable and directly linked to the hydrau-lic conductivity at a given location. As proposed in previousstudies on transport in heterogeneous formations (e.g.,Chiogna et al., 2011; Eckert et al., 2012), we express the

M. Muniruzzaman et al. / Geochimica et Cosmochimica Acta 141 (2014) 656–669 659

relationship between the average grain size and the hydrau-lic conductivity using the approximation of Hazen (1892):

d � 0:01ffiffiffiffiKp

ð7Þ

This approach results in an accurate representation of spa-tially variable local dispersion coefficients that is required todescribe solute transport in heterogeneous flow fields. Asoutlined above (Eq. (3) and Table 1), for multicomponenttransport problems the local dispersion coefficients are usedto calculate the cross-coupled dispersive fluxes.

In order to test the validity of our 2-D multicomponenttransport code, we compared with 1-D transient multicom-ponent simulations using PHREEQC (Parkhurst andAppelo, 1999; Appelo and Wersin, 2007). Such comparisonis possible in a homogeneous medium since the transportequation in such a 2-D domain under steady-state condi-tions and in absence of significant longitudinal dispersivefluxes is mathematically equivalent to a 1-D transient diffu-sive/dispersive problem (e.g., Maier and Grathwohl, 2006;Van Breukelen and Rolle, 2012). The simulations were per-formed for two different scenarios at a flow velocity of1.5 m/day (Rolle et al., 2013a): transport of (i) NaCl (1:1electrolyte, continuously injected through the two centralinlet ports of the setup) in pure ambient water, and (ii)MgCl2 (1:2 electrolyte, continuously injected through thetwo central inlet ports of the setup) in an ambient solutionof NaBr (1:1 electrolyte). In both scenarios we found an

Fig. 1. Multicomponent ionic fluxes in the transverse direction at the endd); Electrochemical migration fluxes (e, f), for transport of NaCl in pureand f). The lines represent the values calculated with the 2-D transport cmulticomponent simulations using PHREEQC.

excellent agreement between the concentration profilesobtained with the 2-D steady-state solution and with the1-D transient multicomponent PHREEQC solution. Wealso compared the different components of the transversefluxes for the ionic species at the outlet of the flow-throughdomain as illustrated in Fig. 1. According to Eq. (1), thetotal flux component in the transverse direction can bedecomposed into the additive contribution of a purely dis-persive flux ðJ dis

i ) and of an electrochemical migration fluxðJ mig

i ):

J i ¼ Jdisi þ Jmig

i ð8Þ

Fig. 1a and Fig. 1b represent the total fluxes, Ji for thetransport of NaCl in pure water, and for the transport ofMgCl2 in an ambient/background solution of buffer elec-trolyte (NaBr), respectively. The purely dispersive compo-nents ðJdis

i ) are shown in Fig. 1c and 1d, whereas thecontribution due to the electrical gradients ðJmig

i ) are pre-sented in Fig. 1e and f.

The code was also validated with experimental data inhomogeneous porous media (Rolle et al., 2013a). The sim-ulation results showed a good agreement with the measureddata. The data could also be reproduced with the multidi-mensional transport codes MIN3P (Mayer et al., 2002)and CrunchFlow (Steefel, 2009) which also have the capa-bility of solving multicomponent ionic transport (resultsnot shown). The validated 2-D code described above was

of the flow-through domain: Total fluxes (a, b); Dispersive fluxes (c,water (a, c, and e), and transport of MgCl2 in NaBr solution (b, d,ode developed in this study and the symbols are the results of 1-D

660 M. Muniruzzaman et al. / Geochimica et Cosmochimica Acta 141 (2014) 656–669

used to interpret the results of the experiments in heteroge-neous porous media performed in this study and illustratedin the following section, for which a 1-D model representa-tion would not be adequate to capture the main features ofmulticomponent ionic transport in the spatially variableflow fields.

4. EXPERIMENTAL SETUP

The experiments were performed in the same flow-through setup used in our previous study (Rolle et al.,2013a). Therefore, here we only briefly summarize the mainfeatures of the experimental system. The setup consists of aquasi 2-D flow-through chamber with inner dimensions of100 cm length, 19 cm height and 1 cm width (Fig. 2a andb). The chamber was equipped with 24 equally spaced(5 mm distance) ports both at the inlet and at the outlet,allowing high-resolution sampling of the injected tracers.All ports were made of Alltech pierced rubber septa andthe injection needles were directly connected to FluranHCA pump tubings (Ismatec, Glattbrugg, Switzerland)with inner diameter of 0.64 mm. In order to establishsteady-state flow conditions, two high-precision peristalticpumps (Ismatec IPC-N24, Ismatec, Glattbrugg, Switzer-land) were used at the inlet and outlet of the flow-throughchamber. The experiments were carried out in a tempera-ture-controlled room at 20 �C.

The tank was filled with glass beads by following a wet-packing procedure, in which the water level was alwayskept above the upper limit of the porous medium to avoidair entrapment within the water-saturated domain (Haberer

Fig. 2. Experimental setups with steady-state color tracer (New Coccine)d); velocity distribution (e, f); and calculated flux-related dilution indexWorch, 1993). The flow direction is from left to right.

et al., 2012). The experiments were carried out in twodistinct heterogeneous porous media. In the first setup(Fig. 2a), denoted as HET A, a low-permeability inclusionmade of fine glass beads (0.25–0.30 mm, Sartorius AG,Gottingen, Germany) was embedded into a morepermeable matrix with glass beads of 1.00–1.50 mm (Sarto-rius AG, Gottingen, Germany). The length and thickness ofthe inclusion were 30 and 4 cm respectively, and it wasplaced directly in line with the center of plume’s injection.The distance of the fine material zone from the inlet was35 cm (Fig. 2a). In the second experimental setup (HET

B, Fig. 2b), two high-conductivity inclusions, made ofcoarse glass beads (2.00–2.50 mm, Sartorius AG, Gottin-gen, Germany), were embedded into the same matrix (glassbeads of 1.00–1.50 mm). The coarse-grained zones wereplaced 1.5 cm away from the center-line of plume’s injectionand their thickness was 1 cm. The length of the inclusionswas 20 cm and the distance from the inlet was 70 cm(Fig. 2b).

Fig. 2c and d depict the hydraulic conductivity distribu-tion and the flow-lines computed with the model. The var-iation of local velocities is shown in Fig. 2e and f for HET A

and HET B, respectively. Fig. 2g and h show the mixingbehavior of the steady-state plumes in the two heteroge-neous flow-through systems using the flux-related dilutionindex, EQ(x) (Rolle et al., 2009; Chiogna et al., 2012) asmetric of mixing:

EQðxÞ ¼ exp �Z

XpQðxÞ lnðpQðxÞÞqxðxÞdX

� �ð9Þ

plumes (a, b); computed streamlines and hydraulic conductivities (c,of the color tracer (with Daq = 3.43 � 10�10 m2/s estimated after

M. Muniruzzaman et al. / Geochimica et Cosmochimica Acta 141 (2014) 656–669 661

where qx = vh is the specific discharge, X is the cross-sec-tional area, h is the porosity, and pQ is the flux-related prob-ability density function:

pQðxÞ ¼CiðxÞR

X CiðxÞqxðxÞdXð10Þ

EQ(x) quantifies the distribution of the mass flux of adissolved species over the volumetric water flux at a givenlongitudinal location x and refers to the color tracer exper-iments performed to visualize the flow and transport pat-terns in the flow-through systems. After the ionicexperiments, a color tracer (New Coccine solution, 90 mg/L; CAS: 2611-82-7) was injected through the two centralinlet ports and a steady-state plume was established inorder to observe the shape of the plume in the heteroge-neous setups (Fig. 2a and b). The trend of EQ(x) inFig. 2g and h shows a change of the slope at the locationsof the inclusions, indicating the variation of mixing proper-ties due to flow focusing and de-focusing in these zones withdifferent grain sizes and permeability.

The transport experiments were performed, by injectingtracer electrolyte solutions through the central injectionports (port 11 and 12, at 5.5 cm and 6 cm from the bottom,respectively) and ambient solutions through the remainingports, after a steady-state flow condition was achieved inthe flow-through systems. Strong 1:1 (NaCl) and 1:2(MgCl2) electrolytes were used as tracer solutions, andMilli-Q water (Millipore, MA, USA) or a 1:1 electrolyte(NaBr) were used as ambient solutions. When the plumesreached steady-state conditions samples were collected atall 24 outlet ports and the ion concentrations were mea-sured by ion-chromatography (Dionex Dx-120, Fisher Sci-entific, Schwerte, Germany). The electrolyte concentrationsused in the different experiments were in the range of0.4 mM to 1.70 mM. The flow-through experiments wereperformed at an average linear horizontal flow velocity of

Table 2Summary of flow and transport parameters in the two heterogeneous ex

Experimental settings and hydraulic properties

Tank inner dimensions (L � H �W) [cm]Number of inlet/outlet ports usedPort spacing [mm]Fine grain diameter, low-K inclusion in HET A [mm]Intermediate grain diameter, porous matrix [mm]Coarse grain diameter, high-K inclusions in HET B [mm]Hydraulic conductivity, low-K inclusion in HET A [m/s]Hydraulic conductivity, porous matrix [m/s]Hydraulic conductivity, high-K inclusions in HET B [m/s]Average horizontal flow velocity a [m/day]Average porosity [�]

Diffusion coefficients

Na+

Mg2+

Cl�

Br�

a the first value is for HET A and the second one is for HET B.b values from Lasaga (1998) at 18 �C, and corrected for temperature a

1.5 m/day for HET A and 1.0 m/day for HET B. Theparameters summarizing the flow and transport propertiesof the experimental setups are reported in Table 2.

5. RESULTS AND DISCUSSION

For each heterogeneous setup, HET A and HET B, twodifferent flow-through experiments were performed: (i)transport of NaCl (1:1 electrolyte) in Milli-Q water, and(ii) transport of MgCl2 (1:2 electrolyte) in NaBr (1:1electrolyte) buffer electrolyte solution. The results obtainedfrom these experiments are interpreted with the 2-D multi-component numerical model presented in Section 3.

5.1. Flow-through experiments with a low-permeability

inclusion (HET A)

In this flow-through setup (HET A, Fig. 2a) we used awell-defined low-permeability zone (K = 6.14 � 10�4 m/s)embedded into a higher permeability matrix(K = 1.27 � 10�2 m/s); this leads to the divergence of thestreamlines around the low-permeability inclusion(Fig. 2c). In the first experiment, a 1:1 electrolyte solution(0.80 mM NaCl) and Milli-Q water were continuouslyinjected through the two central and the remaining inletports, respectively. The steady-state concentration profilesof the cation and of the anion were measured at the outletof the flow-through domain. Fig. 3 depicts the concentra-tion distribution and the vertical profiles of Na+ and Cl�

at three different cross sections (20, 50, and 80 cm fromthe inlet) simulated with the 2-D multicomponent model.

The plumes of both cation and anion show more spread-ing near the low-permeability inclusion (Fig. 3a and b).This behavior is, in fact, induced from the bypassing of flowlines around the low-conductivity zone (Fig. 2c), due to thepermeability contrast (K ratio = 0.05) between the two

perimental setups (HET A and HET B).

Value

100 � 19 � 123/2450.25–0.301.00–1.502.00–2.506.14 � 10�4

1.27 � 10�2

4.12 � 10�2

1.5; 1.00.41

Daq [m2/s]b

1.20 � 10�9

0.63 � 10�9

1.81 � 10�9

1.86 � 10�9

nd viscosity changes of the experimental conditions at 20 �C.

Fig. 3. 2-D concentration distributions (a and b), and cross-sectional profiles at 20 cm (c), 50 cm (d), and 80 cm (e) from the inlet of the flow-through domain (HET A), computed for the case of NaCl transport in Milli-Q water.

662 M. Muniruzzaman et al. / Geochimica et Cosmochimica Acta 141 (2014) 656–669

different grain sizes. Downgradient, the streamlines becomecloser and the computed plumes exhibit smaller spreadingcompared to the central region. These characteristics arealso illustrated by the concentration profiles at differentcross sections. At 20 cm from the inlet, the profiles have aGaussian shape (Fig. 3c) since the plume is travelling in ahomogeneous medium whereas the ionic profiles are dis-torted and show more spreading in the center of the setupdue to the presence of the low-K inclusion (Fig. 3d). Theprofiles of sodium and chloride have a more regular shapeat the cross-section after the inclusion (x = 80 cm, Fig. 3e).The comparison between the measured and the computedprofiles of the ions at the outlet of the flow-through systemare shown in Fig. 4.

Fig. 4. Transverse concentration profiles at the outlet of the flow-through system (HET A) for transport of NaCl in Milli-Q water:measured cation and anion concentrations (symbols) and modelingresults (lines).

The cation and anion appear to travel together in spiteof having significantly different self-diffusion (and thusself-dispersion) coefficients ðDaq

Naþ ¼ 1:20� 10�9 m2/s andDaq

Cl� ¼ 1:81� 10�9 m2/s). Their concentration profiles atthe outlet overlap and a very good agreement existsbetween the measured concentrations and the outcomesof the purely forward multicomponent simulations (i.e.,without any fitting parameters). The observed behavior isdue to the existence of Coulombic interactions, which leadto the cross-coupling of the transverse dispersive fluxes,between the positively and the negatively charged species.These Coulombic effects for Na+ and Cl� can be effectivelyvisualized by mapping the different contributions of trans-verse dispersive fluxes (Fig. 5).

Fig. 5a and b show that the total transverse flux (Ji) hasexactly the same distribution for both Na+ and Cl�. How-ever, these identical total fluxes have different contributionsof the dispersive and the electrochemical migration terms(Eq. (8)). In fact, chloride has larger values of the dispersiveflux component (J dis

i ) compared to Na+ (Fig. 5c and d) sinceit has higher diffusion and transverse dispersion coefficients.Fig. 5e and f map the electrochemical migration fluxesðJ mig

i ) for both the cation and the anion. These electrochem-ical coupling components have the same distribution butopposite signs: positive for Na+ and negative for Cl�. Thus,a positive contribution for Na+ (Fig. 5e) and a negativecontribution for Cl� (Fig. 5f) facilitate, respectively, theenhancement of the transverse flux for the slower species,Na+, and the reduction of the transverse flux for the fasterion, Cl�. Hence, the electrostatic interactions couple thetwo oppositely charged species and lead to an equal trans-verse displacement, which allows attaining identical verticalconcentration profiles for both ions at the outlet (Fig. 4).

Fig. 5. Maps of multicomponent ionic transverse fluxes: Total fluxes (a, b); Dispersive fluxes (c, d); Electrochemical migration fluxes (e, f), fortransport of NaCl in Milli-Q water (HET A). Note that the direction from the core to the fringe of the plume is considered positive for thecalculated fluxes.

M. Muniruzzaman et al. / Geochimica et Cosmochimica Acta 141 (2014) 656–669 663

In the second experiment, we investigated the transportof a 1:2 electrolyte (0.44 mM MgCl2) through an ambientbuffer electrolyte solution (1.70 mM NaBr) in the samesetup, HET A. Fig. 6 illustrates the results obtained fromthe multicomponent simulations. Unlike the previousexperiment (i.e., transport of NaCl in Milli-Q water), the2-D concentration distribution of the cation and the anioninjected through the central inlet ports show a dissimilarpattern (Fig. 6a and b). The concentration maps show thatin the low-permeability inclusion, magnesium has relativelyhigher concentration than chloride, as can be observedfrom the slight color differences in Fig 6a and b.

This pattern is consistent with the calculatedconcentration profiles at different cross sections of the

Fig. 6. 2-D concentration distributions of Mg2+ (a), Cl� (b), Na+ (c), a80 cm (g) from the inlet of the flow-through domain (HET A), compute

flow-through chamber, for which the peak concentrationof the cation, Mg2+, is higher than the one of the anion,Cl� (Fig. 6e–g). Interesting is also the behavior of thebackground electrolyte (NaBr) solution. In fact, Na+

shows comparatively smaller concentrations than Br� inthe core of the plume (Fig. 6c and d), although they wereinjected at a uniform concentration throughout the inletboundary.

The measured concentration profiles of the ionic speciesat the outlet of the flow-through chamber and the results ofthe multicomponent simulations are reported in Fig. 7.From Fig. 7a, it is evident that Mg2+ and Cl� have clearlydistinct profiles, with Mg2+ having a higher peak andsmaller spreading than Cl�.

nd Br� (d); and cross-sectional profiles at 20 cm (e), 50 cm (f), andd for the case of MgCl2 transport in NaBr solution.

Fig. 7. Transverse concentration profiles at the end of the flow-through system (HET A) for transport of MgCl2 in NaBr solution: measuredcation and anion concentrations (symbols) and modeling results (lines).

Fig. 8. Transverse electrochemical migration fluxes of Mg2+ (a), Cl� (b), Na+ (c), and Br� (d) during transport of MgCl2 in NaBr solution inHET A setup. The direction from the core to the fringe of the plume is considered positive for the calculated fluxes.

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The behavior observed at the outlet depends on theCoulombic interactions between the charged species. Sucheffects can be rationalized and effectively visualized bymapping the fluxes of the different ions. For the sake ofbrevity, in Fig. 8 we present only the electrochemical migra-tion component (J mig

i ) of the total fluxes of the cations andanions in the tracer and buffer electrolyte solutions.

The outward, positive migration flux component of Na+

(Fig. 8c) and inward, negative component of Br� displacethese ions, respectively, towards the fringe (Na+) andtowards the centerline (Br�) of the plume. This induces arelative increase of Br� concentration in the plume’s coreand a decrease in the plume’s fringe and an opposite effectfor Na+. Due to the higher concentration of the ambientbuffer electrolyte solution (approximately fourfold), theelectrochemical gradients of Mg2+ and Cl� injected in thecentral ports are almost suppressed by counter balancingthe charge differences with the vertical displacement ofbackground ions (Na+ and Br�). The electrochemicalmigration flux components of Mg2+ and Cl� (Fig. 8a andb) are significantly smaller than those of the backgroundionic species (Na+ and Br�; Fig. 8c and d). Therefore, thedisplacement of Mg2+ and Cl� is dominated by the disper-sive flux components. This explains the differences in themeasured concentration profiles of these ions whosedisplacement approaches the one determined by their self-diffusion and self-dispersion coefficients.

5.2. Flow-through experiments with high-permeability

inclusions (HET B)

The experiments mentioned in the previous section werealso carried out in the second heterogeneous setup, HET B

(Fig. 2b), where two high-permeability inclusions(K = 4.12 � 10�2 m/s) were embedded into a matrix oflower hydraulic conductivity (K = 1.27 � 10�2 m/s). Fig. 9reports the modeling results for the transport of 1:1 electro-lyte (NaCl) in Milli-Q water.

Upstream of the high-conductivity zones, the plume hasthe regular shape characteristic of transport in homoge-neous porous media. As the ionic plumes approach theinclusions, they appear to spread over a larger area(Fig. 9a and b). Due to the permeability contrast (Kratio = 3.24) the flow lines are focused within the high-permeability inclusions (Fig. 2d). Consequently, the smalldistance among the streamlines in the high-conductivityzones facilitates dispersive mass transfer of solute massand enhances the transverse mixing in the fringe area ofthe plume. The situation is opposite along the centerlineof the plume: in this zone the distance among the stream-lines is larger (Fig. 2d) and this leads to a decrease of trans-verse mass exchange. These effects are also reflected in thecalculated transverse profiles at different cross sections(Fig. 9c, d, and e). The profiles at both 20 and 50 cm fromthe inlet show a Gaussian shape whereas the computed

Fig. 9. 2-D concentration distributions (a and b), and cross-sectional profiles at 20 cm (c), 50 cm (d), and 80 cm (e) from the inlet of the flow-through domain (HET B), computed for the case of NaCl transport in Milli-Q water.

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ionic profiles directly at the high-K inclusion (Fig. 9e) showhigher spreading when compared to the upstream cross sec-tions. Fig. 10 illustrates the measured outlet profiles of thecation and the anion. Both Na+ and Cl� have a distortedand clearly non-Gaussian shape with the higher peakconcentrations presenting lower transverse spreading andthe lateral lower concentrations significantly more spreaddue to enhanced mixing in the fringe areas. Also in thissetup the measurements of the outlet ports clearly showthe electrochemical coupling between the two ions, whichtravel together. Similarly to the findings for HET A, alsofor this setup the forward multicomponent simulationsshow a satisfactory agreement with the experimental data.

The different components of the transverse dispersivefluxes calculated with the 2-D code are shown in Fig. 11.

Fig. 10. Transverse concentration profiles at the end of the flow-through system (HET B) for transport of NaCl in Milli-Q water:measured cation and anion concentrations (symbols) and modelingresults (lines).

The total and dispersive fluxes of both Na+ and Cl� areenhanced (higher color intensity) within the coarse-materialinclusions due to flow focusing in these zones.

Similarly to what observed in HET A, the positive compo-nent for Na+ (Fig. 11e) and the negative component for Cl�

(Fig. 11f) of the electrochemical migration flux show themagnitude of Coulombic interactions between the two ions.Such effects cause the overlapping of measured outletconcentration profiles (Fig. 10) as well as identical distribu-tions of the total transverse dispersive fluxes (Fig. 11a and b).

Fig. 12 and Fig. 13 show the modeling and experimentaloutcomes for the transport of MgCl2 in an ambient bufferelectrolyte solution (NaBr) in the heterogeneous setupHET B. The boundary conditions and the initial concentra-tions of the electrolyte solutions (0.44 mM MgCl2 and1.70 mM NaBr) where the same used in the analogousexperiment performed in HET A.

Also in this heterogeneous scenario we observed a clearseparation between the profiles of Mg2+ and Cl� in thepeak area of the plume inside the porous medium(Fig. 12e–g) and at the outlet of the flow-through system(Fig. 13a), as well as the appearance of concentrationgradients for the ions in the buffer electrolyte solution(Fig. 13b).

The 2-D distribution of transverse electrochemicalmigration fluxes (Fig. 14) of the different ions show similarpatterns as the ones obtained in the HET A setup. Thehigher values of the electrochemical flux components forthe background buffer ions show that Na+ and Br� providea substantial contribution in maintaining the electroneu-trality condition. Consequently, the ions in the tracersolutions (Mg2+ and Cl�) are loosely coupled and cantravel with diffusive/dispersive properties close to theirself-diffusion and self-dispersion coefficients.

Fig. 12. 2-D concentration distributions of Mg2+ (a), Cl� (b), Na+ (c), and Br� (d); and cross-sectional profiles at 20 cm (e), 50 cm (f), and80 cm (g) from the inlet of the flow-through domain (HET B), computed for the case of MgCl2 transport in NaBr solution.

Fig. 13. Transverse concentration profiles at the outlet of the flow-through system (HET B) for transport of MgCl2 in NaBr solution:measured cation and anion concentrations (symbols) and modeling results (lines).

Fig. 11. Maps of multicomponent ionic transverse fluxes: Total fluxes (a, b); Dispersive fluxes (c, d); Electrochemical migration fluxes (e, f),for transport of NaCl in Milli-Q water (HET B). Note that the direction from the core to the fringe of the plume is considered positive for thecalculated fluxes.

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Fig. 14. Transverse electrochemical migration fluxes of Mg2+ (a), Cl� (b), Na+ (c), and Br� (d) in the transverse direction for transport ofMgCl2 in NaBr solution in the HET B setup. Note that the direction from the core to the fringe of the plume is considered positive for thecalculated fluxes.

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6. SUMMARY AND CONCLUSIONS

In this paper, we presented a detailed investigation ofCoulombic effects during multicomponent ionic transportin spatially variable flow fields. The experiments wereperformed in two different heterogeneous porous media:one containing a low-conductivity inclusion in a morepermeable matrix (HET A), and the other containing twohigh-conductivity inclusions in a less permeable porousmatrix (HET B). We worked in advection-dominated flowregimes with an average flow velocity of 1.5 m/day forHET A and of 1.0 m/day for HET B.

A two-dimensional multicomponent numerical modelwas developed to capture the Coulombic interactionsamong charged species in the two-dimensional heteroge-neous systems under steady-state flow and transport condi-tions. The model was validated in homogeneous porousmedia against 1-D PHREEQC simulations and previousexperimental results (Rolle et al., 2013a). Successively, the2-D model was applied to quantitatively interpret theresults of the flow-through experiments in the heteroge-neous porous media performed in this study. For the differ-ent experiments in both setups (HET A and HET B), a verygood agreement was observed between the experimentalresults and the outcomes of purely forward numerical sim-ulations. This substantiates the capability of the proposedmodeling approach, based on a multicomponent ionic for-mulation and on the accurate and spatially variable descrip-tion of local dispersion, to precisely capture the physicalprocesses governing the transport and the interactions ofthe charged species in the heterogeneous flow-throughsystems.

We have restricted our investigation to multicomponentsteady-state conservative transport of charged species indilute solutions. Under such conditions, our experimentaland modeling findings show that microscopic Coulombicinteractions occurring at the molecular scale affect themacro-scale transport and the observations at thelaboratory Darcy scale. Such effects most commonly relatedto diffusion of charged species do not vanish under theadvection-dominated flow regimes and in the heteroge-neous porous media tested in this work. To improve the

understanding of electrochemical migration of charged spe-cies in geologic formations under flow-through conditionsfuture experimental and modeling studies should addressthe next level of upscaling from the laboratory to the fieldscale under natural and engineered conditions (e.g.,Saichek and Reddy, 2005; Wu et al., 2012). In such largerscale setups, transport occurs in complex flow fields inwhich heterogeneity, anisotropy and flow topology influ-ence mixing and, thus, affect the magnitude and the impactof Coulombic effects on the migration of charged species.Furthermore, additional work is necessary to investigatethe effects of multicomponent ionic dispersion coupled toaccurate description of geochemical reaction rates (e.g.,Maher et al., 2006; Li et al., 2007, 2014; Molins et al.,2012) and to processes such as sorption and ion exchange(e.g., Gvirtzman and Gorelick, 1991; Bjerg et al., 1993;Bjerg and Christensen, 1993; Appelo, 1994; Liu et al.,2011; Holden et al., 2012, 2013).

ACKNOWLEDGMENTS

This work was supported by the Baden-Wurttemberg Stiftungunder the Eliteprogram for Postdocs. We thank Bernice Nischand Annegret Walz for their assistance with the ion-chromatogra-phy measurements, and Olaf A. Cirpka and Dominik Eckert fortheir help in developing the multicomponent streamline-orientedcode. M.R. acknowledges the support of the Marie Curie Interna-tional Outgoing Fellowship (DILREACT project) within the 7thEuropean Community Framework Programme. Finally, we wouldlike to thank the Associate Editor (C. Steefel) and two anonymousreviewers for their constructive comments and suggestions.

REFERENCES

Appelo C. A. J. (1994) Some calculations on multicomponenttransport with cation exchange in aquifers. Ground Water 32,968–975.

Appelo C. A. J. and Wersin P. (2007) Multicomponent diffusionmodeling in clay systems with application to the diffusion oftritium, iodide, and sodium in Opalinus Clay. Environ. Sci.

Technol. 41, 5002–5007.Appelo C. A. J., Vinsot A., Mettler S. and Wechner S. (2008)

Obtaining the porewater composition of a clay rock by

668 M. Muniruzzaman et al. / Geochimica et Cosmochimica Acta 141 (2014) 656–669

modeling the in- and out-diffusion of anions and cations froman in-situ experiment. J. Contam. Hydrol. 101, 67–76.

Appelo C. A. J., Van Loon L. R. and Wersin P. (2010)Multicomponent diffusion of a suite of tracers (HTO, Cl, Br,I, Na, Sr, Cs) in a single sample of Opalinus Clay. Geochim.

Cosmochim. Acta 74, 1201–1219.Bard A. J. and Faulkner L. R. (2001) Electrochemical Methods

Fundamentals and Applications, second ed. Wiley Weinheim,New York.

Bear J. and Bachmat Y. (1967) A generalized theory on hydrody-namic dispersion in porous media. In IASH Symposium on

Artificial Recharge and Management of Aquifers, Haifa, Israel,vol. 72, 7–16.

Ben-Yaakov S. (1972) Diffusion of sea water ions—I. Diffusion ofsea water into a dilute solution. Geochim. Cosmochim. Acta 36,1395–1406.

Bjerg P. L., Ammentorp H. C. and Christensen T. H. (1993) Modelsimulations of a field experiment on cation exchange-affectedmulticomponent solute transport in a sandy aquifer. J. Contam.

Hydrol. 12, 291–311.Bjerg P. L. and Christensen T. H. (1993) A field experiment on

cation exchange-affected multicomponent solute transport in asandy aquifer. J. Contam. Hydrol. 12, 269–290.

Boudreau B. P. (1997) Diagenetic Models and their Implementation:

Modelling Transport and Reactions in Aquatic Sediments.Springer-Verlag, Heidelberg.

Boudreau B. P., Meysman F. J. R. and Middelburg J. J. (2004)Multicomponent ionic diffusion in porewaters: coulombiceffects revisited. Earth Planet. Sci. Lett. 222, 653–666.

Carrera J., Sanchez-Vila X., Benet I., Medina A., Galarza G. andGuimera J. (1998) On matrix diffusion: formulations, solutionmethods and qualitative effects. Hydrogeol. J. 6, 178–190.

Chiogna G., Eberhardt C., Grathwohl P., Cirpka O. A. and RolleM. (2010) Evidence of compound-dependent hydrodynamicand mechanical transverse dispersion by multitracer laboratoryexperiments. Environ. Sci. Technol. 44, 688–693.

Chiogna G., Cirpka O. A., Grathwohl P. and Rolle M. (2011)Relevance of local compound-specific transverse dispersion forconservative and reactive mixing in heterogeneous porousmedia. Water Resour. Res. 47, W06515, 10.1029 /2010WR010270.

Chiogna G., Hochstetler D. L., Bellin A., Kitanidis P. K. and RolleM. (2012) Mixing, entropy and reactive solute transport.Geophys. Res. Lett. 39, L20405. http://dx.doi.org/10.1029/2012GL053295.

Cirpka O. A., Frind E. O. and Helmig R. (1999a) Streamline-oriented grid generation for transport modelling in two-dimensional domains including wells. Adv. Water Resour. 22,697–710. http://dx.doi.org/10.1016/s0309-1708(98)00050-5.

Cirpka O. A., Frind E. O. and Helmig R. (1999b) Numericalmethods for reactive transport on rectangular and streamline-oriented grids. Adv. Water Resour. 22, 711–728, http://dx.doi.org/10.1016/S0309-1708(98)00051-7.

Cussler E. L. (2009) Diffusion: Mass Transfer in Fluid Systems,third ed. Cambridge University Press, New York, USA.

Davis T. and Duff I. (1997) An unsymmetric-pattern multifrontalmethod for sparse LU factorization. SIAM J. Matrix Anal. A

18, 140–158, doi:10.1137/S0895479894246905.Eckert D., Rolle M. and Cirpka O. A. (2012) Numerical simulation

of isotope fractionation in steady-state bioreactive transportcontrolled by transverse mixing. J. Contam. Hydrol. 140–141,95–106.

Felmy A. R. and Weare J. H. (1991) Calculation of multicompo-nent ionic-diffusion from zero to high-concentration: I. Thesystem Na-K-Ca-Mg-Cl-So4-H2O at 25-degrees-C. Geochim.

Cosmochim. Acta 55, 113–131.

Giambalvo E. R., Steefel C. I., Fisher A. T., Rosenberg N. D.and Wheat C. G. (2002) Effect of fluid-sediment reaction onhydrothermal fluxes of major elements, eastern flank of theJuan de Fuca Ridge. Geochim. Cosmochim. Acta 66, 1739–1757.

Gvirtzman H. and Gorelick S. M. (1991) Dispersion and advectionin unsaturated porous media enhanced by anion exclusion.Nature 352, 793–795.

Hadley P. W. and Newell C. (2014) The new potential forunderstanding groundwater contaminant transport. Ground

Water 52, 174–186. http://dx.doi.org/10.1111/gwat.12135.Haberer C. M., Rolle M., Liu S., Cirpka O. A. and Grathwohl P.

(2011) A high-resolution non-invasive approach to quantifyoxygen transport across the capillary fringe and within theunderlying groundwater. J. Contam. Hydrol. 122, 26–39.

Haberer C. M., Rolle M., Cirpka O. A. and Grathwohl P. (2012)Oxygen transfer in a fluctuating capillary fringe. Vadose Zone J.

11. http://dx.doi.org/10.2136/vzj2011.0056.Hazen A. (1892) Some physical properties of sands and gravels

special reference to their use in filtration. Ann. Rep. State Board

Health Mass. 24, 541–556.Hochstetler D. L., Rolle M., Chiogna G., Haberer C. M.,

Grathwohl P. and Kitanidis P. K. (2013) Effects of com-pound-specific transverse mixing on steady-state reactiveplumes: insights from pore-scale simulations and Darcy-scaleexperiments. Adv. Water Resour. 54, 1–13. http://dx.doi.org/10.1016/j.advwatres.2012.12.007.

Holden A. A., Mayer K. U. and Ulrich A. C. (2012) Evaluatingmethods for quantifying cation exchange in mildly calcareoussediments in Northern Alberta. Appl. Geochem. 27, 2511–2523.

Holden A. A., Haque S. E., Mayer K. U. and Ulrich A. C. (2013)Biogeochemical processes controlling the mobility of major ionsand trace metals in aquitard sediments beneath an oil sandtailing pond: laboratory studies and reactive transport model-ing. J. Contam. Hydrol. 151, 55–67.

Kitanidis P. K. (1994) The concept of the dilution index. Water

Resour. Res. 30, 2011–2026.LaBolle E. M. and Fogg G. E. (2001) Role of molecular diffusion in

contaminant migration and recovery in alluvial aquifer system.Transp. Porous Med. 42, 155–179.

Lasaga A. C. (1979) The treatment of multi-component diffusionand ion pairs in diagenetic fluxes. Am. J. Sci. 279, 324–346.

Lasaga A. C. (1998) Kinetic Theory in the Earth Sciences. PrincetonUniversity Press, Princeton, New Jersey.

Li L., Peters C. A. and Celia M. A. (2007) Effects of mineral spatialdistribution on reaction rates in porous media. Water Resour.

Res. 43, W01419. http://dx.doi.org/10.1029/2005WR004848.Li L., Steefel C. I. and Yang L. (2008) Scale dependence of mineral

dissolution kinetics within single pores and fractures. Geochim.

Cosmochim. Acta 72, 360–377.Li L., Salehikhoo F., Brantley S. and Heidari P. (2014) Spatial

zonation limits magnesite dissolution in porous media. Geo-

chim. Cosmochim. Acta 126, 555–573.Liu C., Zachara J. M., Felmy A. and Gorby Y. (2004) An

electrodynamics-based model for ion diffusion in microbialpolysaccharides. Colloids Surf. A and B 38, 55–65.

Liu C. X. (2007) An ion diffusion model in semi-permeable claymaterials. Environ. Sci. Technol. 41, 5403–5409.

Liu C. X., Shang J. and Zachara J. M. (2011) Multispecies diffusionmodels: a study of uranyl species diffusion. Water Resour. Res.

47, W12514. http://dx.doi.org/10.1029/2011WR010575.Maher K., Steefel C. I., DePaolo D. and Viani B. (2006) The

mineral dissolution rate conundrum: insights from reactivetransport modeling of U isotopes and pore fluid chemistry inmarine sediments. Geochim. Cosmochim. Acta 70, 337–363.

M. Muniruzzaman et al. / Geochimica et Cosmochimica Acta 141 (2014) 656–669 669

Maier U. and Grathwohl P. (2006) Numerical experiments andfield results on the size of steady state plumes. J. Contam.

Hydrol. 85, 33–52.Mayer K. U., Frind E. O. and Blowes D. W. (2002) Multicom-

ponent reactive transport modeling in variably saturatedporous media using a generalized formulation for kineticallycontrolled reactions. Water Resour. Res. 38, 1174–1194.

Molins S., Trebotich D., Steefel C. I. and Shen C. (2012) Aninvestigation of pore scale flow on average geochemical reactionrates using different numerical simulation. Water Resour. Res.

48, W03527. http://dx.doi.org/10.1029/2011WR011404.Parkhurst D. L. and Appelo C. A. J. (1999) User’s guide to

PHREEQC – a computer program for speciation, reaction path,

1-D transport, and inverse geochemical calculations. TechnicalReport 99–4259, US Geol. Survey Water Resources Investiga-tion Report, 1999.

Rolle M., Hochstetler D. L., Chiogna G., Kitanidis P. andGrathwohl P. (2012) Experimental investigation and pore-scalemodeling interpretation of compound-specific transverse dis-persion in porous media. Transp. Porous Media 93, 347–362.

Rolle M., Muniruzzaman M., Haberer C. M. and Grathwohl P.(2013a) Coulombic effects in advection-dominated transport ofelectrolytes in porous media: multicomponent ionic dispersion.Geochim. Cosmochim. Acta 120, 195–205.

Rolle M., Chiogna G., Hochstetler D. L. and Kitanidis P. K.(2013b) On the importance of diffusion and compound-specificmixing for groundwater transport: an investigation from poreto field scale. J. Contam. Hydrol. 153, 51–68.

Rolle M., Eberhardt C., Chiogna G., Cirpka O. A. and GrathwohlP. (2009) Enhancement of dilution and transverse reactivemixing in porous media: experiments and model-based inter-pretation. J. Contam. Hydrol. 110, 130–142.

Saichek R. E. and Reddy K. R. (2005) Electrokinetically enhancedremediation of hydrophobic compounds in soils: a review. Crit.

Rev. Environ. Sci. Technol. 35, 115–192.Steefel C. I. (2009) CrunchFlow. Software for Modeling Multicom-

ponent Reactive Flow and Transport. User’s Manual. LawrenceBerkeley National Laboratory, Berkeley, USA.

Steefel C. I. and Maher K. (2009) Fluid-rock interactions: a reactivetransport approach. Rev. Mineral. Geochem. 70, 485–532.

Thullner M., Van Cappellen P. and Regnier P. (2005) Modeling theimpact of microbial activity on redox dynamics in porousmedia. Geochim. Cosmochim. Acta 69, 5005–5019.

Van Breukelen B. M. and Rolle M. (2012) Transverse hydrody-namic dispersion effects on isotope signals in groundwaterchlorinated solvents’ plumes. Environ. Sci. Technol. 46, 7700–7708.

Van Cappellen P. and Gaillard J. -F. (1996) Biogeochemicaldynamics in aquatic sediments. In Reactive Transport in Porous

Media: General Principles and Application to Geochemical

Processes (eds. P. C. Lichtner, C. I. Steefel and E. H.Oelkers).Mineral. Soc. Amer. pp. 335–376.

Vinograd J. R. and McBain J. W. (1941) Diffusion of electrolytesand of the ions in their mixtures. J. Am. Chem. Soc. 63, 2008–2015.

Wang Y. F. and Van Cappellen P. (1996) A multicomponentreactive transport model of early diagenesis: application toredox cycling in coastal marine sediments. Geochim. Cosmo-

chim. Acta 60, 2993–3014.Worch E. (1993) Eine neue Gleichung zur Berechnung von

Diffusionskoeffizienten geloster Stoffe (A new equation for thecalculation of diffusion coefficients for dissolved substances).Vom Wasser 81, 289–297.

Wu M. Z., Reynolds D. A., Fourie A., Prommer H. and ThomasD. G. (2012) Electrokinetic in situ oxidation remediation:assessment of parameter sensitivities and the influence ofaquifer heterogeneity on remediation efficiency. J. Contam.

Hydrol. 136–137, 72–85.Zhang C. Y., Dehoff K., Hess N., Oostrom M., Wietsma T. W.,

Valocchi A. J. and Werth C. J. (2010) Pore-scale study oftransverse mixing induced CaCO3 precipitation and permeabil-ity reduction in a model subsurface sedimentary system.Environ. Sci. Technol. 44, 7833–7838.

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