Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol22.Issue7.1999:

Numerical methods for reactive transport on

rectangular and streamline-oriented grids

Olaf A. Cirpka

a,c,*

_{, Emil O. Frind}

b

& Rainer Helmig

a,d a

Institut fur Wasserbau, Universitat Stuttgart, Germany b

Department of Earth Sciences, University of Waterloo, Waterloo, ON N2L 3G1, Canada

c

Department of Civil and Environmental Engineering, Stanford University, Stanford, CA 94305-4020, USA

d

Institut fur Computeranwendungen im Bauingenieurwesen, Technische Universit at Braunschweig, Pockelsstr. 3, 38106 Braunschweig,

Germany

(Received 1 October 1997; revised 1 August 1998; accepted 13 November 1998)

Coupling advection-dominated transport to reactive processes leads to additional requirements and limitations for numerical simulation beyond those for non-re-active transport. Particularly, both monotonicity avoiding the occurence of negative concentrations, and high-order accuracy suppressing arti®cial di€usion, are necessary to study accurately the reactive interactions of compounds trans-ported in groundwater. These requirements are met by non-linear Eulerian methods. Two cell-centered Finite Volume schemes are presented for the simu-lation of advection-dominated reactive transport. The ®rst scheme is based on rectangular grids, whereas the second scheme requires streamline-oriented grids the generation of which is explained in an accompanying paper. Although excellent results for conservative transport are obtained by the scheme for rect-angular grids, some arti®cial transverse mixing occurs in the case of multi-com-ponent transport. This may lead to erroneous reaction rates if the compounds interact. The transport scheme for streamline-oriented grids, on the other hand, avoids arti®cial transverse mixing. A quantitative comparison is given by two test cases. A conservative tracer simulation for a ®ve-spot con®guration in a hetero-geneous aquifer shows a high coincidence of the breakthrough curves obtained for the two methods, whereas a test case of two reacting compounds shows sig-ni®cant di€erences. In this test case, a rate of convergence with respect to the overall reaction rates lower than ®rst-order is calculated for the rectangular grid. Ó _{1999 Elsevier Science Ltd. All rights reserved}

Key words: Transport modeling, Grid orientation, Arti®cial di€usion, Finite

volume schemes.

1 INTRODUCTION

In groundwater systems, dissolved compounds undergo advective-dispersive transport, mass-transfer processes and chemical transformations, both biotic and abiotic. Numerous studies have appeared in the literature in recent years investigating the interactions between these processes by experimental as well as numerical means. These studies have greatly enhanced the understanding

of the fate and behavior of contaminants in the sub-surface, the assessment of related risks and the predic-tion of the future evolupredic-tion of contaminapredic-tion scenarios. The occurence of chemical interactions requires mixing of the reacting compounds on the local scale.7 Dependent on boundary conditions and the mass-transfer properties of the reactants, unsucient mixing may lead to a strong limitation of the chemical reactions compared to idealized mixed reactors. Hence, the ac-curate determination of local-scale mixing processes is a key issue for the investigation on the interactions be-tween advective±dispersive transport and reactive pro-cesses.

Ó1999 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0309-1708/99/$ ± see front matter PII: S 0 3 0 9 - 1 7 0 8 ( 9 8 ) 0 0 0 5 1 - 7

*_{Corresponding author. E-mail:} ocirpka@hydro2.stan-ford.edu

In this context, numerical simulation of multi-com-ponent reactive transport is a powerful tool for coupling reactive processes, studied in detail in lab-scale experi-ments, to hydrogeological mechanisms, the parameters of which are generally retrieved by ®eld-scale experi-ments. This provides the opportunity for comparing reactive behavior on di€erent scales. In order to achieve reliable results, it must be ensured that the mixing of the compounds approximated by the numerical methods chosen re¯ects the hydrogeologic parameters rather than numerical errors such as arti®cial di€usion. Therefore accurate methods for the simulation of reactive trans-port are needed.

One of the fundamental requirements for the accurate simulation of reactive transport is that numerical oscil-lations are not acceptable because they result in negative concentrations which lead to unstabilities in the calcu-lation of reactive processes9. This is in contrast to non-reactive transport simulations where small oscillations are unproblematic. Small oscillations typically occur when linear high-order methods are applied to advec-tion-dominated transport problems. It follows that a high order of accuracy alone does not guarantee the correct and stable solution of reactive transport prob-lems. Conversely, linear low-order methods may intro-duce signi®cant arti®cial di€usion thus leading to an overestimation of local-scale mixing processes. There-fore, traditional linear transport schemes are not good choices for simulating advection-dominated transport of interacting compounds.

In this paper two numerical methods are presented for the solution of such problems. The ®rst method is based on the¯ux-corrected transport(FCT) scheme3and formulated for rectangular grids. It may be extended to curvilinear grids by coordinate transformation, but it does not require any special orientation of the grid. The second scheme is based on theslope limiteror MUSCL approach32and requires streamline-oriented grids which are discussed in an accompanying paper8.

The paper is organized as follows: A brief review of existing FVM schemes for advection-dominated trans-port is given in Section 2. In Section 3 the general ap-proach underlying both schemes is explained. In Section 4 the two numerical schemes for conservative transport are described. Some aspects of solving the reactive sub-problem and its coupling to advective-dis-persive transport are discussed in Section 5. Finally the two schemes are compared by application to two test cases for conservative and mixing-controlled reactive transport in Section 6.

2 REVIEW OF FINITE VOLUME METHODS FOR ADVECTION-DOMINATED TRANSPORT

Most numerical schemes for multi-component reactive transport are combinations of existing numerical

meth-ods for conservative transport and for reactions in mixed systems, respectively. Since the equations de-scribing reactive processes are based on concentrations, the most common way to solve these equations for spatially variable domains is to divide the domain into control volumes and calculate the reactive processes independently in each of them. It is a natural choice to use the same spatial discretization for transport, that is applying a Finite Volume Method (FVM). This has been done by various authors18,17 and is adopted in the present study. As an alternative, the Finite Element Method (FEM) may be used for transport thus requir-ing the simulation of reactive processes on a nodal ba-sis.20,33 Strictly Lagrangian12 and Eulerian±Lagrangian schemes34have also been used for reactive transport, but most of these schemes included spatial redistribution of the calculated masses after solution of the transport sub-problem. This spatial redistribution causes undesirable arti®cial di€usion. In most studies the extent of arti®cial di€usion has not been determined.

Accurate modeling of multi-component reactive transport requires the underlying transport scheme to be neither oscillative nor arti®cially di€usive. Oscillations may cause negative concentrations which do not re¯ect any physical behavior. Furthermore, common equations for the description of (bio)reactive processes such as the Michaelis±Menten terms are discontinuous in the neg-ative concentration range. As a consequence, negneg-ative concentrations caused by spurious oscillations may lead to serious stability problems. Oscillations are suppressed if the transport scheme preserves the monotonicity of the concentration distribution.

On the other hand, reactive interactions of com-pounds may be limited by insu€ucient mixing of the compounds. Arti®cial di€usion introduced by the scheme for advective transport may lead to an overes-timation of mixing and related reactions rates. Therefore arti®cial di€usion must be minimized. Since longitudinal mixing is often much stronger than transverse, arti®cial di€usion may be tolerated to a higher extent for the longitudinal than for the transverse direction. It is well known that linear, monotonic Eulerian schemes are at most ®rst-order accurate15and therefore di€usive. As a consequence, these schemes are not well suited to reac-tive transport problems sensireac-tive to di€usive mixing.

The requirement of both monotonicity and second-order accuracy has led to the development of non-linear FVM schemes refered to as total variation diminishing

(TVD) or high resolution methods22. Some of these schemes are based on the reconstruction of concentra-tion distribuconcentra-tions within cells such as in the case of the

and ¯ux-limiter approaches, respectively, are identical. They are dependent on the concentration distribution: in smooth regions no limitation of the Lax±Wendro€ ¯uxes is necessary, whereas near discontinuities the ¯uxes approach those achieved by upstream di€erenti-ation. Le Veque21 interpreted the limited anti-di€usive ¯uxes as correction waves. In the present study, the slope-limiter approach is applied in the scheme on streamline-oriented grids (Section 4.2).

The ®rst non-linear TVD methods were formulated for explicit time-integration and applied only to one-dimensional domains. Unfortunately, the extension to multi-dimensional applications was not straightforward. In ®rst approaches multi-dimensional problems were solved by directional splitting.22 As an alternative the TVD schemes could be written in a semi-discrete form. Higher-order results were achieved either by Crank± Nicolson integration in time thus requiring linearization or by explicit multi-step methods such as the Runge± Kutta scheme. The latter was applied e.g. in the multi-dimensional ENO scheme of Casper and Atkins.6

LeVeque24,23 presented an explicit multi-dimensional extension of the correction-wave approach. However, the limiter function he applied did not include the in-¯uences of diagonally positioned cells and could not guarantee monotonicity.

A very attractive alternative to the introduction of non-linear limiter functions is the Flux-Corrected Transport (FCT) approach3,35. The FCT method has been developed by Boris and Book3and was one of the ®rst high-order monotonic methods. The ®rst extension to multi-dimensions was done by Zalesak.35In contrast to other high-resolution methods the scheme could be transfered to FEM schemes as well.26,25 In the present study a version of the FCT method is applied in the scheme on rectangular grids (Section 4.1).

Note that all of the above-mentioned schemes apply exclusively for the stabilization of advective transport. It was supposed that dispersive transport always leads to additional stabilization. However, this is only the case if the dispersion tensor is a diagonal matrix. As will be shown in Section 4.1 oscillations due to dispersive transport occurs for full dispersion tensors the principal directions of which di€er from those of the grid.

3 GENERAL APPROACH

In the following, two di€erent cell-centered FVM schemes for two-dimensional modeling of transport will be presented, the ®rst of which is used for calculations on rectangular grids with varying grid spacings, whereas for the second scheme streamline-oriented grids con-sisting of quadrilateral elements are used. For both transport schemes the underlying ¯ow-®eld is solved by the mixed-hybrid FEM. The method for streamline-oriented grid-generation is explained by Cirpkaet al.8

In the current setup of both schemes the reactive transport problem is solved by an operator-split ap-proach. Advective transport is solved for each com-pound by a non-linear explicit method which is followed by implicit calculation of di€usive/dispersive transport:

~

in whichkdetermines the cell of interest andmdenotes all cells connected to cell kvia the advective ¯uxes Fa

k;m and the di€usive ¯uxes Fd

k;m, respectively. Note that these ¯uxes are expressed as total mass transfered over the entire timestep, the outward direction being posi-tive.~ck…t‡Dt†is the intermediate concentration at time t‡Dt considering exclusively advective-dispersive transport. Eqn. 1 is solved for each mobile compound. Reactive processes are solved sequentially in a separate step without implicit feedback to the transport calcu-lation: reactive source/sink terms which may contain non-linear terms. Theoperator-splitcoupling has been compared by the authors to implicit coupling methods including di-rect coupling and iterative two-step coupling.9No sig-ni®cant di€erences were found. The modi®cations which are necessary for the implementation of implicit cou-pling schemes will be discussed in Section 5.

4 CALCULATION OF CONSERVATIVE TRANS-PORT

The main di€erences for numerical simulation of transport on streamline-oriented grids compared to ar-bitrarily oriented ones may be explained by the trans-port equation of a reacting compound which is given for two dimensions in streamline coordinates by eqn (3) and in arbitrarily oriented coordinates by eqn (4):

/Rioci

vanishes in the numerical methods for transport because of the operator-split approach,/is the porosity,~qis the speci®c discharge vector with its directional components qx and qy and its absolute value q, n is the spatial co-ordinate in the direction of ¯ow andgtransverse to it,al andat are the longitudinal and transverse dispersivities, respectively, and Dm is the molecular di€usion

coe-cient. The full dispersion tensorDin thex;y-coordinate system can be evaluated by transformation of coordi-nates:

Comparing eqn (3) with eqn (4), it is obvious that advective transport is quasi-one-dimensional on streamline-oriented grids whereas advective ¯uxes occur into both principal directions of arbitrarily oriented grids. This is illustrated in Fig. 1. Therefore using streamline-oriented grids simpli®es the stabilization of advective transport, since one-dimensional approaches can directly be applied. In the scheme presented this is theslope limitermethod32.

Second, the dispersion tensor is diagonal in stream-line coordinates whereas it is a full tensor in arbitrarily oriented. As a consequence, no cross-di€usion terms need to be evaluated on streamline-oriented grids thus guaranteeing second-order accuracy for two-dimen-sional di€erentiation by a ®ve-point stencil. In contrast to this, nine-point di€erentiation is required to approx-imate the ¯uxes related to the full dispersion tensor on arbitrarily oriented grids. This is illustrated in Fig. 2.

As will be shown, adopting streamline-oriented grids does not only simplify the discretization of advective± dispersive transport but also avoids adding arti®cial transverse di€usion which may be necessary in order to achieve monotonicity on an arbitrarily oriented grid. This is of particular interest for the transport simulation of interacting compounds in which reaction rates are dependent on mixing of the compounds. On the other hand, the streamline-oriented grid-generation scheme8is restricted to steady-state ¯ow®elds, and the extension to

three-dimensional applications may be rather compli-cated.

4.1 Calculation of transport on a rectangular grid

For the calculation of transport on a rectangular grid theFlux-Corrected Transport method(FCT)3,35based on a cell-centered Finite Volume discretization was chosen. The basic idea is to combine a low-order monotonic method and a high-order oscillatory method by means of a predictor-corrector loop, in which the low-order scheme acts as predictor. First a time step is solved in-dependently for both methods. In the proceeding steps, high-order ¯uxes are limited in such a way that no new extrema with respect to the low-order solution and the solution of the previous time step occur. Hence mono-tonicity of the low-order solution is preserved. In the following sections the chosen low-order and the high-order methods for dispersive and advective transport, respectively, as well as the implementation of the FCT method, are explained.

4.1.1 Dispersive transport

In the classical block-centered FVM, the ¯uxes are evaluated at the centers of the interfaces and multiplied by the area of the interface. For two-dimensional rect-angular cells of constant thickness Dz this yields the following mass transfered due to dispersive ¯uxes within a timestep of sizeDt:

Fig. 1.Di€erences in the approximation of advective transport on rectangular versus streamline-oriented grids.

withibeing the index in the x-direction andjin they -direction. In the following…i;j†is the index to the cell in row i and column j, …i‡1=2;j† is the index to the interface between the cells …i;j† and …i‡1;j†, and

…i‡1=2;j‡1=2† is the vertex between the cells …i;j†,

…i‡1;j†, …i;j‡1†and…i‡1;j‡1†. The term ``¯ux'' is used for the total mass transfered within a time step rather than the mass transported per area and time. The gradients occuring in eqn (6) perpendicular to an inter-face and parallel to it may be approximated by ®nite di€erences:

o_c

o_x

i‡1=2;j

2…ci‡1;jÿci;j† Dxi‡Dxi‡1

;

o_c

o_y

i‡1=2

…ci;j‡1‡ci‡1;j‡1ÿci;jÿ1ÿci‡1;jÿ1† Dyjÿ1‡2Dyj‡Dyj‡1

;

…7†

with similar expressions for the interface …i;j‡1=2†. For constant coecients in/Dthe ¯uxes may therefore directly be evaluated by inserting eqn (7) or its equiva-lents into eqn (6). If the coecients di€er in the adjacent cells, an averaging procedure is necessary. For the en-tries/Dxx _{and /D}yy _{it can be shown that the} distance-weighted harmonic average re¯ects continuity of both the concentration and the normal ¯ux components:

…/Dxx_† i‡1=2;jˆ

…Dxi‡Dxi‡1†…/Dxx_†

i;j…/Dxx†i‡1;j

Dxi…/Dxx_†

i‡1;j‡Dxi‡1…/Dxx†i;j

;

…/Dyy†_i;j‡1=2ˆ…Dyj‡Dyj‡1†…/D

yy_†

i;j…/Dyy†i;j‡1 Dyj…/Dyy_†

i;j‡1‡Dyj‡1…/Dyy†i;j

: …8†

For the ¯uxes related to the o€-diagonal entries/Dxy_, diculties arise from the choice of locations for their evaluation. Inserting eqn (7) into eqn (6) yields a de-pendence between the cells …i‡1;j‡1† and …i;j†. As illustrated by Fig. 3, the in¯uence of cell…i‡1;j‡1†on cell …i;j† is evaluated at the edges …i‡1=2;j† and

…i;j‡1=2†, whereas the in¯uence of cell…i;j†on cell…i‡

1;j‡1† is evaluated at the edges …i‡1=2;j‡1† and

…i‡1;j‡1=2†. This may result in a non-symmetric system of equations.

Arbogastet al.1applied a combination of the trape-zoidal and the mid-point rule in the evaluation of the ¯uxes, in order to guarantee symmetry. However, this is only applicable for constant coecients in /D and smooth grids. Arbogast et al.2 introduced Lagrangian multipliers as additional unknowns on the edges if the coecients in /D di€er in the two cells related to the edge. Note that for most practical applications the ¯ow ®eld will be non-uniform throughout the entire domain. As a consequnce, Lagrangian multipliers would be necessary at almost every edge, ®nally resulting in a mixed-hybrid FEM formulation based on the RT0

function space2. This system of equation contains ap-proximately twice the number of unknowns than the

original one. Note that the approach of Arbogastet al.2 has been developed for groundwater ¯ow problems rather than dispersive transport. For larger domains the assumption of constant coecients is more realistic for the hydraulic conductivity than the dispersion tensor.

By contrast, symmetric systems of equations are al-ways achieved in the FEM regardless of the anisotropy of dispersion tensors, even on deformed grids. Sym-metric equations may be prefered due to their faster solution. Assume a virtual element the corners of which are identical to the cell midpoints as illustrated in Fig. 4(a). Assuming further a constant value of /Dxy which is evaluated at the vertex …i‡1=2;j‡1=2†, the mass change due to the ¯uxes related to the o€-diagonal entry in the dispersion tensor/Dxy _{is for the FEM:} Fig. 3.In¯uence of cell…i;j†on cell…i‡1;j‡1†and vice versa by cross-di€usion terms. Black dots: interfaces for cross-dif-fusive ¯uxes related to cell…i;j†, grey dots: interfaces for

cross-di€usive ¯uxes related to cell…i‡1;j‡1†.

Fig. 4. Evaluation of ¯uxes related to the o€-diagonal entry

Dxy_{at a vertex. (a) Di€usive and anti-di€usive ¯ux at the vertex}

i‡1=2;j‡1=2. (b) Di€erentiation stencil for cell i;j and choice of the points of evaluation: black dots: ¯uxes related to

Dyy_{, circles: ¯uxes related to D}xx_{, gray dots: ¯uxes related to}

Mxy_d…i‡1=2;j‡1=2† ˆ ÿDtDz

with clockwise numbering of the cells around the vertex and the bilinear shape function N. In the FEM, the approximation of the related temporal concentration change is of course di€erent to the cell-centered FVM. Only for the case of regular spacing in both directions the related volumes are identical. For this particular case eqn (9) can also be retrieved by the FVM.1Nevertheless, in the present study eqn (9) is also applied to grids with irregular spacing.

Equation (9) may be interpreted as summation of two diagonal di€usive ¯uxes across the vertex in which the product of the di€usion coecient times the interfacial area divided by the distance equals 1=2/Dxy _{for the ¯ux} between the lower left and the upper right cell and

ÿ1=2/Dxy _{for the ¯ux between the upper left and lower} right cell, respectively. From the de®nition of /Dxy _in eqn (5) it is clear that its value may be positive or neg-ative. It vanishes only if the principal directions of the grid and the dispersion tensor coincide. For all other cases one relation between diagonally positioned cells will be of a di€usive nature but with a negative di€usion coecient. This is refered to as anti-di€usive relation. It is important to notice that anti-di€usion does not occur in nature, leads to unphysical sharpening of concentra-tion distribuconcentra-tions and may cause oscillaconcentra-tions in the vi-cinity of discontinuities.

Because of the anti-di€usive ¯uxes, the approxima-tion of full dispersion tensors on arbitrarily oriented grids destroys the monotonicity of the scheme. That is, even if a TVD method such as those listed above is chosen for advective transport, the accurate approxi-mation of dispersive ¯uxes by nine-point di€erentiation may still lead to over- or undershooting of the concen-tration distribution. In contrast to this, the dispersion tensor is a diagonal matrix on perfectly streamline-ori-ented grids, and therefore monotonic ®ve-point stencils may be applied without the introduction of cross-dif-fusive errors.

For cell …i;j† there are now eight ¯uxes to be con-sidered. The locations of evaluation are shown in Fig. 4(b). These ¯uxes are calculated by:

Fdx…i‡1=2;j† be evaluated at the vertex. For each cell a di€erent value may be approximated at this location. The e€ective value is calculated by volume-weighted harmonic aver-aging of the cell-related values:

…/Dxy†_{i‡1=2;j‡1=2}

It appears physically reasonable that diagonal ¯uxes only appear if the coecient is of same sign in all related cells. Otherwise the diagonal ¯uxes are deleted. Note that, in contrast to/Dxx_and/Dyy_{, harmonic averaging} is not a direct result from continuity considerations.

In the framework of the FCT method, the low-order sub-scheme must preserve monotonicity. Otherwise the FCT solution is not monotonic. As a consequence, in the low-order scheme the anti-di€usive diagonal ¯uxes need to be eliminated, whereas the di€usive diagonal ¯uxes may remain. In terms of a truncation error anal-ysis this leads to a second-order (di€usive) error of the low-order method. In the high-order method both types of diagonal ¯uxes are accounted for, since monotonicity is not required for this sub-scheme. The low-order and high-order di€usive ¯uxes Ld and Hd, respectively, are

Lyxd…i‡1=2;j‡1=2†

ˆ F

yx

d …i‡1=2;j‡1=2† if …/D

xy_†

i‡1=2;j‡1=2<0;

0 if …/Dxy_†

i‡1=2;j‡1=2>0:

(

…12†

Indices are identical to those of eqn (10).

4.1.2 Advective transport

For the approximation of advective transport, the cor-rection-wave approach of LeVeque24has been modi®ed and implemented into the FCT framework. The intro-duction of correction terms in eqns (15), (17) and (20) is written in pseudo-code: The corrected ¯uxes are on the left side of these equations, whereas the unmodi®ed ¯uxes appear with the same notation on the right side.

Low-order method. As low-order approximation, the multi-dimensional upwind method for explicit time-in-tegration as presented by Colella10is applied. Point of departure is the dimensional upwind method which yields for positive velocity components qx and qy, de-®ned at the interfaces of the cells:

…13†

in which Lx

a…i‡1=2;j† is the low-order advective mass

¯ux into thex-direction from cell …i;j† to cell …i‡1;j†

andLy

a…i;j‡1=2†is the equivalent ¯ux into y-direction

from cell…i;j†to cell…i;j‡1†, respectively. At this state, no diagonal advective ¯uxes are de®ned:

Lxy

a…i‡1=2;j‡1=2† ˆ0;

Lyx

a…i‡1=2;j‡1=2† ˆ0;

…14†

in which the indices are identical to those of the dis-persive ¯uxes. As illustrated in Fig. 5, the ¯ux originated from cell…i;j†may be interpretated as shifting the area of cell…i;j†along the characteristics and redistributing the mass to the intersecting cells. As a consequence, a fraction of the ¯ux from cell…i;j†is oriented to the di-agonally positioned downstream cell…i‡1;j‡1†, par-tially via cell…i‡1;j†and partially via cell…i;j‡1†. The latter may be of importance for non-uniform ¯ow ®elds. Assuming that cell …i‡1;j†is downstream of cell…i;j†

the consideration of the diagonal ¯ux from cell…i;j†to cell…i‡1;j‡1†via cell…i‡1;j†leads to the following corrections:

Lxy

a…i‡1=2;j‡1=2† ˆL

xy

a…i‡1=2;j‡1=2†

‡ci;jui‡1=2;jCyi‡1;‡j;

Lx

a…i‡1=2;j† ˆL

x

a…i‡1=2;j† ÿci;jui‡1=2;jC

y‡

i‡1;j;

Ly

a…i‡1;j‡1=2† ˆL

y

a…i‡1;j‡1=2†

ÿci‡1;jui‡1=2;jCiy‡1;‡j;

…15†

with the correction factor Ciy‡1;‡ j of cell…i‡1;j†related to the outward-directed ¯uxqy_i‡1;j‡1=2 as de®ned by:

C_iy_‡1;‡ _jˆ Dt

2/_i‡1;jDyjq y

i‡1;j‡1=2: …16†

Similar results are obtained if cell…i;j‡1†is assumed to be downstream of cell …i;j† and cell …i‡1;j‡1†

downstream of cell…i;j‡1†:

Lxy

a…i‡1=2;j‡1=2† ˆL

xy

a…i‡1=2;j‡1=2†

‡ci;jvi;j‡1=2Cix;‡j‡1;

Lxa…i‡1=2;j‡1† ˆL

x

a…i‡1=2;j‡1†

ÿci;j‡1vi;j‡1=2Cix;‡j‡1;

Lya…i;j‡1=2† ˆL

y

a…i;j‡1=2† ÿci;jvi;j‡1=2Cxi;‡j‡1; …17†

with the correction factor Cx‡

i;j‡1 of cell…i;j‡1†related

to the outward-directed ¯uxqx

i‡1=2;j‡1 as de®ned by:

Cx‡

i;j‡1 ˆ Dt 2/_i;j‡1Dxiq

x

i‡1=2;j‡1: …18†

In the equations above and in all further discussions of transverse propagation, only the case is considered in which both qx and qy are positive. Any other case in-cluding those of switching signs may be treated by symmetry. Note that in the original CTU formulation10 terms similar to cross-di€usion expressions are used to correct the ¯uxes into the principal directions rather than directly considering diagonally oriented ¯uxes.

High-order method. Colella10 suggested that the con-sideration of the diagonal ¯uxes in the CTU approach may already be viewed as an approximation of the mixed derivatives occuring in the second-order Taylor

expansion of the transport equation. LeVeque24,23 Fig. 5. Principle of the Corner Transport Upwind method

mentioned that this approximation is slightly di€erent to the one obtained by the Lax±Wendro€ scheme19. However, both authors claim second-order accuracy if only the directional second-order derivatives are added to the CTU solution. This results in the following high-order ¯uxes:

…19†

LeVeque24,23 suggested that the transverse propaga-tion of the low-order ¯ux may be applied as well to the second-order correction terms. He did this by modifying cross-di€usion terms. These terms may again be refor-mulated so that diagonal ¯uxes occur directly. This yields to the following additional modi®cations related to the correction termBx

i‡1=2;j:

Similarly the following diagonal corrections are per-formed for the correction termBy_i;j‡1=2:

In the work of LeVeque24,23 the second-order cor-rector terms are limited by dimensional ¯ux-limiters.

Although the resulting scheme is almost oscillation-free, it is not yet monotonic. Furthermore no limitation of anti-di€usive ¯uxes due to cross-di€usion is conducted. As a consequence, in the present method the second-order corrector terms are taken as advective contribu-tion to the anti-di€usive ¯uxes considered in the FCT context.

4.1.3 Flux-corrected transport

The implementation of the FCT methods for two-di-mensional applications follows mainly the approach of Zalesak35. In contrast to the original formulation, however, we assume that di€erences between high-order and low-order ¯uxes do not only occur in the advective terms but also in the di€usive ones since the discretiza-tion of cross-di€usion leads to anti-di€usive ¯uxes which need to be eliminated in the low-order scheme.

Another modi®cation of the present implementation concerns the direct consideration of diagonal ¯uxes both in the advective and di€usive terms. This consideration yields a FCT formulation which is similar to the scheme developed for FEM discretizations26.

First the mass ¯uxes of the high-order and low-order method H and L, respectively, are evaluated indepen-dently from each other according to the procedures described in Section 4.1.1 for the di€usive contribution and Section 4.1.2 for the advective, respectively. Total ¯uxes are calculated by summation of the di€usive and advectice contributions. From these ¯uxes so called anti-di€usive ¯uxesAare evaluated for all edges and diago-nals: to celli‡1;j. These anti-di€usive ¯uxes are individually limited by speci®c factorsTi‡1=2;j,Ti;j‡1=2, Tixy‡1=2;j‡1=2and

Dci;jˆAx;c…iÿ1=2;j† ÿAx;c…i‡1=2;j†

‡Ay;c…i;jÿ1=2† ÿAy;c…i;j‡1=2†

‡Axy;c_{iÿ1=2;jÿ1=2†}

ÿAxy;c_{i‡1=2;j‡1=2†}

‡Ayx;c_{iÿ1=2;j‡1=2†}

ÿAxy;c_{i‡1=2;jÿ1=2†;}

^

cfcti;j ˆc l i;j‡

Dci;j Vi;j

;

…24†

in whichVi;jˆRi;jDxiDyjDz/i;jis the e€ective volume of celli;jandcl

i;j is the low-order solution.

The limitation procedure listed in Appendix A is a straightforward extension of Zalesak's35 method in-cluding diagonal contributions.

4.2 Calculation of transport on a streamline-oriented grid

The second scheme for transport calculations presented is based on streamline-oriented grids containing quad-rilateral cells. The method of grid-generation is ex-plained in detail by Cirpkaet al.8. The transport scheme presented is an adaption of theprincipal direction(PD) technique13,14 to cell-centered Finite Volumes. In order to stabilize advective transport, aslope-limiter22method is used.

4.2.1 Approximation of gradients in streamline-oriented elements

Spatial gradients along the direction of the streamlines and orthogonal to it are required for the approximation of dispersive ¯uxes as well as for the slope-limiter

method described below. Fig. 6(a) shows a perfectly streamline-oriented curvilinear quadrilateral the edges of which are orthogonal to each other. However, the

bilinear elements generated by the grid-generator8 are no longer perfectly adopted to the streamlines (see Fig. 6(b)). Particularly, perfect orthogonality of bilinear element edges can only be achieved in regions of parallel ¯ow. By a transformation of coordinates both types of elements may be transformed to a unit square in local coordinates (see Fig. 6(c)). The direction of ¯ow may be from the left-hand side to the right-hand side.

For the curvilinear element the e€ective lengths for the evaluation of gradients parallel and orthogonal to the direction of streamlines would equal the length of the curves along the streamline and the pseudopotential line, respectively, starting from point C and ending at the edges of the element. Both the streamline and the pseudopotential line through pointCdivide the element into pieces of equal area. As illustrated in Fig. 6(c), these lines point, due to the streamline-oriented grid-genera-tion procedure, into the direcgrid-genera-tion of the local coordi-nates n and g, respectively. Therefore point C may be referred to as center of gravity with respect to the local coordinates. The bilinear approximation leads in global coordinates to linear lines of action which may not be exactly orthogonal to the edges. Their lengths are eval-uated by the following procedure:

· _{Determine the center of gravityC with respect to}

the local coordinates…nC;gC†.

· _{Transform the local coordinates of pointC as well}

as of the intermediate points I…nC;0†, II…1;gC†, III

…nC;1†und IV…0;gC†into global coordinates.

· _{Determine the distances between the intermediate}

points and pointC in global coordinates.

In the following the distances mentioned will be re-ferred to either as the half apparent lengths lk

p;k=2 and lk

k;q=2 of element k with respect to the longitudinal in-terfaces…p;k†and…k;q†, respectively, or as the half ap-parent widths wk

k;m=2 andw k

h;k of elementkwith respect to the transverse interfaces…k;m†and…h;k†, respectively. In this context, the elements p and q are directed

Fig. 6.Approximation of gradients oriented into the direction of ¯ow and orthogonal to it. (a) Perfectly streamline-oriented ele-ment; (b) Approximation by a bilinear quadrilateral; (c) Quadrilateral in local coordinates.C: Center of gravity with respect to the

from element k into the direction of ¯ow, whereas the elements h and m are directed into the direction of pseodopotential lines. For illustration see Fig. 7.

The longitudinal gradientrck;qbetween cellskandq may now evaluated by:

rck;q ˆ

The procedure for the transverse gradient rck;m be-tween cellsk andmis equivalent:

rck;m ˆ culated by division of the ®rst areal moments with re-spect to the local directions MA1

n and M

A1

g , respectively, by the area of the element:

nCˆ

in which J is the Jacobian for the transformation of coordinates. The integrals in eqn (28) can be evaluated analytically. Transformation into global coordinates is done following the isoparametric concept.

4.2.2 Dispersive transport

As stated in the previous section and illustrated in Fig. 6, the approximation of the streamline-bounded cells by quadrilaterals leads to some non-orthogonality. Applying strict transformation of coordinates to the dispersion tensor would yield o€-diagonal entries. This artefact is only due to the linearization of the element edges introduced in the last step of the grid-generation procedure8. In the preceeding steps the streamlines and pseudopotential lines were tracked accurately leading to a multi-point approximation of the element edges. If these lines would have been kept in the description of the elements, the dispersion tensor would be a diagonal matrix in local coordinates. Therefore the authors as-sume that the neglection of cross-di€usion terms is jus-ti®ed also for the quadrilateral elements used since this only corrects for an error introduced by a preceeding step of simpli®cation.

For this purpose the interfaces and the concentration gradients derived in the previous section are treated as if they were orthogonal which would be the case for curvilinear elements. Introducing eqns (25) and (26) into the de®nition of dispersive ¯uxes leads to the evaluation of the di€usive ¯uxes across longitudinal interfaces eqn (29) and transverse interfaces eqn (30), respectively: eqns (25) and (26), respectively, and qk;q is the volu-metric ¯ux across the longitudinal interface between elements k and q. Eqn. (30) includes the e€ective vol-umetrix ¯ux qe for a transverse interface. Since the

volumetric ¯uxes are de®ned at the longitudinal rather than the transverse interfaces, qe is evaluated by

arith-metic averaging of the ¯uxes at the two longitudinal interfaces belonging to each element followed by dis-tance-weighted harmonic averaging of the element-re-lated ¯uxes:

in which qk and qm are the arithmetic averaged volu-metric ¯uxes in the elementsk andm, respectively.

4.2.3 Advective transport

For the calculation of advective transport on streamline-oriented grids theslope limitermethod is applied32. This Fig. 7.Apparent widths and lengths of elementk. Direction of

scheme is an extension of Godunov's explicit method15. For given cell-averaged concentrations, a piecewise lin-ear innercell distribution of the concentration is recon-structed. This concentration distribution is used to solve exactly for the Riemann problem of advective transport. Finally the cell-related concentrations are computed by spatial averaging. It is obvious that this approach pre-serves monotonicity if the reconstruction of the innercell concentration distributions does not lead to new ext-rema.

De®ning a slope sk inside of cell k, the mass trans-ported over the interfacek;qwithin a timestep is given by:

F_ka_;qˆwk~ ;qqk;qDtDz cnk‡s n k

lk

k;qÿvk;qDt 2

!!

; …32†

in whichwk~ ;qis the width of the interface between cells k and q. Note that in eqn (32) vk;qˆqk;q=/ is the seepage velocity whereasqk;qis the volumetric ¯ux. The crucial point of the method is the choice of the slope. Some natural choices shown in Fig. 8 would be the linear interpolation to the upstream or downstream cell midpoints sup_lin and sdwn

lin , respectively, and the slopes

yielding the values of the up- or downstream cells di-rectly at the interface sup

max and s dwn

max. For

non-rectan-gular quadrilaterals these natural slopes may be calculated by:

sdwn_klin ˆ2…cqÿck†

lk k;q‡l

q k;q

;

sdwn_kmaxˆ 2…cqÿck†

max lk k;q;l

q k;q

;

sup_klin ˆ2…ckÿcp†

lp_p;k‡lk p;k

;

supkmaxˆ

2…ckÿcp†

max lp_p;k;lk p;k

:

…33†

In the implementation of the slope limiter method for streamline-oriented grids, a modi®cation of Roe's

Superbee limiter29 is used. The choice of the innercell slope is given by:

if sdwn lin s

up

lin<0 thensˆ0;

elseif jsupmaxj<js dwn

lin jthen sˆs up max;

elseif jsdwnlin j>js up

linjthen sˆs dwn lin ;

elseif jsup_linj<jsdwn_maxjthen sˆsup_lin;

else sˆsdwn max:

…34†

Note that eqn (34) leads to a non-linear dependence of the advective mass ¯ux approximated on the con-centration distribution. However, since the scheme for advective transport is fully explicit, this does not result in signi®cant additional computational e€ort.

5 CALCULATION OF REACTIVE PROCESSES AND COUPLING CONSIDERATIONS

For the solution of the system of di€erential-algebraic equations (DAES) describing the reactive processes, the solver DASSL is used27,4. DASSL is an adaptation of Gear's sti€ method for systems of ordinary di€erential equations to di€erential-algebraic systems which has recently been modi®ed to include sparse matrices and iterative solvers5. In an alternative approach the system of algebraic equations (AES) may be decoupled from the system of ordinary di€erential equations (ODES). This has been applied by various authors18,20,33. For this approach either the AES is solved ®rst and the resulting concentrations are taken as initial conditions for the solution of the ODES, or vice versa.

For certain types of reactive processes the operator-splitcoupling may not be accurate enough. This should be checked for one-dimensional model problems before transfering the scheme to multi-dimensional applica-tions. If implicit coupling turns out to be necessary, the transport schemes presented above need to be veri®ed. These veri®cations will be explained in the following.

Two alternative schemes of coupling may be con-sidered. In the iterative two-step method both the re-active processes and the advective-dispersive transport are solved in independent steps. In contrast to the Fig. 8.De®nition of slopes for elementk. Open circles:

aver-aged concentrations in elements p (upstream), k and q

operator-split scheme the reactive source-sink term is considered in the transport step as zero-order term, whereas the impact of transport on the reactions is considered in the reactive step as explicit source-sink term. The terms of interaction are updated iteratively until a de®ned convergence criterion is reached.

In this scheme the reactive source-sink terms are ta-ken into account in the transport step. This requires the ¯uxes to be evaluated at the center of the timestep (Crank±Nicolson integration). As a consequence, for the method on rectangular grids neither the CTU approach for the low-order method nor the second-order correc-tion terms in the high-order method are applicable. These schemes are restricted to explicit time integration. However, using central di€erentiation in a Crank± Nicolson integration scheme is already of second-order. In the low-order method diagonal upwinding following the scheme of Roe and Sidilkover30may be applied. This scheme is identical to the one of Rice and Schnipke28but transfered to the cell-centered FVM.

For the explicitslope limitermethod as presented for the transport calculations on a streamline-oriented grid, no consideration of source-sink terms is possible. This limitation may be overcome by reformulating the scheme in a semi-discrete form (see e.g. in the book of LeVeque22). Then again accurate and monotonic results can be obtained using Crank±Nicolson time integration. Note that, due to the non-linearity of the slope limiter

method, any (semi)-implicit time integration requires linearization which may be performed by a Newton method.

Comparing the FCT method and the slope limiter

method in the context of (semi)-implicit time integra-tion, it is obvious that the FCT method is more ecient since linearization is restricted to a single predictor-limiter-corrector loop whereas a Newton method for linearization of the slope limiter method may require several iterations. In contrast to this, the slope limiter

method is more ecient in an explicit time integration scheme since the FCT method requires to solve every timestep by two di€erent methods before applying the limiter procedure.

The iterative two-step method for coupling as dis-cussed above guarantees the coupling error to be in a prede®ned range. However, the decoupled treatment of the reactive and advective-dispersive parts of the prob-lem may lead to non-convergent iterations if the dis-cretization in time is not chosen in an adaptive manner. Furthermore the scheme may be inecient due to dou-ble linearization, once by the two-step method and once within the method for solving the reactive sub-problem. Therefore direct coupling, solving the reactive and the advective-dispersive terms at once, may be attractive.

Direct coupling can be performed by discretizing advective-dispersive transport only in time and adding the resulting ODES to the reactive DAES thus leading to a very large DAES of the orderncompnnodewhich may

be solved by a solver such as DASSL. This requires the transport discretization to ®t into the method-of-lines approach. The semi-discreteslope limiter method meets this requirement whereas the FCT method is incom-patible to such an approach.

Comparisons of di€erent coupling schemes have been carried out by the authors for a one-dimensional model problem9. In this study direct coupling performed even better than the iterative two-step method due to con-vergence problems of the latter approach. With respect to CPU-times theoperator-splitmethod performed best. But considering accuracy, direct coupling was still competitive. Unpublished two-dimensional comparisons showed di€erent results, since linearization of the semi-discreteslope limitermethod was very time-intensive for the less well-posed two-dimensional case.

6 APPLICATIONS

6.1 Five-spot con®guration in a heterogeneous aquifer

As test case for conservative transport, a ®ve-spot con-®guration in a heterogeneous aquifer is chosen. The con®ned aquifer is 25 m long, 25 m wide and 1 m thick. The geometric mean of the hydraulic conductivity is 10ÿ3_{m=s, the standard deviation of lnKis set to unity,}

and the correlation length is 2.5 m. Fig. 9 shows the distribution of the hydraulic conductivity generated with the GSLIB library11 for a grid-size of 0.25 m in both directions.

There is no ¯ow across the boundaries of the domain, except 1 m long portions at the lower left and the upper right corners representing an injection and extraction well, respectively. For these corners a constant head is de®ned as boundary condition with the head at the lower left boundary being 1.5 m higher than at the upper

Fig. 9. Five-spot con®guration in a heterogeneous aquifer. Distribution of hydraulic conductivity and de®nition of

right one. Fig. 10 shows the streamline-oriented grid generated for this test case. It consists of 9989 elements in 67 streamtubes. The rectangular grid consists of 100´_{100ˆ10,000 elements and is therefore}

compara-ble to the streamline-oriented one.

The porosity is 0.25, the longitudinal and transverse dispersivities are 0.01 m and 0.001 m, respectively. The tracer concentration in the initial state is zero. From the start of the simulation on, a unit tracer concentration is injected via the in¯ow boundary at the lower left corner of the domain. There is no sorption of the tracer.

Fig. 11 and Fig. 12 show the concentration distri-bution of the tracer three days after the start of

injec-tion for the streamline-oriented and rectangular grid, respectively. These concentration distributions are in rather good agreement. In low-velocity regions, the resolution of the front is poorer for the the streamline-oriented grid than for the rectangular one. This is due to the coarser grid in these regions. On the other hand, the streamline-oriented grid is ®ner in high-velocity re-gions, namely in the vicinities of the in- and out¯ow boundaries. Therefore the propagation of the fronts near these boundaries is better approximated on the streamline-oriented grid. Fig. 11 shows several de®nite ®ngers approaching the extraction well. These ®ngers cannot be resolved on the rectangular grid. As a con-sequence, arti®cial transverse dispersion occurs on this grid.

In Fig. 13 the breakthrough curves obtained at the out¯ow boundary are compared for both grids. The coincidence of the two breakthrough curves is almost perfect. Although there is some arti®cial transverse dispersion on the rectangular grid, this hardly has an e€ect on the breakthrough curves, since the latter are obtained by integration over the entire out¯ow boun-dary. As a consequence, the spatial distribution of the concentration along the out¯ow boundary may di€er from one grid to the other, but the averaged out¯ow concentrations do not. Therefore no superiority of one method over the other can be stated for this test case.

6.2 Reactive transport in a perfect well couple

As reactive test case a pair of an injection and an ex-traction well in a homogeneous aquifer with uniform ¯ow is chosen. The extraction well is located directly downstream of the injection well. As a consequence, the well couple forms a perfectly closed system. It is obvious that in this idealized situation, complete recovery of an injected compound occurs if di€usive exchange across Fig. 10. Five-spot con®guration in a heterogeneous aquifer.

Streamline-oriented grid.

Fig. 11. Five-spot con®guration in a heterogeneous aquifer. Concentration distribution of a conservative tracer for the streamline-oriented grid three days after the start of injection.

Fig. 12. Five-spot con®guration in a heterogeneous aquifer. Concentration distribution of a conservative tracer for the

the dividing streamline is neglected. Hence, this test case is very sensitive to arti®cial transverse di€usion.

A rectangular domain, 80 m long, 21 m wide and 1 m thick is considered. The porosity is 0.25. No ¯ow crosses the two long boundaries of the domain, at the two short boundaries constant piezometric heads with a head-di€erence of 0.6 m are de®ned as boundary conditions. The distance of the two wells is 40 m, the discharge of the wells 1:510ÿ4 _m=s.

Grid spacing for the calculation of the ¯ow-®eld and the transport calculation on the rectangular grid is 1 m thus leading to 1680 cells. The streamline-oriented grid consists of 1622 elements in 22 streamtubes. This grid is shown in Fig. 14. The dividing streamline is marked as bold line.

Calculations are conducted with various dispersivi-ties. The anisotropy factor of the dispersion tensor is set to 10. Molecular di€usion is neglected in all calculations. Compound A is added into the injection well and a di€erent compoundBis injected via the in¯ow boundary of the domain. These two compounds react, forming compound Cby a simple double-linear rate-law:

A‡B!C; …35†

rAˆrBˆ ÿrC ˆ ÿkcAcB: …36†

For the model calculations the input concentrations both ofAandBare 1 mol/l andkis set to 10ÿ3 _l/mol/s.

The total time simulated is 50 d which is sucient to reach steady-state conditions.

Fig. 15 shows the concentration distributions at the end of the simulation for the test case withalˆ1 m and

atˆ0:1 m for the streamline-oriented grid. It is obvious that compoundsAand Bget mixed only in the vicinity of the dividing streamline. Therefore the mass of com-poundCwhich is formed by the reaction ofAandB, is a measure of the obtained transverse di€usion, both physical and arti®cial.

Fig. 16 shows the total masses of compoundCin the domain obtained at the end of the simulations as a function of the physical transverse dispersivity at. Ob-viously higher masses are obtained using the rectangular grid, indicating arti®cial transverse mixing of the scheme. Even for the test case with no physical exchange a production of 29.0 kmol ofCis calculated within the domain, using the rectangular grid.

The relative error of the rectangular grid compared to the streamline-oriented decreases with increasing dispersivities. However, for a transverse dispersivity of 0.1 m which is 10% of the grid spacing, the di€erence in the production of compoundC equals still 9.6%.

Calculations on the rectangular grid were repeated with re®ned grids under assumption of strictly advective transport. Theoretically, the mass of compound C pro-duced under this assumption should be zero. The cal-culated total masses of compoundCobtained at the end of simulations are plotted in Fig. 17. An apparent order Fig. 15.Reactive transport in a perfect well couple. Concen-tration distribution 50dafter start of the injection adopting

the streamline-oriented grid.alˆ1 m,atˆ0:1 m.

Fig. 14. Reactive transport in a perfect well couple. Stream-line-oriented grid adopted for the test case. Bold line: dividing

streamline.

Fig. 13. Five-spot con®guration in a heterogeneous aquifer. Breakthrough curves obtained at the out¯ow boundary for the

of accuracy with respect to the measure chosen was obtained by ®tting an exponential model. The value calculated of 0.614 is much less than the second-order accuracy claimed by Collela10 and LeVeque24,23. How-ever, in the context of a ¯ux-limiting procedure, second-order accuracy can only be expected in smooth regions. The reaction of compounds A and B takes place at a sharp interface along the dividing streamline. Hence the erroneous production of compoundC is a measure for the accuracy in the region e€ected by the limiting pro-cedure to the highest extent. Unfortunately, reactive systems of interacting compounds tend to self-sharpen-ing. Therefore the poor order of accuracy obtained should be expected for any reactive systems in which the grid is oriented diagonally to the principal directions of ¯ow.

The arti®cial transverse mixing introduced by adopting the rectangular grid may be explained by the shifting-and-averaging model for advective transport in a conservative discretization scheme. As shown in Fig. 5 the concentration of a cell is partially shifted to a di-agonally positioned cell. Unless the entire cell is shifted to an adjacent or diagonally positioned cell, the aver-aging procedure included in the scheme leads to arti®cial mixing. On a streamline-oriented grid this e€ect is re-stricted to arti®cial mixing into the longitudinal direc-tion, whereas on an arbitrarily oriented grid additionally arti®cial transverse mixing occurs.

In the case of conservative transport, smearing of the transverse pro®le does not change the mass balance. In contrast to this, for the case of interacting compounds, transverse smearing leads to an enhancement of reaction rates und thus changes in the mass balance. Although the second-order correction procedure allows to stabi-lize a smeared transverse pro®le of a conservative compound if the direction of ¯ow does not change, it cannot avoid mixing and related reactions of interacting compounds. If the direction of ¯ow changes, additional arti®cial transverse mixing occurs. Note that the most signi®cant arti®cial transverse mixing is detected in re-gions of convergent ¯ow.

A careful review of the ®rst test case (Figs. 11 and 12) shows that the introduction of arti®cial transverse dif-fusion due to grid-orientation is even worse for domains with spatially varying hydraulic conductivities, since these heterogeneities cause rather non-uniform ¯ow ®elds including rapid changes in the ¯ow direction and regions of convergent ¯ow. For situations in which transverse exchange dominates the reactive behavior, this will cause erroneous mass balances. This is shown in a related paper7considering biotransformations in het-erogeneous domains.

7 CONCLUSIONS

Standard Eulerian methods su€er either from arti®cial di€usion or numerical oscillations both of which cannot be accepted for accurately studying the interactions of advective-dispersive and reactive processes. Since transverse mixing is physically much more limited than longitudinal mixing, the avoidance of arti®cial trans-verse di€usion is of prime importance.

This requirement is met by the application of streamline-oriented grids. The latter conclusion has al-ready been stated in earlier descriptions of theprincipal directiontechnique13,14. However, the original technique was based on standard FEM discretization for advec-tive transport thus requiring ®ne discretization guar-anteeing the grid Peclet number not to exceed a value of two. By using non-linear Finite Volume methods the latter requirement could be overcome in the present study.

Fig. 17.Reactive transport in a perfect well couple. Total mass of compoundC 50d after start of the injection calculated on the rectangular grid under assumption of strictly advective

transport as function of grid re®nement.

Fig. 16.Reactive transport in a perfect well couple. Total mass of compoundC50dafter start of the injection as function of

Using a cell-centered FVM discretization technique makes it possible to approximate rapid concentration changes, which may be an e€ect of reactive processes, by means of discontinuous distributions. In the reactive test case presented, the location of the discontinuity is de-termined by the dividing streamline. As a consequence, a grid capturing the dividing streamline yields an excellent approximation of reactive behavior.

The method for the generation of streamline-oriented grids8is restricted to two-dimensional applications un-der assumption of a steady-state ¯ow ®eld. In the case of transient ¯ow ®elds the method presented for rectan-gular grids may be used. Although this method was prone to overestimate transverse mixing, it minimizes arti®cial di€usion to an extent which may be acceptable in many cases. Particularly, in the case of transient ¯ow ®elds, transverse mixing of dissolved compounds is less limited than in the case of steady-state ¯ow, since lon-gitudinal mixing acts partially into the transverse di-rection.

The e€ect of oscillatory behavior due to the anisot-ropy of dispersion in combination with a non-principal grid orientation has been missed by many authors dealing with advection-dominated transport. If stan-dard nine-point di€erentiation is applied to anisotropic dispersion on arbitrarily oriented grids, negative con-centrations may result even if the simulation of advec-tive transport is stabilized. In the present study formulating the cross-di€usion ¯uxes as diagonally-oriented terms allows the elimination of anti-di€usive contributions in the low-order method used in a ¯ux-corrected transport (FCT) context. Using the FCT method for stabilization of advective-dispersive trans-port on arbitrarily oriented grids is therefore superior to

slope limiter and ¯ux limiter methods taking into ac-count only advective ¯uxes. This is di€erent for streamline-oriented grids for which second-order accu-racy of the dispersive ¯uxes can be obtained by mono-tonic ®ve-point di€erentiation.

For a situation in which only integrative data are considered, such as in the test case of conservative transport for a ®ve-spot con®guration shown in Sec-tion 6.1, arti®cial transverse dispersion may be of minor signi®cance. However, the importance of elimi-nating numerical transverse dispersion in the case of reactive transport is illustrated in the reactive test case shown in Section 6.2. Further examples showing the dramatic e€ect of erroneous transverse mixing in the presence of aquifer heterogeneities are given in a related paper 7.

APPENDIX A LIMITING PROCEDURE OF THE FLUX-CORRECTED TRANSPORT METHOD

Summation of all positive/negative anti-di€usive ¯uxes at a cell

De®nition of the maximum/minimum allowed anti-dif-fusive ¯uxes at a cell

Q‡i;jˆVi;j cmaxi;j ÿc

with the de®nition of the maximum/minimum allowed concentration in the cellcmax

i;j andc

Calculation of the ratioQ

De®nition of correction factors

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