tnu

7, T

Keywords:Reactive transportNumerical schemesNumerical diffusionHigh heterogeneityLocal dispersion model

of ns wriat

method, which we modied to improve its accuracy. The advective nature of the transport problem ren-

ion-doia is tsharp

geneous geological formations the Darcian velocity eld showsstrong spatial variability at a hierarchy of scales (e.g. [2,3]). Thisvariability leads to a disordered spatial distribution of the concen-tration with sharp interfaces which enhance mixing, thereby exert-ing a strong control on the reactions (e.g. [3]). The accuratereproduction of these features in numerical simulations is essentialto correctly model multi-species solute reactions and biochemical

relative simplicity and accessibility, these numerical schemes havebeen shown to be plagued by numerical (articial) diffusion (e.g.[5,6,4]), which can be partially alleviated by increasing the orderof accuracy. However, high-order schemes are prone to developspurious oscillations in high gradient regions, which can be some-how controlled by reducing both grid size and time-step, or byusing higher order nonlinear schemes, such as essentially non-oscillatory (ENO) [7] or weighted essentially non-oscillatory(WENO) methods [8]. These methods are also available on generalunstructured meshes to deal with complex geometries [911].

Corresponding author.

Advances in Water Resources 52 (2013) 178189

Contents lists available at

a

lseE-mail address: francesca.boso@ing.unitn.it (F. Boso).plume. This situation is typically encountered in many branchesof physics and engineering when the transport of an agent is con-trolled by a spatially disordered velocity eld (see e.g. [1]). In thepresent work we focus on passive and reactive transport in highlyheterogeneous geological formations, but the results of our analy-sis can be extended also to other cases in which transport occurs ina heterogeneous velocity eld .

Modeling multi-species reactive solute transport in advection-dominated environments requires special care in treating theadvective component of the governing equation. In highly hetero-

which are common to most grid-based numerical methods, deteri-orate the accuracy of the numerical schemes, in particular in prox-imity of large concentration gradients, such as those observed atthe plume fringes [5]. When more than one agent is transported,such as in the case of two or more aqueous species reacting uponmixing, errors may increase further because of the nonlineardependence of the reaction rates on the local concentration ofthe reacting species.

Most applications use low-order Eulerian schemes applied to axed grid of either nite elements or nite volumes. Despite their1. Introduction

A distinctive character of advectcesses in heterogeneous porous medtion of solute concentration with0309-1708/$ - see front matter 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.advwatres.2012.08.006ders the numerical solution very challenging with the solutions provided by classic Eulerian methods thatare plagued by numerical diffusion and spurious oscillations. Our analysis shows that TVD is severelyaffected by numerical diffusion, while the modied SB-FV method shows the tendency to underestimatedilution to an extent that increases with r2Y . In addition, we show that MOC is not mass-conservative, SPHis computationally demanding and cannot handle anisotropic dispersion, while RWPT develops spuriousconcentration uctuations, which can be attenuated by increasing the number of particles at theexpenses of an increase of the CPU time. Moreover, we investigate the effect of uniform and non-uniformlocal dispersion models on the overall plume dilution. These results help to consciously choose thenumerical scheme according to investigations objectives and heterogeneity degree.

2012 Elsevier Ltd. All rights reserved.

minated transport pro-he disordered distribu-interfaces across the

processes as well. In this situation, obtaining accurate numericalsolutions of the transport equation is challenging because the tra-ditional methods used to approximate the advective term of thetransport equation add numerical diffusion, often accompaniedby spurious oscillations of the solution [4]. These pathologies,Available online 30 August 2012istics (MOC), are available in widely used packages. The other three schemes are the Random Walk Par-ticle Tracking (RWPT), the Smoothed Particle Hydrodynamics (SPH) and a Streamline-Based (SB-FV)Numerical simulations of solute transporformations: A comparison of alternative

Francesca Boso , Alberto Bellin, Michael DumbserDepartment of Civil and Environmental Engineering, University of Trento, Via Mesiano 7

a r t i c l e i n f o

Article history:Received 23 February 2012Received in revised form 13 July 2012Accepted 17 August 2012

a b s t r a c t

We compare the accuracysolutes in porous formatioTwo schemes, the Total Va

Advances in W

journal homepage: www.ell rights reserved.in highly heterogeneousmerical schemes

rento 38123, Italy

ve numerical schemes in modeling transport of nonreactive and reactiveith heterogeneity increasing from low (r2Y 0:2) to very high (r2Y 10).

ion Diminishing (TVD) and the EulerianLagrangian Method of Character-

SciVerse ScienceDirect

ter Resources

vier .com/ locate/advwatres

ter RWhen simulations are three-dimensional, or conducted in a MonteCarlo framework, and formations are highly heterogeneous, gridrenement and smaller time-steps are often unfeasible solutions,because of the high computational burden and the large memoryrequirements associated with the use of very ne grids.

In Lagrangian methods mass conservation is imposed ondeformable volumes which move along streamlines according tothe local velocity, such that the total time variation of concentra-tion depends on local dispersion and reaction, solely [5]. In doingthat global mass conservation is granted, while local mass conser-vation depends on the accuracy with which the velocity eld is re-solved [12]. Among the available Lagrangian methods, the classicalRandomWalk Particle Tracking (RWPT) subdivides the solute massinto a large number of particles which are moved according to thevelocity eld to represent the advective component of the trans-port equation; a Brownian component is added in the form of par-ticle jumps to mimic the effect of local dispersion. Other methods,such as Smoothed Particle Hydrodynamics (SPH), refer to mobileuid particles and discretize the Fickian local dispersion operatorby means of an integral interpolation [13].

Finally, EulerianLagrangian methods use a combination ofLagrangian and Eulerian schemes applied to the advection and dis-persion terms, respectively [6,5]. The Method of Characteristics(MOC), for example, relies on particle tracking for computing anintermediate concentration eld on a xed reference grid, whichis then updated by solving the dispersion term with an Eulerianscheme [14]. A periodic re-mapping of particle conguration is ap-plied in order to reduce the computational effort, which cumulatesnumerical diffusion to an extent that is difcult to quantify [15].

Other EulerianLagrangian methods apply Eulerian algorithmsto streamline-oriented grids of elements, thus minimizing grid-orientation effects and reducing numerical diffusion at least inthe transverse direction where the concentration time variationis only due to local dispersion [5,16]. A comparison betweenstreamline-based and grid-based solutions of the transport equa-tion for a nonreactive solute in a two-dimensional set up can befound in the work by [6]. Although possible, the extension of thisapproach to a three-dimensional ow eld is cumbersome. Analternative approach is to split the transport equation into twoparts: a one-dimensional transport equation along the streamlines,supplemented by a purely diffusive equation which accommodatesthe dispersive term through operator splitting. This approach hasbeen used by [17], who discretized the dispersive termwith a nitedifference scheme, whereas [18] modeled dispersion as SPH-likemass exchange across streamlines. The main difculty here is toobtain an adequate resolution throughout the domain, sincestreamlines become highly irregular as the heterogeneity of theformation increases [19].

In the present work we compare the performance of venumerical schemes used to model passive and reactive solutetransport in heterogeneous formations and propose a modicationof the SB scheme presented by Herrera et al. [18], to improve theaccuracy of the numerical concentration eld. First we analyzehow numerical diffusion inuences dilution of a passive tracer,and successively, by using the same set of numerical simulations,we analyze the impact of numerical diffusion on the transport oftwo reactive aqueous species reacting upon mixing. Moreover,we analyze the impact of the local dispersion model on the overalldilution experienced by the plume. We simulate transport in two-dimensional heterogeneous formations with a log-conductivityvariance ranging from 0.2 to 10. Working with high heterogeneityis particularly challenging because of the large deformation thatthe plume experiences, which leads to large concentration gradi-

F. Boso et al. / Advances in Waents along extended plumes fringes.In Section 2 we briey describe the numerical methods, in Sec-

tion 3 we present the numerical setup, while in Section 4 we dis-cuss the results for transport of both conservative and reactivesolutes. Finally, conclusions are drawn in Section 5.

2. Mathematical model and numerical schemes

2.1. Mathematical model

The governing equation for multi-species solute transport in aheterogeneous stationary velocity eld is as follows:

@ci@t

u rci r Drci rin ; i 1; . . . ;m 1

where ci is the concentration of the ith species, n is the porosity,which is assumed constant (uniform) through the computationaldomain, u is the spatially variable and stationary velocity eld, Dis the local dispersion tensor with components Djk; j; k 1; . . . ; mand m is the number of aqueous species belonging to the reactivesystem. Hereafter m indicates space dimensionality. If the uid isincompressible, the velocity eld is divergence free, such that thefollowing condition applies: r u 0, which in combination withthe Darcys law

u Kn rH 2

leads to the following mass-conservation equation:

r KrH 0 3where K is the hydraulic conductivity tensor and H is the piezomet-ric head (see e.g. [20]). Eqs. (1)(3) are written at the Darcys scale,i.e. a scale at which the porous medium can be considered as a con-tinuum equivalent [20]. The interplay between molecular diffusionand the heterogeneous water velocity prole within the porescauses solute mass transfer at the sub-Darcys scale, which is mim-icked in Eq. (1) as a hydrodynamic dispersive mass ux:FD D rci, where ci is the mean concentration of the species idened over a support volume of size corresponding to the Darcysscale.

The local dispersion is often parameterized as follows [21,20]:

Djk Dm;i aT uj jdjk aL aTujukuj j 4

where Dm;i is the effective compound-dependent molecular diffu-sion (or pore diffusion coefcient), corrected for the effect of tortu-osity, while aL and aT are the longitudinal and transversedispersivities, respectively. Other non-linear models are available,which better represent the dependence of dispersivity from the lo-cal velocity in laboratory experiments (see e.g. [22]), but in thepresent work we limit ourselves to the comparison of model (4)with the model of constant dispersion coefcient often imple-mented in eld-scale simulations. The dispersion tensor becomesdiagonal if one axis of the orthogonal reference system is parallelto the velocity vector. For reactive tracers, additional expressionsfor the reactive source/sink terms, ric1; c2; . . . ; cm, either linear ornon-linear, are required.

In addition to a nonreactive tracer, we choose to model the caseof a bimolecular instantaneous equilibrium reaction of two aque-ous species, A and B, to produce a precipitate C according to the fol-lowing stoichiometric reaction: aA bB! cC #, where a; b and care the stoichiometric coefcients. For an instantaneous bimolecu-lar reaction whose product is a precipitate, the mass action law re-duces to the following algebraic expression ca1 c

b2 Keq, where c1

and c2 are the concentrations of the species A and B, respectively,and Keq is an equilibrium constant (e.g. [23]). By subtracting the

esources 52 (2013) 178189 179transport equation of c1 from that of c2 - which are obtained byspecializing Eq. (1) for i 1 and i 2, respectively and assumingthat the two aqueous species share the same molecular diffusion

ter Rand that the stoichiometric coefcients are unitary, we obtain theclassical Advection Dispersion Equation for a ctitious nonreactivetracer with concentration t c1 c2, supplemented by the follow-ing two speciation equations [23,24]:

c1;2 12 tt2 4Keq

q 5

where the upper sign between the two terms within the bracket ap-ply to c1 and the lower to c2. This approach belongs to a well estab-lished class of methods, called chromatography, used in chemistryand chemical engineering to simplify multispecies transport prob-lems [25], and has been brought into the hydrologic communityby the pioneering work of [26], followed by several others (seee.g. [23,27,24]).

2.2. Numerical schemes

Numerical methods for solving the ADE are typically classiedas Eulerian or Lagrangian, according to the adopted spatial refer-ence system. These methods attempt to overcome the many dif-culties arising in advection-dominated transport processes byusing different strategies, the main challenge being the accuratereproduction of sharp moving fronts and the reduction of numeri-cal diffusion in mixing-controlled reactions.

In the present work we applied ve numerical schemes to thesame heterogeneous velocity eld in weakly as well as in highlyheterogeneous formations. We considered three among the mostwidely used numerical schemes: the Total Variation DiminishingFinite Volume Method, the Method of Characteristics and the clas-sical Random Walk Particle Tracking method. The rst twoschemes are included into the popular MT3DMS package distrib-uted by USGS and hence are widely diffused among practitioners[14]. In addition, we considered two Lagrangian schemes recentlyproposed by [15,18]. For the sake of completeness, subsequentlywe provide a short description of these numerical schemes. Wedecided to consider elds of increasing heterogeneity becausesome pathologies may emerge only when the numerical schemeis applied to highly heterogeneous formations [19]. In all cases sat-urated ow has been solved by applying the classic nite volumesscheme MODFLOW [28] to a very ne grid, which we veried capa-ble of reproducing the strong non-uniformities of the velocity eldtypical of highly heterogeneous formations, as we show inSection 3.

2.2.1. TVD and MOC schemesThe literature offers several nite volume TVD schemes con-

structed in such a way that the Total Variation TV RX jrcjdx ofthe numerical approximate solution diminishes as time increases.Among the several TVD schemes available, we adopt here the ULTI-MATE TVD scheme, in which the concentration at the interfaces be-tween adjacent cells is obtained through a third-order polynomialinterpolation of nodal concentrations. This scheme is computation-ally effective (both in term of CPU time and memory require-ments), at least for small to moderate heterogeneity, and readilyavailable within the package MT3DMS [14]. The tendency of higherorder polynomial interpolations to develop spurious oscillations iscontrolled through ux limitation, thus ensuring that TV dimin-ishes, whereas the order of accuracy is raised by the polynomialreconstruction of the concentration within the cell. The gain inaccuracy introduced by the polynomial reconstruction is expectedto become less effective at large heterogeneity, as sharp edges aremaintained by high velocity gradients. This numerical scheme is

180 F. Boso et al. / Advances in Wamass conservative and the accuracy achieved along sharp frontsdepends on grid renement and time step. Its use is recommendedin dispersion-dominated scenarios where the high order of accu-racy is particularly benecial to obtain accurate reproductions ofthe concentration also for relatively coarse grids. In advectiondominated transport problems the use of TVD can be in principlequestioned because it deliberately adds numerical diffusion acrosssharp fronts, as typical of most Eulerian schemes.

MOC solves the ADE (i.e., Eq. (1) for i 1 and r1 0) in twosteps. In the rst step an intermediate concentration is computedthrough particle tracking assuming that only the advective termcontributes to the variation of the solute concentration. Thisapproximated concentration is then updated by discretizing the lo-cal dispersive ux by nite differences. Although particle trackingis free of numerical diffusion, errors are introduced when the inter-mediate concentration is computed as weighted average of theconcentration of the particles contained within the computationalcell of volume DV . The accuracy can be increased by reducing DV ,which in turn calls for an increase of the number of particles, asdiscussed in Section 2.2.2. In order to contain memory require-ments and CPU time, the number and positions of the particlesare dynamically redened at each time-step. MOC has been shownto minimize numerical diffusion also in advection-dominated sce-narios, but unfortunately the scheme is not mass-conservative.Mass balance errors are expected to increase with the heterogene-ity of the velocity eld and when the computational grid isirregular.

Further details concerning these two numerical schemes andtheir implementation can be obtained from the users manual ofMT3DMS ([14] and references therein).

2.2.2. Random Walk Particle Tracking (RWPT)The classical RWPT is one of the most intuitive methods for

modeling transport of solutes in heterogeneous porous formations.Details on this meshless method, which is virtually free of numer-ical diffusion method, can be found in the review paper by [29].

The mass of the solute is divided into a large number Np of par-ticles, such that each particle carries a small fraction Dm M0=Npof the total injected massM0. The number of particles is selected asa tradeoff between computational burden and accuracy which, foran instantaneous injection, depends on the ratio between the vol-ume of the source area V0 and of the computational cell DV [29,30].

An important and somewhat limiting characteristic of thismethod is that each particle conserves its initial mass since nomass exchange occurs between adjacent particles. The particlesare then moved independently at small time steps Dt by successivejumps, each one composed by a drift and a Brownian component:

Xpt Dt Xpt AXp; tDt BXp; t ntDt

p6

where Xpt is the position of the pth particle at time t;A is the driftvector which accounts for the effect of both the velocity eld andthe divergence of the dispersion tensor D; B is the displacementmatrix which depends upon velocity and dispersion, and nt is astandard random deviate array, whose components have zero meanand unit variance.

The concentration at a given position x and time t is then com-puted as follows:

cx; t nptDmDV

7

where npt is the number of particles that at time t are within thesampling volume DV , centered at x. In order to obtain an accuraterepresentation of the point concentration the dimensions of DVshould be much smaller than the respective integral scales of thelog-conductivity, which calls for using a large number of particles.

esources 52 (2013) 178189In fact, as discussed in [30], the minimum concentration than canbe resolved with the forward scheme (6), for an instantaneousinjection of a solute mass M0 c0 V0n within V0, is given by:

ter Rcmin c0 V0=NpDV, such that, to maintain the same sensitivity inthe computation of the concentration, a reduction of DV shouldbe accompanied by an increase of the total number of particlesNp. Consequently, a large number of particles is needed when theratio V0=DV is large, i.e. for wide source areas or small, point-like,DV , as for example when the local distribution of the concentrationshould be accurately reproduced. CPU time grows linearly with thenumber of particles [15], while some benet can be obtained bykeeping constant the length of the particle jumps and letting thetime step vary according to the local velocity, instead of xing itto a constant value [29]. Back-particle tracking may be used whenthe ratio V0=DV is large, to alleviate the computational burdenand the memory requirements [30]. The error associated to the esti-mate (7) of the solute concentration reduces as Np increases, withthe exact solution of the ADE (1) that is obtained for Np !1.

Moreover, since velocity is known along interfaces between thecells of the grid where ow is solved, a velocity mapping is re-quired in order to reconstruct a continuous information to be usedfor computing A and B in Eq. (6). We chose hybrid interpolation inorder to fulll both local uid and solute mass conservation [12,29]and avoid particle accumulation in low-permeability regions [31].

This scheme is natively mass conservative and free of numericaldiffusion. The main limitations are related to local uctuations ofconcentration occurring when the concentration is mapped on anEulerian grid, which are usually enhanced by non-linear reactionrates, and to the difcult implementation of reaction terms otherthan simple linear sorption [31].

2.2.3. Smoothed Particle HydrodynamicsSmoothed Particle Hydrodynamics (SPH) is a Lagrangian mesh-

less method whose nodes are moving uid particles with a niteuid volume. Advection is taken into account by the movementof the particles along the streamlines, whereas dispersion is mod-eled as Fickian solute mass exchange among particles. As a result,SPH is suitable for pure advection simulations since it can providestrictly bimodal concentration distributions by switching off massexchange among particles. The concentration cxi; t of the particlelocated at the position x xi at time t is approximated through thefollowing Monte Carlo integration scheme [13]:

cxi; t 1NpxiXNpxij1

cxj; tWxi xjxj=Npxj 8

where W is a suitable kernel function with compact support H cen-tered on x xi, and Npxi is the number of particles within H. Inaddition, the particle densityxj assumes the following approximateexpression:

xj XNpxjk1

Wxj xk 9

The integral approximation of the dispersive uxes is borrowedfrom the SPH solution of the heat conduction equation [32,33] andreads [15,34]:

dcxi; tdt

2XNpxij

1xij

Dij cxj; t cxi; t rij

j rijj2 rW rij

10where dcxi; t=dt is the total time rate of change of cxi; t. In addi-tion, rij is the distance vector between the two locations xi and xj,whereas rW is the gradient of the kernel function W. Finallyxij 2xixj=xi xj and Dij 2DiDj=Di Dj are the harmonicmeans ofx and D at xi and xj, respectively. The use of the harmonic

F. Boso et al. / Advances in Wamean assures the symmetry of the mass exchange among particles.Under these conditions the numerical scheme is virtually free ofnumerical diffusion.The integral approximated by Eq. (8) introduces an error whichincreases with the smoothing volume H of W. Moreover, the errorassociated to the numerical Monte Carlo integration also dependson the number and on the distribution of the particles within H[13]. In particular, it increases when the particles are unevenly dis-tributed within H, with the weights xxj that partially attenuatethis undesired effect. This calls for an accurate and detailed discret-ization of the ow eld, especially at high r2Y , such as to obtain analmost uniform, although disordered, distribution of particleswithin H. In this case the expected error in the Monte Carlo inte-gration is of the order 1=

N

p, where N is the total number of parti-

cles belonging to the integration domain [35]. In highlyheterogeneous three-dimensional formations both the number ofparticles and the number of grid cells needed to accurately repro-duce the velocity eld may be prohibitively large. Notice that thenumber of particles is not proportional to the injected mass asfor RWPT, but depends on the size of the computational domain.

Following [13] we use for W the following cubic B splinefunction:

Wq Kmhm

23 q2 12 q3; for 0 6 q < 116 2 q 3; for 1 6 q < 20; for q > 2

8>: 11where q jrijj=h and Km is a normalization constant which dependson the space dimensionality m. The smoothing length h over whichW is dened is related to the mean particle distance and is the ref-erence length of the numerical scheme. Similarly to RWPT, particletracking is performed by using a linear interpolation of the velocityinside the computational cells [12] and a second-order explicit Tay-lor scheme. Furthermore, uid particles are introduced at a densityproportional to local ow from the inow boundary of the domain.

The stability of the solution requires that the time step is xedaccording to the following condition: Dt 6 h2=D, where is anempirical coefcient [15,33]. The main problem with this numeri-cal scheme is that it produces spurious oscillations for anisotropiclocal dispersion and, consequently, its use is currently limited toisotropic dispersion [18,36]. Furthermore, it is computationallyexpensive, with the most demanding step being the search forneighboring particles, although alleviated by searching withinpredened macro-cells as suggested by [37]. Notwithstanding,SPH allows to accurately reproduce sharp changes of soluteconcentration when advection dominates over local dispersion[15]. CPU time is proportional to the product between the numberof particles and the average number of particles per smoothingarea [15].

2.2.4. Streamline-based numerical methodIn an attempt to obtain a Lagrangian scheme with the charac-

teristics of SPH, but a much smaller computational burden andthe capability to handle anisotropic dispersion, [18] proposed atwo-step streamline based (SB) method. As in classical streamlinemethods a Reactive AdvectionDispersion Equation (RADE) (1) isrst written for each species i in a orthogonal local coordinate sys-tem along a streamline [38]:

@ci@t

u @ci@s

r bDrci rin 12where s is the longitudinal curvilinear coordinate measured alongthe streamline and u juj is the local Darcian velocity. Further-more, the tensor of local dispersion bD is set as diagonal [20] andEq. (12) is solved by using an operator splitting algorithm. At therst step of the procedure suggested by [18] the longitudinal com-b

esources 52 (2013) 178189 181ponent of the dispersion tensor is set to D11 DL DT , with all theother components set to zero such as to reduce Eq. (12) to a one-dimensional ADE in the curvilinear coordinate s:

ter R@ci@t

u @ci@s

aL aTu @2ci@s2

13

where the Scheidegger dispersion model (4) was considered [21].The alternative model with spatially constant local dispersion coef-cients can be obtained by replacing u with the effective meanvelocity U in the second right hand term of Eq. (13). U is obtainedby dividing the total ux per unit thickness crossing the down-stream boundary by the product between the width L2 of the com-putational domain and the porosity n. Notice that, for isotropic localdispersion, Eq. (13) reduces to a purely advective transport equa-tion, whose solution with an Eulerian scheme is error prone.

In the second step of the procedure an isotropic dispersionequation is solved, which adds the remaining part of the dispersiveux and the gain/loss of solute mass due to the reaction rate ri:

@ci@t

Dm;i aT u @2ci

@s2 @

2ci@g2

@2ci@f2

! rin

14

where g and f are the two local moving coordinates orthogonal to s.Eq. (14) is written for a three-dimensional ow eld. For a two-dimensional ow eld the term @ci=@f on the right hand side ofthe equation drops out and the concentration depends on the twoorthogonal spatial coordinates s and g. Similarly to Eq. (13) the casewith constant local dispersion can be obtained by replacing u withU in the right hand side of Eq. (14).

The one-dimensional problem (13) can be solved numericallyby one of the several, implicit or explicit, currently available Eule-rian schemes applied to a xed number of nodes positioned alongthe streamline. In the present work, we adopted an explicit TVDscheme suitable for the solution of Eq. (13) with variable-coefcients [39]. The solution of Eq. (14) is obtained by using theSPH scheme presented in Section 2.2.3. Both numerical schemesare explicit in time and the total time variation is therefore ob-tained by simply adding the two variations.

This splitting algorithm enjoys two main advantages with re-spect to SPH: a strong reduction of the computational burdendue to the xed position of the nodes, which eliminates the needto search for the neighbor particles at each time step, and, mostimportantly, a better handling of anisotropic local dispersion.Streamlines are obtained by means of the particle tracking algo-rithm (6) with A u and B 0. The computational nodes alongthe streamlines are chosen at a constant spacing ds, which is setequal to a fraction of the smoothing length h.

The advantage of the strong reduction of the CPU time obtainedby relying on xed particles are in part counterbalanced by the lar-ger memory needed to store their position. Moreover, preliminarysimulations showed that in highly heterogeneous formations, fre-quent ow focusing followed by expansion leads to a non-uniformparticle distribution which reduces the accuracy of the Monte Car-lo integration. To minimize this adverse effect, and also to reducememory occupation, we let the smoothing length to vary in spaceaccording to the following expression: hi 2 1=xi 1=m, where theindex i refers to the particle at the position xi. For the purpose ofspatial integration, numerical Monte Carlo techniques are em-ployed [35,40]; each nodal value is weighted according to the localparticle density.

With the intent to improve further the accuracy of Monte Carlointegration we propose to replace in Eqs. (8) and (10) xi with theinverse of the volume assigned to each particle, as in the classicalSPH implementation [34]. This requires to perform the second stepof the SB procedure by using the following expressions of theconcentration:

182 F. Boso et al. / Advances in Wacxi; t XNpxij1

cxjVxjWxi xj 15and its time variation:

dcxi; tdt

2XNpxij

VxjDijcxj; t cxi; t rijj rijj2 rW rij

16According to this modied numerical scheme at the node xj

along the streamline we assign the volume Vxj Axjds, whereds is the constant spacing of the nodes along the streamline. Thecross-sectional area Axj is computed by imposing mass conserva-tion along the streamline:

Aajuaj Axjuxj 17where u is the local velocity and a is the position from which thestreamline originates. Hereafter, we indicate this modied schemewith SB-FV, to evidence that each node is characterized by a nitevolume, which depends on its position along the streamline. Noticethat the numerical solution of the one-dimensional ADE (14) takesnow explicitly into account the variation of the cross-sectional areaalong the streamline [39, Section 9.1], and is obtained by employingthe TVD Finite Volume method described in Section 2.2.

In Section 4.1 we compare SB and SB-FV in term of global massconservation.

3. Numerical set-up

In order to compare the above numerical schemes under thesame conditions we considered transport of nonreactive and reac-tive solutes in a multi-Gaussian log-conductivity eld with meanhYi and variance r2Y , both constant. Furthermore, we adopted theclassical exponential model of spatial variability CYr1; r2 r2Y expr0, where r0

r21 r22

q=IY is the dimensionless, with re-

spect to the integral scale IY , two-point separation distance withcomponents r1 and r2 in longitudinal and transverse directions,respectively. For the numerical simulations we considered fourtwo-dimensional log-conductivity elds with the same ensemblemean and r2Y 0:2;1;4 and 10, obtained by rescaling a correlatedRandom Space Function (RSF) y with hyi 0 and r2y 1 throughthe following expression: Yx hYi

r2Y

qyx. In doing that,

the resulting four elds share the same spatial structure but showuctuations of different amplitude. The log-conductivity eld yxwas generated by Hydro_Gen [2], an evolution of the sequentialsimulator algorithm of [41].

We conducted two-dimensional numerical simulations oftransport of both non-reactive and reactive solutes, the latter fortwo aqueous species reacting instantaneously upon mixing. Wesolved the ow (2) and Darcys (3) equations by using MODFLOW[28] with harmonic intercell transmissivity in a computational do-main L1 40IY long and L2 20IY large, with very ne square cellsof size IY=20 and permeameter-like boundary conditions of imper-vious longitudinal boundaries at x2 0 and x2 L2 and constantheads at the other two sides (x1 0 and x1 L1) set such as to in-duce a mean head gradient of J 0:03. It has been shown by [42]that with this grid resolution all numerical schemes converge tothe same solution for the ow eld. The resulting velocity eld isthen used as advective component in the solution of the transportEq. (1) with the numerical schemes described in Section 2.2. Sincein all cases the velocity eld is the same, the observed differencesin the numerical solutions can be attributed to the numericalscheme used to solve transport.

In order to better evidence the impact of numerical diffusion ondilution in a rst set of simulations we considered the local disper-

esources 52 (2013) 178189sion spatially constant (uniform) and isotropic, i.e. DL DT D andPeL PeT Pe U IY=D 1000, where Pe is the Pclet number.The hypothesis of spatially constant D is consistent with the

rst-order analysis of ow and transport (see e.g. [43,44]), but itcannot be given for granted in highly heterogeneous formationswhere the model of local dispersion is expected to have a signi-cant impact on dilution. For this reason, successively we comparedthe results obtained with this model with those obtained by usingthe Scheidegger model (4), which postulates a linear dependenceof local dispersion on the local velocity. As discussed in [20] theScheidegger model is able to represent the increase of local disper-sion with the Pclet number of the grain: Peg ud50=Dm, forPeg > 100 where d50 is the mean grain size, observed in columnexperiments.

We performed the simulations for a uniform instantaneousinjection of a passive tracer with a unitary dimensionless concen-tration within the initial volume V0, which in the two-dimensionalset up used in the present work assumes the form of a squared areaof side L IY centered at x 5IY ;10IY. Due to the small horizon-tal transverse dimension of V0 transport develops under non-ergodic conditions. The simulated concentrations z are thereforedimensionless with respect to the initial concentration. The same

F. Boso et al. / Advances in Water Rconcentrations, once rescaled with the following expression:t tin t0z t0, can be substituted into the speciation Eqs. (5)to obtain the concentration of two aqueous species, A and B, react-ing instantaneously upon mixing to produce the precipitate Caccording to the following simplied stoichiometry: A B ! C #.These concentrations are normalized with respect to

Keq

p. The ini-

tial conditions of the nonreactive tracer transport problem in termof the ctitious concentration t are of constant tin c01;in c02;inwithin the initial volume V0 and of constant t0 c01;0 c02;0 in theambient water, with c0i ci=

Keq

pthe values of c0i;in and c

0i;0, with

i 1;2 as shown in Table 1, such that the reaction occurs alwaysin the direction of precipitation [23].

A comprehensive comparison of different numerical schemesrequires that equivalent conditions are established, to be deter-mined according to the objectives of the analysis. Since our aimis to verify to what extent the numerical schemes are affected bynumerical diffusion, we leave aside considerations concerningthe computational burden and compare the numerical schemesin their optimal conguration at similar discretization levels. Inaddition, due to the diversity of numerical schemes involved inthe analysis, establishing a common ground for a fair comparisonin term of computational burden is a difcult task, which would re-quire also recoding the commercial software such as TVD andMOC; this is beyond the scope of the present work. However, someconsiderations concerning the computational burden of the

Table 1Parameters used for the bimolecular reactive transport model: c01;in and c

02;in are the

dimensionless concentrations of the two aqueous species in the water injectedinstantaneously within the volume V0 ; c01;0 and c

02;0 are the dimensionless concen-

trations of the same aqueous species in the ambient water. The initial concentrationsvalues are the same for all the numerical methods. Specic parameters of thenumerical schemes are as follows. SPH: Np is the mean number of particle within thesupport volume of the weighting function W with uniform smoothing lengthh 1:75 dx, where dx is the isotropic mean distance between adjacent particles; SB-FV: hmin and hmax are the minimum and maximum values of the variable smoothinglength; RWPT: Np is the total number of particles and cmin is the minimumconcentration that can be resolved in the numerical simulations.

Numerical scheme Parameter Value

All c01;in; c02;in 10, 0.1

All c01;0; c02;0 0.1, 10

SB-FV hmin=IY 0.025SB-FV hmax=IY 0:28;0:53;1:91;16:8 SPH h=IY 0.05SPH Np 50RWPT Np 4:84 106RWPT cmin=c0 8:26 105numerical schemes will be presented later. Therefore, we opti-mized the parameters of each numerical scheme independentlywithout balancing their computational burden.

In order to compare the numerical schemes at similar levels ofdiscretization, TVD, MOC and RWPT are applied on the same uni-form staggered grid used for the solution of the ow equation(i.e. a square grid with side D IY=20), which is also the smoothinglength we adopted in SPH and the uniform distance ds of adjacentnodes along the streamlines in the SB-FV method. For SB-FV,streamlines are traced starting from the inlet boundary at x1 0at a constant transverse mean distance equal to D IY=20. Further-more, in order to increase particle density where ow is divergent,we added particles in low density areas and traced them forwardand backward such as to increase the density of the nodes whereit was too low. The number or particles used in SPH is xed suchas to obtain convergence of the Cumulative Distribution Function(CDF) of the concentration, which we estimate as the CumulativeFrequency (CF) of the sample composed of all the concentrationsat the positions occupied by the particles. Given the niteness ofthe sample, the CDF can show unphysical discontinuities at highconcentration values. The strategies adopted to control this prob-lem include increasing the number of particles, reducing thesmoothing length in zones with high contrasts in the spatial distri-bution of the concentration [45] and adding an approximate Rie-mann solver (Rusanov ux), as discussed in the work [37]. Wedecided to increase the number of particles to a level above whichadditional particles did not change the empirical concentrationCFD.

Finally,we assigned similar values to the concentrationdetectionlimit of RWPT and the concentration threshold of MOC. The concen-tration convergence limit for TVD and MOC is set equal to 106.

4. Results

Here we present the results of the transport exercises con-ducted by applying the numerical schemes described in Section 2.2to the same set of velocity elds obtained by solving numericallyEq. (3) for r2Y 0:2;1;4 and 10. All the results are presented indimensionless form.

4.1. Nonreactive tracer

In case of an instantaneous injection of a nonreactive tracer thetotal mass of the plume should be conserved at all times t > 0. Amethod is mass conservative if the relative difference of the totalmass computed from the numerical concentration with respectto the initial (injected) mass M0:

eV t Mt M0 =M0 18is zero for all t > 0. The total mass assumes the followingexpression:

Mt ZXncx; tdx 19

where X is the computational domain. The integral (19) is com-puted numerically by using the concentrations provided by thenumerical schemes at the nodal points. For TVD and MOC we useda classical trapezoidal quadrature scheme, while for SPH, SB and SB-FV we used the same Monte Carlo integration scheme adopted inthe solution of the ADE:

Mt hcinX XNi1

ki citnX 20

esources 52 (2013) 178189 183where ki 1=xi for SPH and SB and ki Vxi for SB-FV. This indi-cator was used also by [15] to show that SPH is mass conservative.

whereX is the domain of integration. Furthermore, the trajectory ofthe plumes center of mass Rt is computed as follows:

Rit 1MtZXnxi cx; tdx i 1; . . . ; m 22

An interesting property of RWPT is that the spatial moments(21) are very well approximated by the moments of inertia Sijtof the cloud of points marking the position of the particles at thetime t:

Sijt 1NpXNpk1

Xit; ak Xit

Xjt; ak Xjt

i; j 1; . . . ; m

V

ter RWe considered rst the two streamline-based methods, SB andSB-FV, who share the same conguration of nodal points. In Fig. 1we show eV versus time for both SB and SB-FV and r2Y 1. Becauseof the low accuracy of the post-processing step for the computa-tion of the spatial integrals, SB does not conserve the total masswith a maximum error of 3%. This is because in SB-FV we solve aone-dimensional conservative ADE form, which states solute massconservation along a streamtube with a variable cross-section.Cross-sectional variations are governed by the deterministic vol-umes assigned to each node. Then further dispersive solute massexchanges are handled by SPH, which is inherently mass-conservative.

Fig. 2 shows eV versus the dimensionless time t0 tU=IY for allthe models analyzed in the present work, except SB which is re-placed by SB-FV in all the considerations that follow. For a correctinterpretation of this gure one should notice that RWPT is strictlyconservative, since the initial massM0 is equally subdivided amongNp particles which conserve the mass received while traveling inthe computational domain, such that mass conservation impliesnding all the released particles within the computational domain.As shown by [15], and conrmed here, mass conservation holds forSPH simulations as well, since mass exchange within the smooth-ing volume H is inherently mass conservative. Mass is also con-served for TVD which, however, shows the signs of an earlyreduction of the mass within the computational domain due tooutow from the downstream boundary, associated to anticipated,with respect to the other methods, early arrivals. Finally, Fig. 2shows that MOC is not mass conservative, with the total amount

Fig. 1. Relative mass error eV for SB and SB-FV for the r2Y 1 eld.

184 F. Boso et al. / Advances in Waof mass within the computational domain showing oscillations ofincreasing amplitude with r2Y . This may be due to successiveremapping of particles, which is needed in order to maintain anadequate resolution of concentration, especially across the edges[14]. While the global mass balance error of MOC is relativelylow for r2Y 0:2 (i.e., 0.3%), it increases to the remarkable valueof 5% for r2Y 10. This value should be considered large becauserelated to a global quantity that it is expected to remain stableand casts some doubts on the accuracy of this numerical schemefor high variance simulations.

We consider now the time evolution of the spatial second ordermoments, which are good descriptors of the plumes deformation(spreading) and are sensitive, though to a lesser extent with re-spect to other indicators, to numerical diffusion. Notice that for thisreason the second-order plume moments are not good indicatorsof dilution. The second order spatial plume moments assume thefollowing general expressions:

Sijt 1MtZXncx; t xi Rixj Rjdx i; j 1; . . . ; m 210 5 10 15-0.008

-0.004

0.000

0.004

0.008

V(

t)

(d) Y2=10(c) Y

2=4

(b) Y2=1(a) Y

2=0.2t=tU/I Y

SPH TVD MOC RWPT SB-FV

0 5 10 15-0.008

-0.004

0.000

0.004

0.008

V(t

)

t=tU/I Y

0 5 10 15-0.008

-0.004

0.000

0.004

0.008

V(t

)

t=tU/IY0 5 10

-0.050

-0.025

0.000

0.025

0.050

V(t

)

t=tU/IY

Fig. 2. Relative mass error eV for: (a) r2Y 0:2; (b) r2Y 1; (c) r2Y 4; (d) r2Y 10.In all cases e 0 indicates that the mass is conserved.

esources 52 (2013) 17818923with

Xit 1NpXNpk1

Xit; ak i 1; . . . ; m 24

where Xit;ak is the ith component of the position of the particle kthat originated at time t 0 at the position ak within the source V0.It can be shown, in fact, that for Np !1, Sij converges to the exactSij given by Eq. (21). The moments computed through the expres-sions (23) and (24) are free of numerical diffusion because RWPTdoes not resort to a discretization of the advective term of Eq. (1).

In the simulations conducted with RWPT we used a large num-ber of particles (Table 1), which were enough to stabilize both S11and S22 at all times considered in the simulations. The plume mo-ments obtained by RWPT are therefore assumed as benchmarksince they reach the maximum accuracy permitted by thediscretization of the velocity eld and are not affected by numeri-cal diffusion. The second order moments S11 and S22, obtained bysimulating transport of a nonreactive tracer with the ve

15

22

15

F. Boso et al. / Advances in W ter R0 5 10 150

4

8

12

16S 1

1/Y2

I Y2

SPH TVD MOC RWPT SB-FV Pe=Inf

0 5 100.0

2.5

5.0

7.5

10.0

0 5 10 150.0

0.3

0.6

0.9

1.2

(b) Y =1S 2

2/Y2

I Y2

(a) Y =0.2

0 5 100.00

0.15

0.30

0.45

0.60numerical schemes described in Section 2.2 are shown in Fig. 3The red1 dotted lines depict the moments obtained by simulatingtransport with RWPT and in the absence of local dispersion. We ob-serve that the effect of local dispersion on the spatial moments isgenerally weak, although larger in transverse than in longitudinadirection and when r2Y is high. As shown in Fig. 3 both S11 and S22show a pulsating behavior, which indicates that ergodicity is not at-tained. The relative difference between second-order spatial mo-ments, in particular S22, with respect to the benchmark of RWPTprovides a rough estimate of numerical diffusion affecting thenumerical scheme. For a weakly heterogeneous formation(r2Y 0:2) all the numerical schemes produce nearly the same S11with the largest relative difference with respect to RWPT of 3% forSB-FV. The largest relative difference is very small (1.5%) also forS22 obtained by using the TVD method. The relative differences re-main generally low, except for the TVD method, up to r2Y 4although for a given r2Y it is larger for S22 than for S11. For r2Y 10larger relative differences are observed in particular for S22. Fig. 3shows that TVD overestimates both spatial moments, while SB-FVunderestimates them, though to a much lesser extent and only forr2Y 10, while for smaller variances the moments obtained by SB-FV does not differ signicantly from those obtained by RWPT. Similarresults have been obtained when local dispersion is let to vary line-arly with the local velocity.

A frequently usedmeasure of mixing is the dilution index, intro-duced by [46] in analogy with information theory, to quantify theamount of entropy, or disorder, in a system,which in our case is rep-resented by the distribution in space of the solute mass. For a set of

t=tU/IY t=tU/IYFig. 3. Longitudinal (upper row) and transverse (lower row) second central plumemr2Y 4; (d) r2Y 10. A combination of color and line type is used to distinguish th

1 For interpretation of color in Fig. 3, the reader is referred to the web version othis article.a.

l

,

omene num

f0 5 10 150

1

2

3

4

0 5 100.0

0.3

0.6

0.9

1.2(d) Y

2=10(c) Y2=4

0 5 10 150.0

0.2

0.4

0.6

0.8

0 5 100.00

0.07

0.14

0.21

0.28

esources 52 (2013) 178189 185discrete concentration data, i.e. concentrations dened over a givensupport volume DV , which in our case coincides with the volume ofthe cell (for TVD, MOC and RWPT) or of the particle (for SPH and SB-FV), the dilution index assumes the following expression [46]:

Et DV exp XNk1

Pk ln Pk

!25

where Pk cxk; t=Mt and N is the sample dimension, i.e. thenumber of concentration data. In Eq. (25) we consider only thenodes whose dimensionless concentration is larger than the follow-ing threshold: zlim clim=c0 103. The dilution index is a measureof the volume occupied by the solute, whose evolution depends onthe nonlinear interplay between local dispersion and concentrationgradients [46], both controlled by the spatial distribution of thevelocity eld. Consequently, the numerical diffusion introducedby approximating the advective term in Eq. (1) causes local varia-tions of the modeled dilution, which affect the dilution index aswell [47].

Fig. 4 compares the evolution of Et for r2Y ranging from 0.2 to10. As expected, the dilution index increases more than linearlywith time showing in all cases larger values when transport is sim-ulated by TVD. On the other hand, SPH and RWPT provide verysimilar results, regardless of the heterogeneity level. In all casesno appreciable deviations are observed among the dilution indexcomputed with MOC and those obtained with SPH or RWPT, de-spite larger differences in particular at high values of r2Y canbe observed in terms of second moments (Fig. 3) and global massconservation (Fig. 2). This may be due to the relatively smaller sen-sitivity of the dilution index to high and intermediate concentra-tion values, which have been shown to be negatively affected bythe dynamic remapping of particles in MOC [14]. The dilution in-

t=tU/IY t=tU/IY

ts as a function of the dimensionless time t0 tU=IY for: (a) r2Y 0:2; (b) r2Y 1; (c)erical schemes.

ter Rdex computed with SB-FV is slightly larger than for SPH and the

0 5 10 150

1

2

3

4

t=tU/IY

E(t)

/I Y2

SPH TVD MOC RWPT SB-FV

0 5 10 150

2

4

6

8

t=tU/IY E

(t)/I

Y2

(d) Y2=10

(b) Y2=1

(c) Y2=4

(a) Y2=0.2

0 5 10 150

4

8

12

16

E(t)

/I Y2

t=tU/IY0 5 10

0

5

10

15

20

E(t)

/I Y2

t=tU/IY

Fig. 4. Dilution index of the passive tracer plume obtained by using the venumerical schemes investigated in the present work and for (a) r2Y 0:2, (b) r2Y 1,(c) r2Y 4 and (d) r2Y 10.

186 F. Boso et al. / Advances in Waother numerical schemes at early times, but becomes smaller atintermediate and large times. Consequently, except at early timeswhen the difference is negligible, SB-FV underestimates E with re-spect to SPH, RWPT and MOC, to an amount that increases with r2Y .At all r2Y values considered in the present study the discrepancywith the reference values of E provided by RWPT is however lessfor SB-FV than for TVD. This is in agreement with the results shownin Figs. 2 and 3 and conrms that TVD introduces a signicantamount of numerical diffusion, while SB-FV slightly underesti-mates dilution. The performance of SB-FV at large r2Y values is pos-sibly penalized by the difculty to compute correctly the volume ofthe nodes when velocity elds are highly non-uniform. Overall,very small differences are observed in the behavior of the dilutionindex obtained with SPH, RWPT and MOC, which minimize the im-pact of numerical diffusion with respect to TVD, in particular atmoderate to high heterogeneity. On the other hand, the much lar-ger values of E shown in Fig. 4 suggests that TVD is unable to repro-duce accurately dilution for moderately to high heterogeneousformations (i.e., r2Y P 1). However, all the numerical schemesreproduce accurately E in weakly heterogeneous formations.

Fig. 5 shows the dilution index obtained by solving the ADE (Eq.(1) for i 1 and r1 0) with SPH considering two different disper-sion models: dispersion coefcients spatially constant (model A)and linearly dependent on the local velocity according to the Sche-idegger model (4) (model B). The difference between the two mod-els of local dispersion is small to negligible for r2Y 0:2 and r2Y 1,but increases with r2Y and time becoming large in highly heteroge-neous formations, i.e. for r2Y 4 and 10. The impact of the localdispersion model on dilution is evident at large r2Y as an effect ofthe pronounced variability of the velocity eld, which only in themodel B feedbacks to the spatial variability of local dispersion. Ina heterogeneous velocity eld the introduction of velocity-depen-dent local dispersion coefcient has contrasting effects on theplume: in high conductivity zones dilution is enhanced with re-spect to model A, while it is reduced in low conductivity zones.

0 5 10 150

1

2

3

4

t=tU/IY

E(t)

/I Y2

0 5 10 150.0

1.5

3.0

4.5

6.0

E(t)

/I Y2

t=tU/IY

0 5 10 150

3

6

9

12

E(t)

/I Y2

t=tU/IY model A model B

0 5 100

4

8

12

16

(d) Y2=10

(b) Y2=1

(c) Y2=4

E(t)

/I Y2

t=tU/IY

(a) Y2=0.2

Fig. 5. Comparison between the dilution index of the nonreactive tracer plume forlocal dispersion spatially uniform (model A) and depending linearly on the velocity(model B), for: (a) r2Y 0:2, (b) r2Y 1, (c) r2Y 4 and (d) r2Y 10. All the numericalsimulations have been conducted with SPH.

esources 52 (2013) 178189In non-ergodic conditions, these two counteracting effects areunbalanced at early times and the predominance of one of thetwo depends on where the plume is injected. If, as in the simula-tions used to produce Fig. 5, solute mass is predominantly injectedinto lower-than-average Y regions, the second effect dominates atearly times, thus leading to larger dilution for model A. However,successively the situation reverses: the larger the plume, the fasterE grows when subject to non-uniform dispersion (model B), be-cause of the tendency of the solute to enter and travel along highconductivity pathways with larger local dispersion. In this casethe values of E obtained with model B become larger than the esti-mates obtained employing model A at decreasing times as r2Y in-creases (i.e., in Fig. 5 the transition from one situation to theother occurs at t0 15; 0:7 and 0:3 for r2Y 1; 4 and 10, respec-tively). Conversely, for large ergodic plumes, solute injection is bal-anced between low and high conductivity zones and the initialeffect will disappear with E growing faster for model B than formodel A, since early times. Similar simulations have been con-ducted with the other numerical schemes, with analogous resultsprovided by RWPT, and by MOC and SB-FV for r2Y up to four. Sim-ilarly to what showed previously for the model A of local disper-sion SB-FV underestimates E for r2Y > 4, whereas MOC isnegatively affected by mass conservation errors due to the overallstronger dilution experienced by the plume. TVD simulations showvery small difference between the two models of local dispersion,except for r2Y 10, when however the difference is moderate. Thisshows that TVD, in the widely available implementation ofMT3DMS, is negatively impacted by articial diffusion, whichmasks the effect of a different model of local dispersion on dilution.

Finally we use E as an indicator of the overall performance ofthe numerical schemes at different levels of discretization(Fig. 6). We coarsen the cell size from D IY=20 (Fine), used inthe previous simulations, to D IY=10 (Intermediate) andD IY=2 (Coarse). All simulations are performed with r2Y 1 and

essary if we were interested in Break Through Curves only, forexample. Notice that a normalized CPU time per particle wouldnot make sense here because of the different physical meaningand density of the particles. In RWPT particle density is propor-tional to concentration, since particles represent solute mass, whilein SPH particle density is proportional to uid density which israther homogeneous throughout the domain, regardless of solutemass dissolved in the water at least up to the point in which soluteconcentration changes water density.

Second, we comparatively analyze the normalized CPU timesper timestep and cell/volume for TVD, MOC and SB-FV. TVD is ta-ken as benchmark. The normalized CPU time (with respect toTVD) is rather low for MOC (0.6 and 0.7 for the intermediate andne renement level, respectively). However, this is the effect ofthe small injection volume, which reduces the amount of injectedparticles and thus the total CPU time. A larger source results in lar-ger memory requirements, possibly reaching the limitations ofstandard PCs, accompanied by a larger CPU time, reversing the ra-tio with respect to TVD [14]. The normalized CPU times are 1.1 forboth Fine and Intermediate renements of SB-FV. Notice that runtimes for MT3DMS increase up to 10 times when model B of localdispersion is used instead of model A.

4.2. Reactive transport

ter Resources 52 (2013) 178189 1870 5 10 150

2

4

6

8

E(t)

/I Y2

t=tU/IY0 5 10 15

0

2

4

6

8

E(t)

/I Y2

t=tU/IY SPH TVD MOC RWPT SB-FV

0 5 10 150

2

4

6

8

t=tU/IYE(

t)/I Y

2

(b) INTERMEDIATE (c) FINE(a) COARSE

Fig. 6. Dilution Index Et0 for the different numerical schemes at differentrenement levels: (a) coarse, (b) intermediate and (c) ne. The ow eld is thesame for each renement level, and is computed on a log-transmissivity eld withunitary variance.

Table 2Parameters for the transport simulations for the different numerical schemes, withthree levels of renement: Coarse (D IY=2), Intermediate (D IY=10) and Fine(D IY=20). D represents the cell size for the mesh-based method, the mesh size forRWPT, the smoothing length h for SPH and the mean particle distance along thestreamline and among streamlines for SB-FV.

F. Boso et al. / Advances in Wathe Lagrangian methods are applied by reducing proportionally thenumber of particles (see Table 2). All the other parameters aremaintained as in the ne renement level simulations.

The resulting variance of the velocity eld decreases withincreasing cell size. For an accurate evaluation of the impact ofthe grid size on the statistics of the velocity eld we refer to thework by [42]. Here we focus on the overall effect of grids coarsen-ing on dilution index. We observe that increasing the cell size fromD IY=20 to IY=10 does not cause signicant changes in the dilu-tion index provided by all the methods, except TVD, which is muchmore sensitive to grid coarsening than the other methods. At thecoarsest level we observe that TVD grossly overestimates dilution,whereas RWPT and MOC show a less severe overestimation accom-panied by unphysical oscillations due to the reduced number ofparticles used to discretize the solute mass. On the other hand,SPH shows the tendency to reduce E as the reference length getslarger as an effect of the increased distance from the particles,which limits mass exchange, whereas SB-FV is affected by articialdiffusion which is large along the streamline because of a largeinternodal space.

For the intermediate and ne renement cases, we also com-pare the computational time, separately for the Lagrangian andthe other methods. Considering rst the Lagrangian methods weobserve that for RWPT the CPU time increases almost linearly withNp [15]. The ratio between the CPU time of SPH and RWPT rangesbetween 0.6 and 0.7 for the intermediate and the ne renementlevel, respectively. For a fair comparison, simulations are run inthe same computational conditions and for the same time-step,that is, concentration mapping for RWPT is performed at eachtime-step. Notice that this post-processing step would not be nec-

Scheme Parameter Coarse Interm. Fine

All D IY=2 IY=10 IY=20SPH h=IY 1=2 1=10 1=20SPH Np 50 50 50SB-FV hmin=IY 0.25 0.05 0.025SB-FV hmax=IY 1.75 0.6 0.53RWPT Np 4:84 104 1:21 106 4:84 106We consider here the case of an instantaneous injection ofwater containing the species A in an ambient water containingthe species B; the two aqueous species react instantaneously uponmixing. This case is approximated by considering an injectionwater with a very small concentration of the species B in equilib-rium with the species A, which consequently has a high concentra-tion, while the opposite is true for the ambient water, as shown inTable 1. We need to resort to this approximation because the equi-librium reaction requires that the product of the concentration ofthe two aqueous species remains locally constant.

Fig. 7 shows the dilution index computed for the species A,whose concentration within the plume reduces as a combined ef-

0 5 10 150

1

2

3

4

t=tU/IY

E(t)

/I Y2

SPH TVD MOC RWPT SB-FV

0 5 10 150

3

6

9

12

E(t)

/I Y2

t=tU/IY

(d) Y2=10

(b) Y2=1

(c) Y2=4

(a) Y2=0.2

0 5 10 150

7

14

21

28

E(t)

/I Y2

t=tU/IY0 5 10

0

12

24

36

48

E(t)

/I Y2

t=tU/IYFig. 7. Dilution index computed for the reacting species A and for (a) r2Y 0:2, (b)r2Y 1, (c) r2Y 4 and (d) r2Y 10.

0.8jter Rfect of dilution and reaction with the species B, which is moreabundant in the ambient water. At early times the dilution indexremains nearly constant with the decay of the mass of the speciesA that compensates for the increase of E due to the dispersive ux.At later times, when the rate of consumption of the two reactingspecies reduces as shown in Fig. 8, dilution dominates over massconsumption and the dilution index increases with time. Like inthe nonreactive case, TVD overestimates dilution to an extent thatis small to negligible in weakly heterogeneous formations, but be-

(d) Y2=10(c) Y

2=4

0 5 10 150.0

0.2

0.4

0.6

m1,

c/m1,

in

t=tU/IY0 5 10

0.0

0.2

0.4

0.6

0.8

m1,

c/m1,

inj

t=tU/IY

Fig. 8. Relative fraction of mass of species A which is consumed by the reaction as afunction of time for (a) r2Y 0:2, (b) r2Y 1, (c) r2Y 4 and (d) r2Y 10.0 5 10 150.0

0.2

0.4

0.6

0.8

1.0

(b) Y2=1t=tU/IY(a) Y

2=0.2

SPH TVD MOC RWPT SB-FV

m1,

c/m1,

inj

0 5 10 150.0

0.2

0.4

0.6

0.8

1.0

m1,

c/m1,

inj

t=tU/IY1.0 1.0

188 F. Boso et al. / Advances in Wacomes relevant for r2Y > 1.Finally, Fig. 8 shows the total mass of the species A consumed

by the reaction, normalized with respect to the mass of the speciesA injected into V0. Coherently with the previously discussed re-sults, when transport is modeled with TVD the injected mass ofthe species A is consumed more quickly than for the other numer-ical schemes. As expected the largest difference is observed forr2Y 10, when also SB-FV diverges from the prediction of the othernumerical schemes, but to a lesser extent. In general, the consump-tion rate is initially fast and reduces with time until most of the in-jected mass is consumed. This asymptotic state is reached at largertimes for weakly heterogeneous formations, as a result of a lowerconsumption rate due to slower mixing.

5. Conclusions

We compared the accuracy of ve numerical schemes in solvingthe transport equation of non-reactive and reactive solutes in het-erogeneous formations with increasing levels of heterogeneity. Weconsidered the following numerical schemes: the Eulerian TVD Fi-nite Volume method, the EulerianLagrangian Streamline-BasedSB-FV and Characteristics (MOC) methods, and nally the Lagrang-ian Random Walk Particle Tracking (RWPT) and Smoothed ParticleHydrodynamics (SPH) methods. For TVD and MOC we used thewidely used MT3DMS implementation [14], while the other meth-ods have been implemented specically for the present work. Weanalyzed the numerical diffusion that these methods introduce inthe discretization of the advective term, because of its negative im-pact on dilution and mixing-controlled reactions. We modied alsothe standard SB algorithm developed by Herrera et al. [18] to in-crease accuracy in the Monte Carlo estimates of the spatialintegration.

A reliable, though not particularly sensitive, benchmark forevaluating numerical diffusion are the longitudinal and transversespatial central moments evaluated by simulating transport withRWPT, because they are directly computed as inertial momentsof the particle cloud and thereby are not affected by numerical dif-fusion. However, spatial moments are mostly sensitive to thespreading of the plume as an effect of the non-uniformity of thevelocity eld, though local dispersion inuences these moments,especially in the transverse direction. Both SPH and MOC matchthese spatial moments, thanks to the accurate reproduction of par-ticle trajectories and the zero or small amount of numerical diffu-sion. On the contrary, TVD and SB-FV show larger differences,which emerge also at low heterogeneity levels for the EulerianTVD method. We considered also several quantities differently re-lated to dilution and mixing in heterogeneous formations: the dilu-tion index for both the conservative and the reactive species andthe total consumed mass of the injected reactive species.

All indicators show that TVD overestimates dilution, to an ex-tent that increases with the formations heterogeneity. SB-FV pro-vides different estimates of all the considered parameters for thehighest heterogeneity case (r2Y 10), generally underestimatingdilution as an effect of the non-uniform distribution of particlesused in the Monte Carlo integration of the diffusive term. However,SB-FV is in a very good agreement with SPH and RWPT for r2Y 6 4and within this range it can be considered as a valid alternative toSPH when CPU time is an issue, as for example for large three-dimensional or Monte Carlo simulations. Similarly, MOC showssignicant mass balance errors at high heterogeneity, but can bea valid alternative to SPH and the other Lagrangian methods atlow heterogeneity. In case of highly heterogeneous formationsRWPT may be a good choice, but limited to conservative transportor simple reactions, due to the difculty encountered in imple-menting reactions in this purely Lagrangian scheme. Moreover, lo-cal concentration values may be unreliable because of localconcentration uctuations. More complex reaction terms can behandled by SPH, which is in turn limited to isotropic dispersiontensor models [18].

Finally, we considered how the model of local dispersion inu-ences dilution in both weakly and highly heterogeneous forma-tions. Considering local dispersion constant or linear dependenton the local velocity does not make difference in weakly heteroge-neous formations, conrming what suggested by classic results ofrst order analysis in r2Y [43]. However, as heterogeneity increases,the two models show large differences in the dilution index, indi-cating that the dependence of the local dispersion on the localvelocity cannot be neglected for r2Y > 1. The dilution index forone of two reacting species undergoing instantaneous bimolecularreaction and the rate a mass consumption of the same species con-rmed these conclusions.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.advwatres.2012.08.006.

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Numerical simulations of solute transport in highly heterogeneous formations: A comparison of alternative numerical schemes1 Introduction2 Mathematical model and numerical schemes2.1 Mathematical model2.2 Numerical schemes2.2.1 TVD and MOC schemes2.2.2 Random Walk Particle Tracking (RWPT)2.2.3 Smoothed Particle Hydrodynamics2.2.4 Streamline-based numerical method

3 Numerical set-up4 Results4.1 Nonreactive tracer4.2 Reactive transport

5 ConclusionsAppendix A Supplementary dataReferences