a modified lagrangian-volumes method to simulate nonlinearly and kinetically sorbing solute...

Journal of Contaminant Hydrology 120–121 (2011) 89–98

Contents lists available at ScienceDirect

Journal of Contaminant Hydrology

j ourna l homepage: www.e lsev ie r.com/ locate / jconhyd

A modified Lagrangian-volumes method to simulate nonlinearly andkinetically sorbing solute transport in heterogeneous porous media

K. Besnard, J.-R. de Dreuzy ⁎, P. Davy, L. AquilinaGéosciences Rennes, UMR CNRS 6118, Campus de Beaulieu, Université de Rennes 1, 35042 Rennes Cedex, France

a r t i c l e i n f o

⁎ Corresponding author.E-mail address: Jean-Raynald.de-Dreuzy@univ-ren

(J.-R. de Dreuzy).

0169-7722/$ – see front matter © 2010 Elsevier B.V.doi:10.1016/j.jconhyd.2010.03.004

a b s t r a c t

Article history:Received 27 November 2009Received in revised form 15 March 2010Accepted 31 March 2010Available online 14 April 2010

Transport in subsurface environments is conditioned by physical and chemical processes ininteraction, with advection and dispersion being the most common physical processes andsorption the most common chemical reaction. Existing numerical approaches become time-consuming in highly-heterogeneous porous media. In this paper, we discuss a new efficientLagrangian method for advection-dominated transport conditions. Modified from the active-walker approach, this method comprises dividing the aqueous phase into elementary volumesmoving with the flow and interacting with the solid phase. Avoiding numerical diffusion, themethod remains efficient whatever the velocity field by adapting the elementary volume transittimes to the local velocity so that mesh cells are crossed in a single numerical time step. Themethod is flexible since a decoupling of the physical and chemical processes at the elementaryvolume scale, i.e. at the lowest scale considered, is achieved. We implement and validate theapproach to the specific case of the nonlinear Freundlich kinetic sorption. The method is relevantas long as the kinetic sorption-induced spreading remains much larger than the dispersion-induced spreading. The variability of the surface-to-volume ratio, a key parameter in sorptionreactions, is explicitly accounted for by deforming the shape of the elementary volumes.

© 2010 Elsevier B.V. All rights reserved.

Keywords:Reactive transportKinetic sorptionFreundlich nonlinear sorptionLagrangian methods

1. Introduction

The transport and fate of solutes in subsurface environ-ments are conditioned by physical and chemical processes ininteraction, with advection and dispersion being the mostcommon physical processes and sorption the most commonchemical reaction. The sorption of organic compounds andheavy metals has been observed in the Borden (Roberts et al.,1986) and Cape Cod (Garabedian et al., 1991) tracer tests.Sorption reactions were initially modeled by linear and in-stantaneous mass transfers. However, both field and laborato-ry investigations suggest that sorption for several commonsorbing solutes is nonlinear (Ball and Roberts, 1991; Vereeckenet al., 2002) and kinetically-controlled (Goltz and Roberts,

nes1.fr

All rights reserved.

1986; Neuman et al., 1987; Garabedian et al., 1991; vanKooten,1996; Reichle et al., 1998).

On the physical side, natural aquifers are heterogeneous atmultiple scales (Gelhar, 1993). For example, measurementsof hydraulic conductivity andmediumporosity along a transectwithin anaquifer in Illinois varies over fourorders ofmagnitudeover distances of a couple of hundredmeters (Bakr et al., 1978).The widely-scattered heterogeneity produces significant dis-crepancies in the classical advection dispersion equation like alarge spreading of contaminants (Dagan, 1989; Gelhar, 1993).Its influence on the sorption reaction at large scale is difficultto establish because of the lack of highly-efficient numericalmodes that handle both thenonlinear, complex and kinetically-controlled sorption and the wide-ranging physical heteroge-neity. In this paper, we address the numerical issue by settingup a new efficient algorithm for solutes undergoing advection-dominated transport coupled with sorption.

Several numerical approaches have been developed tomodel the transport of reactive solutes. They can be classified

mailto:[email protected]

http://dx.doi.org/10.1016/j.jconhyd.2010.03.004

http://www.sciencedirect.com/science/journal/01697722

90 K. Besnard et al. / Journal of Contaminant Hydrology 120–121 (2011) 89–98

according to whether the advective transport term is treatedwith an Eulerian approach or with a Lagrangian approach. Thefirst methods used an Eulerian scheme (Huyakorn and Pinder,1983; Sun, 1996)well suited to the dispersive termbut prone tonumerical dispersion as regards the advective term. Thisnumerical dispersion has been used in a first type of mixing-cell methods to emulate the physical dispersion (Bajracharyaand Barry, 1994) by fixing the spatial discretization so that thenumerical dispersion is equated to the physical dispersion. Thisapproach, however, holds only for homogeneous media. Afurther improvement of the mixing-cell method consistsin selecting a Lagrangian method for the advective term andan Eulerian scheme for the dispersive term (Bajracharyaand Barry, 1994). This method is conceptually close to themethodof characteristics (KonikowandBredehoeft, 1978). Bothmethods retain the advantages of the Lagrangian andthe Eulerian methods for modeling transport. The chemicalreactions are handled at once within the time step so that thephysical and the chemical processes aredecoupledat the scale ofthe time step. The method of characteristics induces somenumerical dispersion through an averaging step in the com-putation of concentrations. Several other pure Lagrangianapproaches based on particle tracking have been developed(Tompson and Gelhar, 1990; Delay et al., 2005; Salamon et al.,2006; Valocchi and Quinodoz, 1989). For inert transport, allparticles are punctual and carry the same solute mass. Forreactive transport, particles can be associated with differentchemical species (Benson andMeerschaert, 2008) or consideredas much larger entities containing several chemical species. Inthe finite-cell method (Sun, 1999), the liquid and solid phasesare modeled by mobile and immobile cells in interaction. Eachcell carries a concentration and reacts with its neighbors alongits path in the medium. In the smoothed-particle hydrodynam-ics (SPH)method (Tartakovsky et al., 2007), theparticles are thebasic elements from which the solutions of the flow andtransport processes are approached. The solute concentration isderived from neighboring particles with a kernel-basedweight-ing scheme and reactivity is performed on the reconstructedconcentrations. Herrera et al. (2009) combine a numericalscheme for flow and a SPH method for solute transport.

We develop a new Lagrangian method adapted from theactive-walker method (Freimuth and Lam, 1992; Kayser et al.,1992) and from the precipitons used by geomorphologs formodeling erosion/sedimentation dynamics (Crave and Davy,2001). In this Lagrangian framework, we prefer the term ofLagrangian volume to that of active walker. The method isdesigned for advection-dominated transport conditions andsorption reactions. We present successively its general prin-ciple (Section 2), its application to the kinetically-controllednon-linear Freundlich isotherm case (Section 3), its imple-mentation, validation and domain of relevance (Section 4).We finally discuss its advantages and drawbacks compared toother numerical methods and its possible extension to re-action between chemical species in solution.

2. Model formulation

2.1. General Eulerian equations for sorbing solute transport

The chemical reactivity of organics and metals is primarycontrolled by their sorption capacities (Bedient et al., 1999;

Oelkers and Schott, 2009). Sorption is a reaction in which thesolute is sorbed on a solid phase characterized by its solidconcentration S expressedperunit surfaceareaof solid (mg/m2)and its concentration in solution C expressed per unit volumeofwater (mg/m3) (Sugita and Gillham, 1995). The chemical flowΦ between the solid and the liquid phases depends both on Cand S. The chemical reaction and the physical transportprocesses occur simultaneously leading to the followingcoupled system of governing equations for C and S:

∂C x; tð Þ∂t + γ xð Þ⋅

∂S x; tð Þ∂t =

∂∂x D xð Þ ∂C x; tð Þ

∂x

� �−∂ u xð ÞC x; tð Þð Þ

∂x∂S x; tð Þ

∂t = Φ C; Sð Þ ð1a and bÞ

with u the velocity (m/s), γ the ratio rock-surface/rock-volume(m−1) and D the dispersion coefficient (m2/s). This is the usualEulerian formulation of the sorbing solute transport equations.Physical heterogeneities are accounted for by the velocity spa-tial distribution and by the variation of the ratio of the solutionvolume to the surface area in contact with the solution. In thisarticle, we handle the advection-dominated case by removingthe dispersion term. We discuss possible extensions of themethod to non-negligible dispersions in Section 5.

2.2. Principles of the Lagrangian volume method for solvingsorbing solute transport equations

The originality of the numerical method proposed in thispaper relies on the use of a modified Lagrangian-volumemethod that generalizes the particle-tracking scheme tochemically active solutes. The aqueous phase entering thesystem is decomposed into a succession of identical elemen-tary volumes of water (Lagrangian volumes) characterized bytheir solute concentration. Lagrangian volumes are displacedwith the fluid velocity according to the classical rules ofparticle-tracking methods and are in chemical interactionwith the solid phase. Consequently, their solute concentrationvaries according to the chemical reactions occurring alongtheir paths. The Lagrangian-volume model can handle alltypes of initial and injection conditions. The injectionconditions are imposed by changing the concentration of theLagrangian volumes. An existing contamination in the aquiferis modeled by setting the sorbed concentration of the aquiferto a non-zero value.Wedescribe now the different steps of thenumerical method, including the advection of the Lagrangianvolumes within a mesh cell and the way of solving thechemical equations within the mesh.

2.3. Lagrangian volume displacement in a mesh cell

Solute is injected in the domain by way of identicalLagrangian volumes dV(W) from inflow limits and source terms.Their injection rate r, and thus the time scale between twosuccessive launchings t0, is fully defined by the water flow q0 as:

r = 1 = t0 = q0 = dV Wð Þ ð2Þ

Because of the possible nonlinearity of the chemical reaction,the Lagrangian volumes must be handled chronologically andcannot be dealt with independently the way they are in most

91K. Besnard et al. / Journal of Contaminant Hydrology 120–121 (2011) 89–98

inert particle-tracking methods. Indeed, the reaction rate de-pends on the sorbed concentration of the mesh cell, itself afunction of the concentration of the previous Lagrangian vol-umes that crossed themesh cell. Lagrangian volumesmust thenbe tracked simultaneously. To this end, particle-based methods,including the finite-cell method (Sun, 1999), use a constanttime step. To speed up the method we use an iterative method,wherein Lagrangian volumes are listed according to the time atwhich they leave themeshcell inwhich theyare. The Lagrangianvolumewith the smallest outgoing time goes through the meshcell. Either it leaves the mesh and is then removed from the list,or it enters the next mesh cell and is then reclassified in the listaccording to the time at which it will leave this new mesh cell.

Even if the tracking process without dispersion is deter-ministic, the method remains fundamentally stochastic. Theflow conservation is obtained after a sufficiently large num-ber N of Lagrangian volumes. N is typically of the order of 10to 100. For example, Lagrangian volumes ending up close to amesh cell corner do not divide but enter as a whole the nextmesh cell given by the flow line of their center of mass. Asthe flow model is stochastic, the reactive transport model isalso stochastic. The variation of the sorbed concentration dSshould be small enough with respect to its absolute value Sso that the concentration variations depend only on thenumber of elements passing through the mesh cell (and noton their time ordering). Stochastic averaging in a mesh cell isobtained over a time scale ts=Nt0, where t0 is the char-acteristic time scale of contact between a Lagrangian volumeand the solid. dS/S≪1 requires that t0 and thus the volume ofthe Lagrangian volume be significantly smaller than the meshcell volume. For the tests reported hereafter, dV(M)/dV(W) is ofthe order of 100.

2.4. Lagrangian volume reactivity with the solid

The chemical reaction occurs all along the path of theLagrangian volume through the mesh cell and changes bothits solute concentration C and the concentration S sorbed onthe solid phase. The variation of the solute concentration Cis the integral of the chemical flux, Φ, over the duration ofthe Lagrangian volume crossing the mesh cell dt(M). dt(M)

depends on the volume of the Lagrangian volume dV(W) andon the flow q as dt(M)=dV(W)/q of the mesh cell.

To solve the reactive transport problem within the frame-work of amesh, we consider the reactive transport within eachmesh cell as a homogeneous problem where (i) the soluteconcentration is uniformwithin the Lagrangian volume, (ii) theconcentration sorbed on the solid is uniform over themesh cellafter each Lagrangian volume crossing, (iii) the physical andgeometrical mesh cell properties including the porosity,Φ, andthe surface-to-volume ratio γ are constant (Fig. 1), and (iv)the velocity is given by an interpolation scheme derived fromthe discretization scheme used to compute the head field.Concentrations as well as chemical fluxes represent averagesat either the Lagrangian volume or the mesh cell scale, and aredenoted in the following by the subscript (W) for the Lagrangianvolume and (M) for themesh cell. Under these assumptions, thesolute mass exchanged during dt writes:

dM = Φ Wð Þ tð Þ⋅dA Wð Þ⋅dt = Φ Wð Þ tð Þ⋅γ Wð Þ⋅dV Wð Þ⋅dt ð3Þ

The mass conservation leads to:

dM = dA Mð Þ⋅dS Mð Þ = γ Mð Þ⋅ϕ Mð Þ⋅dV Mð Þ⋅dS Mð Þ

dM = −dV Wð Þ⋅dC Wð Þð4Þ

According to Eqs. (3) and (4), the basic equations to beintegrated over dt(M) are:

dC Wð Þ tð Þ = −γ Mð Þ⋅Φ Wð Þ tð Þ⋅dt

dSðMÞ tð Þ = 1ϕ Mð Þ

dV Wð ÞdV Mð Þ

Φ Wð Þ tð Þ⋅dtð5Þ

Consequently, the variation of the aqueous concentrationwithin a Lagrangian volume crossing a mesh cell between t0and t0+dt(M) is given by the integral (6), starting with theinitial conditions C(W)(t0) and S(M)(t0):

C Wð Þ t0 + ;d; t Mð Þ� �

= C Wð Þ t0ð Þ−∫ t0 +dt Mð Þt0

γ Mð Þ⋅Φ Wð Þ tð Þ⋅dt ð6Þ

The chemical flux comes from the sorption reaction handled.

3. Implementation of the kinetically-controlledFreudlich-type sorbing solute transport

We illustrate the Lagrangian-volume method with kinet-ically-controlled Freundlich sorption types.

3.1. Sorption equations

The Freundlich reactive behavior is described by a nonlinearisotherm (Weber et al., 1991; Fetter, 1993):

S = Ka⋅Cn ð7Þ

where C is the solute concentration in the fluid (mg/m3), S isthe solute concentration sorbed on the solid (mg/m2), Ka is aconstant defined on an area basis that generalizes the dis-tribution coefficient characterizing the sorption capacity and nis the Freundlich coefficient describing the sorption intensity.Values of n commonly observed are smaller than 1 (Fetter,1993; Abulaban andNieber, 2000). In the linear case (n=1), Ka

is a surface distribution coefficient (m−1) equal to the ratioof themass of solute sorbed per unit surface area of solid phaseto the mass of solute in solution per unit volume of solution.When the reaction has not reached equilibrium, the chemicalflow of Eq. (1b) writes:

∂S x; tð Þ∂t = α⋅ Ka⋅C

n x; tð Þ−S x; tð Þ� � ð8Þ

where α is the kinetic constant of reaction (s−1).

3.2. Chemical reaction at the Lagrangian volume scale

To compute the variation of the aqueous concentrationC(W) in a Lagrangian volume as it goes through a mesh cell inthe case of the Freundlich kinetic sorption, we write the in-tegral (6) as a series of integrals over the time intervals [ti,ti+Δt], where Δt is defined as the time needed by the Lagrangianvolume dV(W) to cross its surface contact dA(W) with the solid.

Fig. 1. Definition of notations at themesh cell scale. Filled and void blue cylinders represent the volume and contact surface of a Lagrangian volume. Filled and voidgrey tubes represent the volume and contact surface of the mesh cell integrated over the Lagrangian volume path.


We determine the chemical flux Φ at the fluid/solid interfacefrom Eq. (3). According to the kinetically-controlled Freun-dlich-type chemical reaction given by Eq. (8), the variation ofthe sorbed solute concentration at the Lagrangian volumescale is:

dS Wð Þ tð Þ = α⋅ Ka⋅CnWð Þ tð Þ−S Wð Þ tð Þ

� �⋅dt ð9Þ

where S(W) is the solute concentration sorbed on the solidsurface contacted by the Lagrangian volume dA(W). At thisscale, mass conservation writes:

dC Wð Þ tð Þ = −γ⋅dS Wð Þ tð Þ ð10Þ

where γ is the surface-to-volume ratio. Integrating the massconservation (10) over Δt leads to:

S Wð Þ ti + Δtð Þ = S Wð Þ tið Þ + 1γðMÞ

⋅ C Wð Þ tið Þ−C Wð Þ ti + Δtð Þ� �

ð11ÞThe decoupling of the chemical and physical processes at

the Lagrangian volume scale leads to the simple followingintegral obtained by combining Eqs. (9) and (11):

C Wð Þ ti + Δtð Þ = C Wð Þ tið Þ−α∫ ti +Δtti

× γ Mð Þ Ka CnWð Þ tð Þ + C Wð Þ tð Þ− C Wð Þ tið Þ + γ Mð Þ S Wð Þ tið Þ

� �h i⋅dtð12Þ

In the linear case (n=1), the integral (12) has the fol-lowing analytical solution:

C Wð Þ ti + Δtð Þ = 1R Mð Þ

× CT tð Þ− CT tið Þ−R Mð Þ C Wð Þ tið Þ� �

⋅exp −αR Mð ÞΔt� �h i

ð13Þ

where R(M)=1+γ(M)Ka is the classical retardation factorfound in a mesh cell in the case of linear and instantaneous

sorption and CT(ti)=C(W)(ti)+γ(M)S(W)(ti) is the total con-centration (amount per volume) of the solute present both inthe Lagrangian volume and on the solid surface contacted bythe volume at the beginning of the time step. In the generalnon-linear case (nb1), Eq. (12) can only be solved numer-ically. Among the available numerical methods, we choose afourth-order Runge–Kutta scheme that turns out to provide agood balance between time efficiency, precision and simplic-ity of programming (Shampine and Watts, 1977). Startingwith the initial conditions C(W)(ti) and S(W)(ti), the solutionof Eq. (12) gives C(W)(ti+Δt) and the mass conservation (11)gives S(W)(ti+Δt). The variation of C(W) as the Lagrangianvolume crosses a mesh cell is thus a series of integrationsgiven by Eq. (12). S(M) is updated only when the Lagrangianvolume leaves the mesh cell by using the mass conservationequation.

3.3. Optimization of computations at the mesh cell scale

3.3.1. Sources of computational costAs a transient-state problem, the transport problem ismuch

more time-consuming than the steady-state flow computation.The computational cost of the transport model comes, first,from themanagementof the list of Lagrangianvolumes, second,from the computation of the reactive transport of a Lagrangianvolume at the mesh cell scale, and third, from the number ofmesh cells crossed by Lagrangian volumes. At the mesh cellscale, the computational cost is linearly proportional to thenumber of time steps required for the Lagrangian volume tocross the mesh cell. At each time step within the mesh cell,Eq. (12) is integrated by the Runge–Kutta method. A straight-forwardmethod to speedup the algorithmwouldbe to increasethe volume of the Lagrangian volume dV(W) as the number oftime steps is inversely proportional to it. This would, however,decrease themodel accuracy. Inorder to speedup thealgorithmwithout loss of accuracy, we propose two approximations fortravel times within the mesh cell respectively much smalleror much larger than the transport characteristic time, i.e. whenthe sorption reaction is respectively under strong kinetic orthermodynamic control.


3.3.2. Strong kinetic control of sorption at the fluid/solidinterface

In the general case, the sorbed concentration S(M) isdiscretized within the mesh cell at the characteristic scaledA(M). Under strong kinetic control, the approximation com-prises replacing the discretized sorbed solute concentrationwithin themesh cell with a unique variable equal to the initialsorbed concentration S(M)(t0) when the Lagrangian volumecrosses the mesh cell. Under this assumption, the variation ofthe aqueous solute concentration C(W) during the time spentin themesh cell can be computed by integrating the followingequation during one time step Δt:

dC Wð Þdt

= −α γ Mð Þ⋅ϕ Mð ÞdV Mð ÞdV Wð Þ

Ka⋅CnWð Þ−S Mð Þ t0ð Þ

� �+ C Wð Þ−C Wð Þ t0ð Þ

� �" #

ð14ÞNote that Eq. (14) is a simple non-linear differential

equation of a single unknown C. As for Eq. (12), Eq. (14) canbe solved analytically in the linear case (n=1), whereas, inall other cases, Eq. (14) is solved numerically by using thepreviously presented Runge–Kutta method.

Under this assumption, the chemical reaction is solved onlyonce in the mesh cell whatever the volume of the Lagrangianvolume. Fig. 2 shows an example of the good agreement be-tween the two methods. With this assumption, the algorithmis around 200 times faster. We have numerically comparedthe difference between the solution obtainedwithout andwiththis approximation. We found that the accuracy is within 5%provided that the following condition is satisfied:

ϕ Mð ÞdV Mð ÞdV Wð Þ

b1

C n−1ð ÞT Kaγ Mð Þ αΔtð Þ 1+0:33nð Þ : ð15Þ

This condition fundamentally establishes an upper limit tothe time step Δt depending on the sorption characteristics αand Ka and on the rate of volumes between the Lagrangian

Fig. 2. Comparison of breakthrough curves expressed in mass rate (M/M0)per day at the cross-section located at 50 m downgradient from the injectionsource for the two types of integration at the Lagrangian volume scale and atthe mesh cell scale presented in Section 3. The parameters of the simulationare flow velocity, U=0.5 m/day, fluid–solid contact length at the Lagrangianvolume scale, dA(M)=2.5 10−3 m, length of a mesh cell, Δx=1 m, surface-to-volume ratio, γ=4000 m−1, and the parameters of the chemical reactionare n=1,=0.1 days−1 and Ka=10−3 m.

volume and the mesh cell. As a result, we use this approxi-mation in all mesh cells, the velocity ofwhich verifies Eq. (15).

3.3.3. Local equilibrium sorption model at the mesh cell scaleIn somemesh cells, the sorption reaction ismuch faster than

the advective transport so that the chemical reaction is almostat equilibrium. The approximation consists then in using a localequilibrium model. Consequently, the relationship between Sand C is simply the Freundlich non-linear isotherm as definedbyEq. (7). This isotherm is solved startingwith the initial valuesof the liquid and solid solute concentrations. We consider thatthis approximation remains valid as long as at least 95% ofthe reaction is completed within the duration of the contactbetween the Lagrangian volume and the solid Δt, which isexpressed for n=1 by ΔtN3/αR(M) and, by extension, in thegeneral non-linear case by:

αΔt 1 + γ Mð ÞKanCn−10

� �N 3: ð16Þ

In the general non-linear case, the linear retardation factorR(M) is replaced with an initial concentration dependent factor.Because the rate of reaction decreases with the magnitude ofthe concentration, larger initial concentrations lower the initialreactivity and increase the duration necessary for the equilib-rium to be reached.

4. Numerical implementation and validation of the model

For the demonstration and validation of the method, weuse a simple configuration made up of a regular 2D grid. Wepresent in the first section the solution to the flow problemthat provides the velocity field and in the following sectionseveral validation cases summarized in Table 1. We validatethe Lagrangian-volume algorithm previously presented in ahomogeneous medium against available analytical solutionsand predictions. The last section shows an application of themethod to a non-linearly and kinetically sorbing solute trans-port case in a heterogeneous medium.

4.1. Solution to the flow problem

We check the model on a three-dimensional regular cubicgrid. Flows q on the grid are calculated using Darcy's law, q=−K ⋅∇h, and the continuity equation,∇q=0, with K the localhydraulic conductivity and h the hydraulic head (de Marsily,1986). Simple permeameter-like boundary conditions areconsidered. Constant fixed head values are assigned to twoopposite faces of the domain while the four perpendicularfaces are impervious. These equations are discretized on thenetwork using a classical finite difference method. The linearsystem is solved by a classical LU decomposition (Saad, 1996)which proved to be far more efficient than iterative methods,at least for grids made up of less than 106 cells (de Dreuzy andErhel, 2002; Beaudoin et al., 2006). We consider a structuredgrid and solve the flow problem with a simple discretizationscheme to check the reactive transport algorithm. The trans-port algorithm, however, does not depend on the specificationof the computation of the flow problem. It can be implementedwhatever the velocity field, and in particular whatever the gridtype and the discretization scheme (finite difference, finiteelement or finite volume).

Table 1Analytical solutions for reactive transport models.


4.2. Numerical results against analytical solutions andpredictions in homogeneous porous media

The equations of the reactive transport problem are givenby Eqs. (1a and b) and (8). To check the algorithm, all prop-erties of the medium, i.e. K, γ, α and Ka, are constant through-out the medium to simulate a homogeneous medium. Wealso select a simple initial condition: no solute sorbed in themedium at the beginning of the experiment.

4.2.1. Linear kinetic sorption (n=1, αb∞)In this section, we validate the kinetic part of the algo-

rithm by comparing our results to an analytical solution. Foran instantaneous injection, an analytical solution for themean plume velocity can be derived from the spatial momentanalysis of Quinodoz and Valocchi (1993). We inject a singleLagrangian volume with a non-zero concentration C0≠0 atthe beginning of the test (t=0) and, after that, Lagrangianvolumes do not have any additional concentration (C0=0).The computeddimensionlessmeanplumevelocityU−1d⟨X(t)⟩/dt and dimensionless longitudinal variance are in very goodagreement with the analytical solution (Figs. 3 and 4).

Fig. 3. Validation of the dimensionless mean plume velocity versus time inthe linear kinetic sorption case (n=1, α=0.05 days−1) for a homogeneousmedium against the analytical solution of (Quinodoz and Valocchi, 1993)

bV tð Þ N = Ut 1 + R R−1ð Þexp −αRtð Þ 3−αRt½ � + R−1ð Þ3exp −2αRtð ÞR2 1 + R−1ð Þexp −αRtð Þ½ �2 . The parameters of the

simulation are flow velocity, U=1m/day, fluid–solid contact length at theLagrangian volume scale, dA(M)=5 10−3 m, length of a mesh cell, Δx=0.5 m, surface-to-volume ratio, γ=4000 m−1 and Ka=10−3 m and there-fore R=5.

4.2.2. Nonlinear equilibrium sorption (nb1, α=∞ )In the case of non-linear equilibrium sorption, the only

analytical prediction is given by Jaekel et al. (1996) on theasymptotic behaviorofmeanpositionof the solute plume. Theiranalysis predicts that the concentration at a fixed positiondecays asymptotically as a power lawof exponent 1/(1−n). Asshown in Fig. 5, the numerical results (dash-dotted lines) areconsistent with their prediction as long-term log–log plots ofbreakthrough curves tend to straight lines of slope 1/(1−n)(solid lines).

4.3. Example of non-linear kinetic sorption in 3D heterogeneousmedia (n≤1, αb∞)

Finally, we present the performance and an illustration ofthe Lagrangian-volume method in the most general case of anon-linearly and kinetically sorbing solute transport in a three-dimensional heterogeneous porous medium. The porosity isassumed constant and the variability of the flowvelocity is onlycaused by the spatial variability of the hydraulic conductivity,K(x). The hydraulic conductivity is assumed to be lognormallydistributed and spatially correlated. The hydraulic conductivityis represented by:

K xð Þ = KGexp f xð Þð Þ ð17Þ

Fig. 4. Mean plume longitudinal variance versus time in the linear kineticsorption case for a homogeneousmedium (n=1,α=0.01 days−1): comparisonof the numerical result and of the analytical solution. Simulation parameters areU=1m/day, dA(M)=5 10−2 m, Δx=0.5 m, γ=4000 m−1, Ka=2.10−4 m.

Fig. 5. Breakthrough curves at evolving distances X from the injection sourcein the nonlinear equilibrium sorption case (n=0.9, α=1) for a homoge-neous medium. Simulation parameters are U=1 m/day, dA(M)=5 10−3 m,Ka=10−3 SI, γ=8000 m−1. The asymptotic behavior of the breakthroughcurve follows a power-law in agreement with the power law of exponent 1/(1−n) indicated by the solid lines (Jaekel et al., 1996). Concentrations arenormalized to the input concentration C0.

Fig. 6. Example of the mean plume travel distance versus time for the non-linear kinetic sorption case in a three-dimensional heterogeneous porousmedium (σf=0.4, λ/L=0.1, n=0.8, α=0.1 days−1). Simulation parametersare a hydraulic gradient of 6 10−4 and Ka=10−3 SI.


whereKG is the geometricmeanofK and f(x) is a realization of anormal, second-order stationary spatially-correlated randomfield of mean zero and variance σf

2. The spatial correlation ismodeled by an isotropic exponential covariance function forf(x):

C ξð Þ = σ2f e

−ξ=λ ð18Þ

where ξ is the lag vector and λ is the correlation length. Thehydraulic conductivity field is derived from Eq. (17) bysynthesizing realizations of the random field f(x). A spectralmethod was adopted (Gutjahr, 1989). Realizations weregenerated in the frequency domain, by filtering an indepen-dently and identically distributed Gaussian field with thesquare root of the correlation function (18) spectrum andthen by reverting to space domain by means of Fouriertransforms. Effects of truncation and grid size were verifiedand a satisfactory reproduction of the target spectrum andcorrelation function was obtained (de Dreuzy et al., 2007).The chemical reactive coefficients, α and Ka, are homoge-neous (α=0.1, Ka=0.001). The ratio of the surface ofcontact between fluid and solid to the volume of fluid γij isfixed as if pores were cylindrical tubes:

γij =2

8μρgπKij

� �1=4

ð19Þ

where Kij is the permeability in the mesh cell connectingnodes i and j, ρ is the fluid density, µ is the fluid viscosityand g is the acceleration of gravity. According to Eq. (19)and since the permeability distribution is lognormal, thesurface-to-volume distribution is also lognormal. To getaverage results, we have run 50 simulations of 60×20×20domain cells, i.e. of 24,000 cells. For one simulation, thecomputational time on a Pentium 2.53 GHz is 0.5 h onaverage. We discuss the good performance of the method inSection 5.

The non-monotonous variations of the mean plume dis-placement (Fig. 6) are characteristic of nonlinear kinetically-controlled sorption coupled with advective transport (Micha-lak and Kitadinis, 2000). The surprising backward movementof the mean solute position comes from the temporary for-mation of a double peak. The first peak corresponds to theparts of solutes that have not been sorbed yet and the secondpeak is made up of the part of solutes that have already beensorbed and desorbed. The backward movement comes fromthe change in the distribution of the mass in solution fromthe first peak to the second peak. The double peak effect isenhanced by the non-linearity.

Fig. 7 finally displays a qualitative illustration for a 2D fieldof the Lagrangian volumes (b) with the reconstructed con-centrations (c) on the grid cells of the 2D heterogeneouspermeability field (a). It shows the large spreading of solutesdue to the sorption-induced kinetic retardation. The kineticcontrol thus induces a strong spreading of the solute plume.As long as this sorption-induced spreading remains muchlarger than the spreading induced by physical dispersivity,the transport conditions can be considered as advection-dominated and dispersion can be neglected. By extension ofEq. (16), the characteristic sorption time is of the order of[α(1+γ(M)KanC0

n−1)]−1 while the characteristic physicaldispersion length is classically taken as the longitudinal dis-persivity αL. The sorption-induced spreading remains dom-inant over the dispersion-induced spreading as long as α(1+γ(M)KanC0

n−1)b ⟨U⟩ /αL wherebUN is the mean velocity.

5. Discussion

5.1. Comparison with the finite-cell method (FCM) by Sun (1999)

This Lagrangian-volume method shares common pointswith the finite-cell method (Sun, 1999). The finite-cell meth-od is also a Lagrangian method that tracks “mobile cells” ofwater carrying solute concentrations. Mobile cells exchangemass with immobile cells representing the solid phase inorder to simulate sorption. Both methods differ in two

Fig. 7. (a) Example of a 2D heterogeneous velocity field obtained from a permeability field characterized by σf=1.5 and λ/L=0.1. The computation grid contains256⁎256 cells. Flow boundary conditions are constant heads on the vertical system edges and no flow on the horizontal system edges. Sorption parameters aren=0.9, α=1 day−1 and γKa=1 SI. At time 0, the medium concentration is 0 and an instantaneous concentration of solute is injected all along a vertical lineparallel to the domain inlet. Lagrangian volumes are characterized by dVp Mð Þ

dVW= 40. (b) Shows the Lagrangian volumes as points the color of which is proportional to

their solute concentration. Concentration increases from dark blue to light blue. (c) Displays the reconstructed concentrations as the volume average in eachmeshcell. Concentrations increase from blue to red.


respects. The first difference lies in the handling of thesurface-to-volume ratio. It is implicit in the FCM as it is theaverage ratio of the immobile to the mobile cells in a meshcell. We handle it explicitly in the Lagrangian volumemethod by deforming the Lagrangian volume according tothe surface-to-volume ratio of the mesh cells. The seconddifference between the proposed Lagrangian-volume meth-od and the finite-cell method is also a difference with allthe previously-mentioned approaches. All approaches im-pose a single constant time step. In a highly heterogeneousmedium, the solution of the flow equation leads to a widely-scattered velocity distribution ranging over several orders ofmagnitude, typically two to four orders of magnitude. Forthe sake of accuracy, the time step must remain small so thatthe fastest particles do not cross more than one mesh cell ina single time step whereas the slowest particles will remainfor a very large number of time steps in a low-velocity meshcell. To solve this problem, we track the Lagrangian volumeson the mesh and compute the time they need to cross amesh cell according to advection so that they cross the meshcell in a single time step whatever the mesh cell velocity(Delay and Bodin, 2001).

5.2. Numerical efficiency

We calculate and compare the complexity of the Lagrang-ian volume (LV) method and the Finite-Cell Method (FCM)(Sun, 1999). Because both methods are based on a particle-tracking scheme, their complexity can be computed like thecomplexity of the particle-tracking algorithm. The complexityof the particle-tracking algorithm is equal to the numberof particles Np times the number of mesh cells visited by aparticle when crossing the system Θ(Ns), Ns being thedimension of the grid, times the number of operationsnecessary for the particle to cross a mesh cell Θ(Nmesh).Both LV and FCM methods have the same term Np Θ(Ns) infactor of the complexity and differ only by the number ofcalculations necessary for a particle to cross a mesh cellΘ(Nmesh).

In the LVmodel, a Lagrangian volume crosses amesh cell inNR steps. On average, in 10% of the cases, NR is equal to theporous volumeof themesh cell dV(M) divided by the volumeofthe Lagrangian volume dVW (Fig. 1). In 90% of the cases, one ofthe approximations detailed in Sections 3.2 and 3.3 is validand NR is equal to 1. Thus, NR = 0:1 dVp Mð Þ

dVW+ 0:9. At each step

(i.e. forNR steps), the chemical flowmust be computed, whichrequires the numerical solution of the chemical Eq. (8) ofcomplexity ΘX. When the Lagrangian volume leaves the meshcell, the next mesh cell that it crosses and its exit time fromthis next mesh cell are determined. The exit time is classifiedin the list of exit times of all the Np Lagrangian volumes. Thecomputational cost results from the search of the right placewithin the list and is of the order of Θ(logNp). Thus,

Θ Nmeshð ÞLV = 0:1dVP Mð ÞdVW

+ 0:9�

⋅ΘX + Θ logNp

� �ð20Þ

where ΘX is the complexity of the numerical solution of thechemical Eq. (8). Np is of the order of 104, so log(Np) is around10, which is much lower than ΘX. Usually dVp(M)/dVW isaround 5.102, so that Θ(Nmesh)LV≈50 ⋅ΘX.

For FCM (Sun, 1999), we also calculate Θ(Nmesh)FCM thenumber of computations necessary for a mobile cell to crossa mesh cell. For the time step Δt, the distance covered by thecell is on average U ⋅Δt, where U is the characteristic velocitywithin the mesh. The number of steps taken by the mobilecell to cross themesh cell is equal to NR = Δx

UΔt . As a mobile

cell cannot cross more than one mesh cell in a single timestep, the maximum time step Δt is equal to Δx/Umax. At eachstep (i.e. for NR steps), the chemical flow mustalso be computed, which requires the numerical solution ofthe chemical Eq. (8) of complexity ΘX. Thus, on average, thecomplexity of FCM is:

Θ Nmeshð ÞFCM = Umax⋅⟨1U ⟩

� ⋅ΘX ð21Þ


For heterogeneous media, the velocity distribution is verybroad with a long tail of low velocities so that UmaxbU

−1N

reaches 6.103 for a lognormal distribution of permeability ofstandard deviation σ equal to 2. Comparison of Eqs. (20) and(21) reveals that the complexity of the FCM method dependson the velocity distribution whereas the LV method does not.For σ=2, the complexity of LV is 120 times lower than thecomplexity of FCM and the rate increases with the heteroge-neity. The comparison with the method of characteristics leadsto the same conclusion as it also requires a smallfixed time step.

5.3. Extension of the Lagrangian-volume method to reactions insolution

In presence of a non-negligible dispersion, the advective–dispersive transport of the Lagrangian volumes leads locallyto an additional fluid compressibility. It results from the pos-sible presence in a mesh cell of more water volumes thansaid mesh cell can effectively contain. This led Sun (1999) toconstrain the number of fluid finite cells according to thevolume of the mesh cells. We do not investigate this issuehere and assume a negligible dispersion in cases of domi-nant sorption-induced spreading. We plan, however, in fu-ture studies to address this issue in the following way. Wewill first perform a decoupling of the advective and dispersiveoperators and deal with them in Lagrangian and Eulerianframeworks, respectively. This will yield a reference solutionwith which we will compare the results of the full Lagrangiantreatment of the advective and dispersive transport termsinducing a spurious additional dispersion. We will test thedifferences between both methods on small simulations. Weexpect that, for highly heterogeneous media, differences willbe low first because of the large velocity variability, andsecond because of the averaging over a larger number ofLagrangian volumes. If the differences remain limited, largersimulations using only the Lagrangian volumes will be per-formed for tackling the upscaling issue. The objective is todeal with all kind of chemical reactions as long as they donot change the physical properties of the medium. With thisflexible decoupling method, physical and chemical processesmight be simultaneously modeled, whatever the complexityof the chemical process.

6. Conclusion

We have set up a new method based on active-walkersfor simulating solutes undergoing advective transport andchemical sorption. It is a Lagrangian method that consists intracking elementary volumes of water called Lagrangian vol-umes, the chemical composition of which is in interactionwith the solid phase. The Lagrangian volumes are trackedsimultaneously on the grid following advection. The Lagrang-ian-volume method has been implemented in order to benumerically efficient for heterogeneous media. In heteroge-neous media, the difficulty arises from the widely-scatteredvelocity field imposing a small time step. We adapt the timestep to the properties of the mesh cell and of the Lagrangianvolume. The complexity of the algorithm becomes indepen-dent of the velocity distribution and of the heterogeneity.We have implemented and validated this Lagrangian-volumemethod in the case of kinetically-controlled nonlinear sorp-

tion, the nonlinearity being modeled by a Freundlich iso-therm. This model will be used for the derivation of themacroscopic reactive transport laws in heterogeneous mediafrom the description of chemical and physical processes atthe laboratory scale.

Acknowledgements

The French National Research Agency ANR is acknowl-edged for its financial funding through the MOHINI project(ANR-07-VULN-008) and for its contribution to the develop-ment of numerical methods through theMICAS project (ANR-07-CIS7-004). Authors thank Alexander Tartakovsky for fruit-ful discussions.

References

Abulaban, A., Nieber, J.L., 2000. Modeling the effects of nonlinear equilibriumsorption on the transport of solute plumes in saturated heterogeneousporous media. Advances in Water Resources 23 (8), 893–905.

Bajracharya, K., Barry, D., 1994. Note on common mixing cell models. Journalof Hydrology 153 (1–4), 189–214.

Bakr, A., Gehlar, L., Gutjahr, A., MacMillan, J., 1978. Stochastic analysis ofspatial variability in subsurface flows 1. Comparison of one- and threedimensional flows. Water Resources Research 14, 263–271.

Ball,W., Roberts, P., 1991. Long-termsorptionof halogenatedorganic-chemicalsbyaquifermaterial. 1. Equilibrium.Environmental Science&Technology 25(7), 1223–1237.

Beaudoin, A., deDreuzy, J.R., Erhel, J., 2006. A comparison between a direct and amultigrid sparse linear solvers for highly heterogeneousflux computations.In: Wesseling, E.O.a.J.P.P. (Ed.), European Conference on ComputationalFluid Dynamics, ECCOMAS CFD 2006.

Bedient, P.B., Rifai, H.S., Newell, C.J., 1999. Ground Water Contamination:Transport and Remediation, 2nd Edition. Prentice Hall.

Benson, D.A., Meerschaert, M.M., 2008. Simulation of chemical reaction viaparticle tracking: diffusion-limited versus thermodynamic rate-limitedregimes. Water Resources Research 44.

Crave, A., Davy, P., 2001. A stochastic “precipiton”model for simulating erosion/sedimentation dynamics. Computers & Geosciences 27 (1), 815–827.

Dagan, G., 1989. Flow and Transport in Porous Formations. Springer Verlag.(465 pp.).

de Dreuzy, J.-R., Beaudoin, A., Erhel, J., 2007. Asymptotic dispersion in 2Dheterogeneous porous media determined by parallel numerical simula-tions. Water Resources Research W10439.

de Dreuzy, J.R., Erhel, J., 2002. Efficient algorithms for the determination ofthe connected fracture network and the solution of the steady-state flowequation in fracture networks. Computers and Geosciences 29 (107–111).

deMarsily, G., 1986. Quantitative Hydrogeology: Groundwater Hydrology forEngineers. Academic Press, Orlando, FL. (464pp.).

Delay, F., Ackerer, P., Danquigny, C., 2005. Solution of solute transport in porousor fractured formations by randomwalk particle tracking: a review. VadoseZone Journal 4, 360–379.

Delay, F., Bodin, J., 2001. Time domain random walk method to simulatetransport by advection–dispersion and matrix diffusion in fracturednetworks. Geophysical Research Letters 28 (21), 4051–4054.

Fetter, C.W., 1993. Contaminant hydrogeology.Macmillan PublishingCompany,New York. (458pp.).

Freimuth, R., Lam, L. (Eds.), 1992. Active walker models for filamentarygrowth patterns. Modeling Complex Phenomena. Springer, New-York.302–313pp.

Garabedian, S.P., LeBlanc, D.R., Gelhar, L.W., Celia, M.A., 1991. Large-scalenatural gradient tracer test in sand and gravel, Cape Cod, Massachusetts,2, Analysis of spatial moments for a nonreactive tracer. Water ResourcesResearch 27 (5), 911–924.

Gelhar, L.W., 1993. Stochastic Subsurface Hydrology. Prentice Hall, Engel-wood Cliffs, New Jersey.

Goltz, M.N., Roberts, P.V., 1986. Interpreting organic solute data from a fieldexperiment using physical nonequilibrium models. Journal of Contam-inant Hydrology 1, 77–93.

Gutjahr, A.L., 1989. Fast Fourier transforms for random field generation, NewMexico Tech project report 4-R58-2690R.

Herrera, P.A., Massabó, M., Beckie, R.D., 2009. A meshless method to simulatesolute transport in heterogeneous porous media. Advances in WaterResources 32 (3), 413–429.


Huyakorn, P.S., Pinder, G.F., 1983. Computational Methods in Subsurface Flow.Academic Press, Inc. (743pp.).

Jaekel, U., Georgescu, A., Vereecken, H., 1996. Asymptotic analysis of nonlinearequilibriumsolute transport inporousmedia.WaterResourcesResearch32(10), 3093–3098.

Kayser, D., Aberle, L., Pochy, R., Lam, L., 1992. Active walker models: tracksand landscapes. Physica A: Statistical Mechanics and its Applications 191,17–24.

Konikow, L., Bredehoeft, J., 1978. Computer model of two-dimensional solutetransport and dispersion in groundwater. Techniques ofWater ResourcesInvestigations, 7. U.S. Geological Survey, p. 90 (chap. C2).

Michalak, A.M., Kitadinis, P.K., 2000. Macroscopic behavior and random-walkparticle tracking of kinetically sorbing solutes. Water Resources Research36 (8), 2133–2146.

Neuman, S., Winter, C., Newman, C., 1987. Stochastic-theory of field-scaleFickian dispersion in anisotropic porous-media. Water Reosurces Research23 (3), 453–466.

Oelkers, E.H., Schott, J. (Eds.), 2009. Thermodynamics and kinetics of water–rock interaction: Reviews in mineralogy and geochemistry, 70.

Quinodoz, H.A.M., Valocchi, A.J., 1993. Stochastic analysis of the transportof kinetically sorbing solutes in aquifers with randomly heteogeneoushydraulic conductivity. Water Resources Research 29 (9), 3227–3240.

Reichle, R., Kinzelbach, W., Kinzelbach, H., 1998. Effective parameters inheterogeneous and homogeneous transport models with kinetic sorption.Water Resources Research 34 (4), 583–594.

Roberts, P., Goltz,M.,Mackay, D., 1986. ANatural gradient experiment on solutetransport in a sand aquifer. 3. Retardation estimates and mass balances fororganic solutes. Water Resources Research 22 (13), 2047–2058.

Saad, Y., 1996. Iterative Methods for Sparse Linear Systems. PWS PublishingCompany.

Salamon, P., Fernandez-Garcia, D., Gomez-Hernandez, J.J., 2006. A reviewand numerical assessment of the random walk particle tracking method.Journal of Contaminant Hydrology 87 (3–4), 277–305.

Shampine, L., Watts, H., 1977. The art of writing a Runge–Kutta code, Part 1.Mathematical Software III. Academic Press, New-York (257–275pp.).

Sugita, F., Gillham, R.W., 1995. Pore scale variation in retardation factor as acause of nonideal reactive breakthrough curves 1. Conceptual model andits evaluation. Water Resources Research 31 (1), 103–112.

Sun, N.-Z., 1996. Mathematical Modeling of Groundwater Pollution. SpringerVerlag, New York.

Sun, N.-Z., 1999. A finite cell method for simulating the mass transport processin porous media. Water Resources Research 35 (12), 3649–3662.

Tartakovsky, A.M., Meakin, P., Scheibe, T.D., Wood, B.D., 2007. A smoothedparticle hydrodynamics model for reactive transport and mineral precip-itation inporous and fractured porousmedia.Water Resources Research 43(5), 654–672.

Tompson, A.F.B., Gelhar, L.W., 1990. Numerical simulation of solute transportin three-dimensional, randomly heterogeneous porous media. WaterResources Research 26 (10), 2541–2562.

Valocchi, A., Quinodoz, H., 1989. Application of the random walk method tosimulate the transport of kinetically adsorbing solutes. GroundwaterContamination. IAHS Press, Wallingford, Oxfordshire, England. 35–42pp.

van Kooten, J., 1996. A method to solve the advection–dispersion equationwith a kinetic adsorption isotherm. Advances in Water Resources 19 (4),193–206.

Vereecken, H., Jaekel, U., Schwarze, H., 2002. Analysis of the long-termbehavior of solute transport with nonlinear equilibrium sorption usingbreakthrough curves and temporal moments. Journal of ContaminantHydrology 56 (3–4), 271–294.

Weber,W.J.,McGinley, P.M., Katz, L.E., 1991. Sorption phenomena in subsurfacesystems—concepts, models and effects on contaminant fate and transport.Water Research 25 (5), 499–528.

a modified lagrangian-volumes method to simulate nonlinearly and kinetically sorbing solute...

Documents