Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol22.Issue6.1999:

A characteristic-conservative model for Darcian

advection

Ashok Chilakapati

1

Paci®c Northwest National Laboratories, Mailstop K9-36, P.O. Box 999, Richland, Washington 99352, USA

(Received 31 March 1998; revised 10 September 1998; accepted 14 September 1998)

A numerical method based on the modi®ed method of characteristics is developed for incompressible Darcy ¯ow. Fluid elements modeled as grid cells are mapped back in time to their twisted forms and a strict equality of volumes is imposed between the two. These relations are then cast in terms of potentials using Darcy's law and a nonlinear algebraic problem is solved for potentials. Though a general technique for obtaining Darcy ¯ow, this method is most useful when the solute advection problem also is solved with the modi®ed method of characteristics. The combined technique (referred to as the characteristic-conservative method) using the same characteristics to obtain both velocities and concentrations is then a direct numerical approximation to the Reynolds transport theorem. The method is implemented in three dimensions and a few sample problems featuring non-uniform ¯ow-®elds are solved to demonstrate the exact mass conservation property. In¯ow and out¯ow boundaries do not cause any problems in the im-plementation. In all cases, the characteristic-conservative method obtains veloc-ities that preserve ¯uid volume and, concentrations that achieve exact local and global mass balance; a desirable property that usually eludes characteristics based methods for solute advection in multidimensional, nonuniform ¯ow-®elds. Ó_{1999 Elsevier Science Limited. All rights reserved}

Key words: darcy ¯ow, solute transport, volume balance, modi®ed method of characteristics, ELLAM.

1 INTRODUCTION

The ¯ow of groundwater is described by Darcy's law and this ¯ow is often assumed to be incompressible and una€ected by the variations in concentrations of the

solutes in groundwater. IfCis the concentration [M=L3_],

qis the source/sink strength [1/T, positive for a sink and

negative for a source],~V is the Darcy velocity ®eld [L/T],

w is the potential [L], / is the porosity, kh _{is the}

hy-draulic conductivity [L/T], then the equations describing

the advection of soluteCcan be written as (Bear2),

~

V ˆ ÿkh_{rw~ ;!Darcy}0_{s Law; …1a†}

~

r ~V ˆ ÿq; …1b†

/oC

o_t ‡r …C~ ~V† ˆ ÿqC~: …1c†

~

C is the injection concentration for the source and for

the sink, it is equal toC. Here we takekh_{to be at most a}

diagonal tensor if not isotropic. If the solute undergoes an instantaneous adsorption governed by a linear

iso-therm, / in eqn (1c) is replaced by R/ where R is the

retardation factor.

Several accurate ®nite di€erence and ®nite element numerical methods exist for the numerical solution of the elliptic problem in eqns (1a) and (1b). Here a new method based on the explicit conservation of volume of ¯uid elements modeled as grid cells is described. A

similar technique was originally proposed by Hirt12et al.

for computing the dynamics of an incompressible Na-vier-Stokes ¯uid with a free surface. But the technique is general and can be exploited here to compute Darcian ¯uxes when a strict conservation of volume of the ¯uid elements is necessary. One such case is when the method of characteristics is used to solve the advection of con-taminants in groundwater. Schemes based on the

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 3 6 - 0

1

E-mail: a_chilakapati@pnl.gov.

method of characteristics for the advection equation have been successfully applied in the past to the

con-taminant transport problem (Douglas and Russell7,

Russell16, Ewing et al.8, Chiang et al.5, Celia et al.4,

Chilakapati6, Healy and Russell9,10, Huang et al.13,

Arbogast and Wheeler1, Binning and Celia3, among

others).

While they help preserve sharp fronts, avoid oscilla-tions and enable the use of large time step sizes, char-acteristics based methods usually su€er from local and global mass balance errors. Local mass balance implies a correct spatial redistribution of solute due to advection, while global mass balance implies that the total mass of solute is conserved. These errors arise primarily from the diculty in evaluating the mass storage integral and to a lesser extent from the diculties in dealing with boun-dary ¯uxes. The mass storage integral is the integral over the region in space from which a solute advects into a grid cell in a single time step (see eqn (2a)). For example,

in Fig. 1(A),X…t1†andX…t2†are the regions in Eulerian

space that a ¯uid element occupies at times t1 and t2

(t2>t1). So, X…t1† is the region in space from which

solute at timet1advects into the regular grid cellX…t2†at

timet2, during a time step of lengtht2ÿt1. In Fig. 1(A),

X…t1† is approximately identi®ed by backtracking the

streamlines from the surface of the grid cellX…t2†, for the

duration oft2ÿt1, and then joining the end points. The

volume occupied byX…t1† is usually distributed among

di€erent regular grid cells in some complex fashion. Since the solute concentration de®ned in these grid cells

is often di€erent, the mass of the solute insideX…t1†will

have to be computed by explicitly evaluating the con-tribution from these di€erent pieces of the regular grid

cells that make upX…t1†. This is referred to in this paper

as an ``exact'' evaluation of the mass storage integral. In simple one-dimensional ¯ow-®elds, exact integration is possible such that an exact local and global mass

bal-ance can be achieved (e.g. Chilakapati6, Healy and

Russell9). But it is a challenge to obtain good mass

balance, when the ¯ow-®eld is multidimensional (as in Fig. 1). Some of the problems in this case are the fol-lowing.

1. An exact evaluation (as de®ned above) of the mass storage integral is usually expensive but a numeri-cal approximation will clearly allow errors in the mass of solute placed in a grid cell, thus losing

lo-cal mass balance (Chilakapati6, Healy and

Rus-sell10).

2. While the identi®cation ofX…t1†associated with a

given grid cell will always be approximate (since the ¯ow-®eld is usually numerically

approximat-ed), theX…t1†associated with adjacent regular grid

cells should neither overlap nor leave a gap be-tween them. While this is not a major issue in two dimensions, a careful identi®cation of volumes would be necessary in three dimensions. Other-wise, there would be neither local nor global mass balance.

3. Even when both the items above have been satis-factorily addressed, there is still the need for

vol-ume-balance between X…t2† and X…t1†. This is a

restatement of the fact that the volume of the ¯uid element should be invariant in an incompressible ¯ow. Otherwise, an excess or smaller mass of sol-ute would advect into the regular grid cell,

depend-ing on whether the volume of X…t1† is greater or

smaller than the volume ofX…t2†. This will destroy

local mass balance (that can cause overshoot/un-dershoot of concentrations) even while global mass balance may be attained. This is precisely the point addressed in this paper.

Russell and Trujillo17 and Healy and Russell9 have

proposed a forward-tracking approach wherein the

known solute mass in a regular grid cell at time t1 is

distributed to di€erent grid cells at timet2. This is shown

in Fig. 1(B), whereX…t1† is now a regular grid cell and

X…t2†is the irregular region in space. Fluid fromX…t1†at

time t1 will end up in X…t2† after a duration of t2ÿt1.

The forward-tracking approach clearly works with all

Fig. 1. (A) Backtracking. X…t2† is the regular grid cell. Its

traceback-region, the twisted cellX…t1†is identi®ed by

``back-tracking'' the streamlines originating on the surface of X…t2†

for the duration …t2ÿt1†. (B) Forward-tracking. X…t1† is the

regular grid cell. Its trace-forward-region, the twisted cellX…t2†

the mass at timet1, thus achieving global mass balance, when the boundary ¯uxes are also carefully accounted. But a correct redistribution of the mass encounters problems similar to those listed above for the back-tracking approach. The ®rst item above is now replaced by the following question. What fraction of the mass in

the regular grid cellX…t1†at time t1 will end up in

an-other regular grid cell at time t2? This will require a

mapping of the irregular regionX…t2†, onto the regular

grid, to exactly identify the pieces of all the regular cells

that make up X…t2†. This is similar to the mapping of

X…t1† in the backtracking approach. A numerical

ap-proximation will clearly loose local mass balance (Healy

and Russell9,10). Likewise, the other two items above

also apply here so that a preservation of the volume of the ¯uid element is needed for good local mass balance, whether backtracking or forward-tracking is used to identify it.

The method described here addresses this mass bal-ance problem by coupling the solution technique for the ¯ow problem to that of transport, so that the combined ¯ow and transport solution is both volume and mass preserving. Characteristics are the means to achieve this coupling in the characteristic-conservative method, en-abling the method to obtain volume preserving velocities and mass preserving concentrations. The method mod-els ¯uid elements as grid cells and obtains pore velocities by explicitly requiring that the volume of ¯uid elements be invariant under the deformation caused by the ¯ow pattern. This introduces three novelties into the com-putation of the ¯ow-®eld.

(a) On discretization, the linear ¯ow problem is

ren-derednonlinearas opposed to linear in potentials.

(b) The computed ``potentials'' can be di€erent for di€erent solutes if they undergo di€erent retardation. (c) The computed ¯ow-®eld is a function of the time step size used in the transport problem.

While this seems unphysical, it is a direct result of (a) the use of characteristics to identify the volume occupied by deformed ¯uid elements; (b) requiring a volume balance of ¯uid elements; and, (c) the use of the same character-istics to advance the solution of eqn (1c) in time. Sec-tion 2 describes brie¯y the characteristic-conservative method for the combined solution of the ¯ow and ad-vection problem. A backtracking procedure is used to

identify the deformed ¯uid elementX…t1†at timet1. The

backtracking procedure, the identi®cation of deformed

¯uid elementsX…t1†, and the computation of its volume is

presented in Section 3, and the stability of the ensuing numerical scheme is examined. The same procedure ap-plies when a forward-tracking approach is used to iden-tify the deformed ¯uid element. Section 4 presents some results and contrasts the mass preservation property of the velocity ®eld computed here, against a velocity ®eld obtained by a conventional cell-centered ®nite-di€erence scheme. The paper concludes with some comments on implementation and CPU aspects of this method.

2 CHARACTERISTIC-CONSERVATIVE METHOD

Consider an element of ¯uid in an incompressible ¯ow-®eld. As the ¯uid element moves through the domain of interest, the volume of a ¯uid element remains the same at all times, though the shape of its surface may vary depending on the ¯ow. The characteristic-conservative scheme for solute transport is a direct numerical ap-proximation of the transport theorem where the ¯uid elements are followed along their volume-preserving streamlines and the changes to the solute mass are

evaluated within this material volume. IfX…t1†andX…t2†

represent the regions in Eulerian space occupied by a

¯uid element at times t1 and t2 then eqn (1c) may be

integrated to yield (Fig. 1(A)),

…2a†

Given the initial conditionC…t1†, the above equation can

be used to evaluate an average concentration C…t2† in

X…t2†. However this does not assure preservation of

mass unless the volume of X…t1† equals the volume of

X…t2†. The need for this equality becomes clear if we

consider the above equation without sources/sinks, and

with a uniform initial concentration C…t1† C. C…t2†

can in this case be written as,

C…t2† ˆ

Clearly, the analytic solutionC…t2† Ccan be achieved

if and only if

The solution of eqn (2a) while satisfying eqn (2c), de-®nes the characteristic-conservative scheme.

A numerical implementation of this scheme proceeds by a discretization of the rectangular domain in the

three cartesian directions X, YandZ to form a tensor

grid. Physical properties like porosity /, and hydraulic

conductivitykh_{are assigned to each cell./is taken to be}

uniform throughout this paper and the case of

nonuni-form / is brie¯y discussed. Each grid cell represents a

¯uid element X…t2† at time t2, the current time. The

volumeX…t1†occupied by this ¯uid element at an earlier

timet1is identi®ed by a ``traceback''. That is, following

grid cell back in time for the duration of the time step

…t2ÿt1† and then joining the end points. This shape

encloses a ``twisted grid cell''X…t1†(or traceback-region

in Fig. 1(A)). The time step size is controlled so that the twisting is not excessive and that every regular grid cell is mapped to a distinct twisted grid cell at the previous time and that an approximate volume can be de®ned for the twisted cell. In the limit when the time step size is zero, the twisted cell and the regular cell become iden-tical, i.e. there is no twisting. Knowing the surface of the twisted cell, its enclosed volume can be evaluated. When the streamlines are approximated to be straight lines, this volume is a function of the pore velocities on the surface of the regular grid cell and the time step size used. In general it will be a function of the velocities along the path of the streamlines. Approximating ve-locities with cell centered potentials using Darcy's law and invoking the ``Volume-Balance'' criterion that the volume of the twisted cell be equal to the volume of the regular grid cell, an equation relating the unknown po-tentials is obtained. Such an equation can be written for each of the regular grid cells to obtain as many equa-tions as the number of unknown potentials.

The twisted cell. The ®rst step in formulating the equations is to identify and approximate the surface of the twisted cell. For this we traceback the points on the surface of the regular grid cell. By traceback, we mean follow the streamlines originating on the surface of the regular cell back in time. This requires the solution of

the following initial-value ODE problem. If t2 is the

current time, t1 is the previous time and~r…t2† is the

position vector of a point on the surface of the regular cell,

The traceback locations~r…t1† for several points on the

surface of the regular cell are then joined by straight lines in the same order as in the regular cell, to identify the twisted surface. The approximation of the twisted cell improves as additional points on the surface of the regular cell are traced back. But there is a problem of excessive twisting which can occur if the streamlines that are being followed numerically are very close to each other. Since the streamlines we follow are approximate, the numerical traceback may result in a situation where the streamlines starting at two di€erent points on the surface of the regular cell may intersect (Fig. 2). This would be an unphysical approximation, hence to guar-antee that the numerical streamlines do not intersect, we require all the points on the surface of the regular grid cell to maintain their relative positions on the surface of the twisted cell when traced back. This can be achieved by using a small enough time step size.

3 VOLUME OF THE TWISTED CELL

LetX…t†refer to a ¯uid element at timet. At the current

timet2it coincides with a regular grid cellX…t2†and at a

previous timet1it occupies the volumeX…t1†, the twisted

cell. Its volume at any time is computed as,

Z

X…t†

/dX: …3a†

So the principle of volume balance (eqn (2c)) is,

Z

3.1 Volumes with corner points

In the simplest implementation of this method the four corners of a regular cell in two dimensions may be traced back to give a twisted cell that is a quadrilateral. See Fig. 4(A). The volume (area) of the twisted cell is uniquely de®ned when the traceback points are joined by straight lines to form the twisted cell.

In three dimensions the four corner points of a face will not, in general, traceback to a plane. The twisted face is a union of two triangular planes in two di€erent ways depending on which diagonal is chosen. So the surface of the twisted cell is a union of triangular planes. Once a diagonal is chosen for a cell-face, the same di-agonal needs to be chosen for the adjacent cell which shares that face. Since the adjacent regular grid cells share a face we need their twisted counterparts to also share the twisted face. Otherwise there may either be gaps between the two twisted cells or they may intersect. The twisted cell itself can be visualized as a union of tetrahedra and its volume is the sum of the volumes of these tetrahedra. Since the adjacent twisted cells share a face, the expressions for their volumes should re¯ect it.

Identifying the twisted cell with the indices…i;j;k†of its

regular cell we de®neoddpas,

Fig. 2. Excessive twisting resulting from the intersection of numerical streamlines is not allowed. Time step size is reduced

oddpˆmod…i‡j‡k;2†: …3c†

Clearly any two adjacent cells will have di€erentoddp

and so can be used to identify the neighbors with which it shares a cell-face. Fig. 3(A) shows the two adjacent

regular cells that share the face 2486 (oracge) and their

decomposition into tetrahedra. Fig. 3(B) shows the twisted forms of these two adjacent cells now sharing the twisted face 20₄0₈0₆0 _{(or a}0_c0_g0_e0_{). It shows that the}

de-composition of adjacent hexahedrons into tetrahedra is such that there are no gaps or intersections between the two. The eight corners of the adjacent twisted grid cells, {10₂0₃0₄0₅0₆0₇0_{8} and {a}0_b0_c0_d0_e0_f0_g0_{h}, are obtained by}

tracing back the eight corners of the corresponding ad-jacent regular grid cells, {12345678} and {abcdefgh}.

Now the volume of the twisted cells 10₂0₃0₄0₅0₆0₇0_{8 and}

a0_b0_c0_d0_e0_f0_g0_h0_{can be written as,}

volume…10₂0₃0₄0₅0₆0₇0₈0_†

ˆ/ ÿvol…10₂0₃0₅0_†

ÿvol…80206050† ÿvol…80203040†

ÿvol…80₇0₃0₅0_{† ‡vol…8}0₂0₃0₅0_†

for oddpˆ0; …3d† volume…a0_b0_c0_d0_e0_f0_g0_h0_†

ˆ/vol…h0_g0_f0_d0_†

‡vol…a0g0c0d0† ‡vol…a0g0f0e0†

‡vol…a0_b0_f0_d0_{† ÿvol…a}0_g0_f0_d0_†

foroddpˆ1: …3e†

In two dimensionsoddpis immaterial and eqns (3d) and

(3e) are identical. Here vol…pqrs† refers to the signed

volume of a tetrahedron with vertices …xp;yp;zp†,

…xq;yq;zq†,…xr;yr;zr†and…xs;ys;zs†.

vol…pqrs† ˆ1

6

…xqÿxp† …yqÿyp† …zqÿzp† …xrÿxp† …yrÿyp† …zrÿzp† …xsÿxp† …ysÿyp† …zsÿzp†

: …3f†

Depending on the location of the cell in the grid, either eqn (3e) or eqn (3d) is used for the volume.

When the ¯ow-®eld is not very heterogeneous or

when the time step sizeDt is small, a one-step forward

Euler scheme can be used to integrate eqn (2e).

~r…t1† ˆ~r…t2† ÿ~V…~r…t2††…t2ÿt1†=/: …3g†

This means that the streamline between the two

loca-tions~r…t2†and~r…t1†has been approximated as a straight

line, which is reasonable wheneverj~r…t2† ÿ~r…t1† jis not

large. Using this equation for~r…t1† …x;y;z†in eqns (3f)

and (3e) or eqn (3d) can be evaluated for the volume of

the twisted cell, as functions of ~r…t2†, …t2ÿt1†;/ and

equal to the bulk volume of the regular grid cell,g1is a

linear function of the velocities at the eight corners, g2

has terms with products of two velocities and, g3

in-volves terms with products of three velocities. Equating

Voltwisted in eqn (3h) to the volume of the regular grid

cell and using the fact that g0 is equal to the volume of

the regular grid cell,

Thex;yandzvelocities at the corners are written as the

weighted averages of the face-centered (normal to the face) velocities in the four faces that share that corner. Central di€erence approximation to the Darcy's law is then used to replace these velocities by potentials. For

example, the face-centered X-velocity vxi‡1=2;j;k is

ap-proximated from eqn (1a) as,

vxi‡1=2;j;k ˆ

Similarly theYandZ-velocities are approximated.wi;j;k

is the potential in the grid block fi;j;kg. eqn (3j) turns

g1into a linear,g2 into a quadratic and,g3 into a cubic

function ofw.

Equation (3i) can be written for each grid cell so that the solution of a system of nonlinear algebraic equations

yields w. It is convenient to use a Picard iteration to

solve the nonlinear system eqn (3i) since it allows us to

avoid a direct evaluation of nonlinear termsg2andg3. It

follows from eqn (3h) that the contribution of the

nonlinear terms at iteration min eqn (3i) is given by,

gm2

So the equation solved forwat iterationmis,

gm

Fewer than 10 Picard iterations are usually sucient to

achieve a relative error in volume to about 10ÿ6_{. Note}

also from eqn (3i) that a decrease in the time step size

Dt, diminishes the contribution from the nonlinear terms

so that faster convergence is obtained. At each iteration,

eqn (3l) is written for every grid cell and a matrixG is

formed.Ghas the structure of a 27-point ®nite di€erence

operator. It is nonsymmetric when the grid is nonsquare or nonuniform or when the conductivity is nonuniform. An incomplete Cholesky preconditioner with an OR-THOMIN accelerator is used to solve the system of

equations (Oppe et al.15). If / is spatially variable but

still smooth enough to allow a straight line approxi-mation for streamlines then eqn (3l) may be extended for this case as,

But the exact evaluation of R

X…t1†mÿ1/dX when / is nonuniform, in multiple dimensions is much more CPU

intensive than when/is uniform (Moore14). The reason

is that eqn (3d) or eqn (3e) cannot be used and the contributions to the volume from each of the regular

cells (each with a possibly di€erent /) that are

inter-sected by the twisted cell have to be added up.

Stability. The stability of the numerical scheme is easy to examine in two dimensions. In two dimensions,

g1 reduces to,

Herevx1 refers to thex-velocity at the corner denoted 1

in Fig. 4(A). The other velocities have similar meanings.

g1 is seen to be the net ¯uid leaving the grid cell. For a

uniform, isotropic kh _{and a uniform, square grid, the}

discretization ofg1according to eqn (3j) yields a 9-point

stencil shown in Fig. 4(B). It is seen that the adjacent cells have no e€ect on each other, which is contrary to what is to be expected. The odd cells and even cells are decoupled. Such a ®nite di€erence scheme for the La-place operator is known to give oscillatory solutions (for

example see Hirsch11). The odd points and even points

converge to di€erent solutions.

The reason this scheme does not give larger weights to the adjacent cells compared to the diagonal neighbors is that the formulation in eqn (3n) includes velocities at

onlythe corner points. The subtractions in eqn (3n) tend

to cancel out the contribution of the adjacent cells leaving alone the contributions from the diagonal neighbors. Larger weights to the adjacent cells can be

obtained ifg1involves velocities not only at the corners

this means is that, tracing the eight corners (in three dimensions) of a regular grid cell to identify the twisted cell is insucient. The regular grid cell should be given more degrees of freedom to twist. The determination of the shape of the twisted cell needs to include more points on the surface of the regular grid cell and not just the eight corner points.

3.2 Re®ned volume computation

Following the above discussion we allow the regular grid cell to twist into more complex shapes. For exam-ple, in 2-dimensions, we divide the cell to four subcells and evaluate the sum of their areas as the area of the regular grid cell. This amounts to adding four degrees of freedom, one each at the center of the four edges of the regular grid cell. See Fig. 4(C). The area of the twisted cell is,

area…10205040† ‡area…20306050† ‡area…40508070† ‡area…50609080†:

Equating this to the area of the regular grid cell

area…1397†, the new expression for g1 in place of

eqn (3n) is,

Dy‰ÿvx1ÿ2vx4ÿvx7‡vx3‡2vx6‡vx9Š=4

‡Dx‰ÿvy1ÿ2vy2ÿvy3‡vy7‡2vy8‡vy9Š=4:

…3o†

With thisg1the ®nite di€erence operator has the weights

shown in Fig. 4(D) leading to a stable and more accu-rate (since the twisted cell has a better de®nition) scheme.

In three dimensions, the regular grid cell is divided into eight subcells and the sum of the volumes of the eight twisted subcells is taken to be the volume of the twisted cell. 26 points on the surface of the regular grid cell and one center point are traced back. See Fig. 5 for the locations of these 27 points. The volume of the twisted cell in three dimensions is given by (for uniform

/),

Fig. 4.(A) The construction of the twisted cell 10₂0₄0_{3 from the regular cell 1243 in two dimensions. (B) The ®nite di€erence operator} for potentials in (A), has odd-even decoupling that makes it oscillatory. (C) The re®ned shape of the traceback-region in two di-mensions when both the corners and the centers of the edges are traced back. (D) A stable ®nite di€erence operator results from the

/_‰volume…10₂0₄0₅0₁₀0₁₁0₁₃0₁₄0_†

Note that the ®rst four subcells in the brackets do not share a face among them, just like the later four subcells in the next pair of brackets. So for the ®rst four subcells in the brackets, the twisted subcell may be identi®ed as

in Fig. 3 under oddpˆ1. Their volumes are evaluated

with eqn (3e). For the latter four twisted subcells in the brackets, the twisted subcell is identi®ed as in Fig. 3

under oddpˆ0. Their volumes are obtained with

eqn (3d). The procedure for formulating the equations

in cases of uniform and nonuniform / is the same as

before. The expression for g1 which is a linear function

of velocities at the 26 locations on the surface is given in Appendix A. The velocities are replaced by cell centered potentials as before by using a central di€erence

ap-proximation to Darcy's law. The matrix G is usually

nonsymmetric and has the structure of a 27-point op-erator. The linear system is solved as before.

Sources/Sinks. Here the equation for the balance of volumes has to include the volume of ¯uid entering/ leaving the well-block during the time step.

Voltwisted ˆVolregular‡q^_wDt; …3q†

whereq^w is the well ¯ow-rate for that well-block. Flow

diverges out of the injection well-block and converges into the production well-block. This means that the traceback-region for the injection well-block is con-tained within the regular well-block. The volume of the

traceback-region ought to be smaller than the regular

well-block volume by the amount of ¯uid injected (q is

negative for an injection well). Also eqn (3q) requires

that the time step sizeDtshould be small enough so that

the Voltwisted for the injection well-block does not

be-come negative. That is, we need the volume of the ¯uid

entering the injection well-block during the time_Dtto be

smaller than the volume of that well-block. For a pro-duction well the volume of the traceback-region is greater than the regular well-block volume by the amount of ¯uid that ¯ows out of that well-block. All the ¯uid that goes out comes from the production well-block. This requires that the volume of ¯uid produced

through a production well-block during the time Dt be

smaller than the volume of that well-block. To sum-marize,

T) into the well-block

lo-cated at …i;j;k† and Dxi

w is the x-width of that

well-block.Dyj

w andDzkw are similarly de®ned. This is similar

to a CFL constraint onDt. Fig. 6 shows the shape of a

twisted injection well-block. This is obtained by placing

a unit strength (qˆ)1.0) source at the center of 33

3 m3_{. A 333 grid is used. The source is placed in}

the cell (2,2). Constant (0.0) potential conditions are

used on all the six faces.kh_{1.0,/1.0, and}

Dt ˆ0.8 days.

Treatment of boundaries. Dirichlet or Neumann boundary conditions for the potential and a Dirichlet boundary condition for concentration can be handled in a straightforward fashion. Since points on the boundary are traced back, the twisted surface from which ¯uid

originates at timet1to reach the boundary at timet2, can

be identi®ed. If the ¯ow enters, then a part of the cor-responding twisted cell lies outside the problem domain. If the time step size is large enough then the entire twisted cell and possibly the twisted cells corresponding to several interior grid cells could also lie outside the

Fig. 6.The twisted cell for an injection well-block. Fig. 5.Locations of the 26 degrees of freedom for the grid cell

problem domain. If the ¯ow leaves a grid cell on the boundary then the corresponding twisted cell lies inside the problem domain. If it is a no¯ow boundary then the boundary surface will undergo no twisting. All these cases are illustrated in Fig. 7(A) in Section 4. The vol-ume of the twisted cells that may partly/entirely lie outside the problem domain are evaluated in the same way as interior twisted cells. The volume of ¯uid en-tering a grid cell from outside is that portion of the

volume of the corresponding twisted cell that isoutside.

The mass of solute entering the grid cell is simply the product of its twisted cell volume that is outside the problem domain and the speci®ed concentration of the solute outside. A grid cell with a larger in¯ow will have more of its twisted cell outside the problem domain. So the distribution of the incoming mass among the grid cells on the boundary, is appropriately weighted ac-cording to the ¯ow-®eld. Since we explicitly conserve ¯uid volume, the total twisted cell volume that is outside the problem domain is equal to the total amount of ¯uid that enters the problem domain. For the grid cells at the

out¯ow boundaries, all the twisted cells are inside the problem domain. The volume of the problem domain

that isnotpart of any twisted cell, is the volume of ¯uid

that leaves through the out¯ow boundary. In the ab-sence of sources and sinks, this volume is equal to the total volume of the twisted cells that is outside the problem domain. When the in¯ow solute concentration is the same as the concentration of the solute everywhere in the problem domain, this ¯uid balance assures us that the mass of solute leaving the domain is equal to the mass entering.

4 RESULTS

Fig. 7(A) shows a sample problem where a unit ¯ow rate

(1 m3_{/day) is maintained in the axial direction (X-axis)}

while a no¯ow condition is imposed on the transverse

faces (Yˆ0,YˆLy). A 77 uniform grid is used on a

domain with LxˆLy ˆ1:0 m. The nonuniform but

isotropic hydraulic conductivity (meters) for each cell is

Fig. 7.(A) The 77 grid and the value ofkin each cell are shown. (B) The magnitude and direction of the ¯ow is shown by the size of the arrows and their direction. (C) The regular cells (- - - edges) and their twisted counterparts (± edges) are shown here. The points (*) are the traceback locations of points on the regular cells. Twisted cells are obtained by joining these points by straight

shown in Fig. 7(A). A uniform porosity/ ˆ1.0, and a

time step sizeDt ˆ0.04 days are used. The convergent/

divergent ¯ow-®eld obtained by the characteristic-con-servative model is shown in Fig. 7(B). Fig. 7(C) shows for each regular grid cell (location of a ¯uid element at 0.04 days), its twisted cell (location of that ¯uid element

at 0.0 days). Since the ¯ow enters atX ˆ0:0, parts of the

twisted cells at the inlet boundary lie outside the prob-lem domain. All the other twisted cells lie inside the problem domain. The maximum relative error in vol-umes obtained by the characteristic-conservative

meth-od is near zero at 10ÿ8_.

To illustrate the mass preservation property of the obtained ¯ow-®eld, the above ¯ow problem is also solved with the conventional cell-centered, ®nite-di€er-ence scheme. Identifying twisted cells and computing their volumes as before we get the maximum relative error in volumes to be 9%, and the standard deviation to be 5%. So the use of this ¯ow-®eld in eqn (2b) would clearly cause local mass balance errors of up to 9% in a single time step. Following the discussion in the previous section regarding the treatment of boundaries, the total twisted cell volume lying outside the problem domain

(x<0) should equal the volume of ¯uid that enters the

xˆ0 boundary in one time step, i.e. 1:0Dtˆ0:04 m3

here. Since the Y-boundaries have a no¯ow condition,

the out¯ow rate atxˆ1:0 would also be 1.0 m3_{/day. So}

the total volume inside the problem domain, that is not

part of any twisted cell should be equal to 1:0Dtˆ

0:04 m3_{, which is the amount of ¯uid that leaves thexˆ}

1:0 boundary in a single time step. The

characteristic-conservative method obtains 0.040000032 m3 _{for the}

total twisted cell volume outside, and 0.040000033 m3

for the volume in the problem domain that is not part of any twisted cell. Both these numbers are in excellent agreement with the exact amount of ¯uid that would be entering/leaving the boundaries. The cell-centered

®nite-di€erence method obtains 0.037682152726 m3 _{as the}

in¯ow and 0.038240686413 m3 _{as the out¯ow, both of}

which cause about 5% error in a single time step. The total volume of all the twisted cells turns out to be about 0.99944 compared to 0.999999999153 obtained by the volume-preserving ¯ow-®eld in Fig. 7(C). Thus even if the overshoot/undershoot of concentrations due to local mass balance errors is tolerated, there would still be a 0.06% global mass balance error in a single time step.

The above problem is solved again with a source of

unit rate (qˆ)1.0/day) at the grid location (2,2) and a

sink of unit rate (qˆ1.0/day) at the grid location (6,6).

Constant potential (ˆ0.0 m) conditions are used on the

boundaries xˆLx;y ˆLy. No¯ow conditions are used

on the boundariesxˆ0;yˆ0. Uniform conductivityk

ˆ1.0, porosity/ ˆ1.0, and a time step size of_Dtˆ0.02

days are used. It follows from eqn (3r) that the maxi-mum time step size that may be allowed (for the twisting not becoming excessive as de®ned in Section 2, and

Fig. 2) is…1=7† …1=7† ˆ0:020408. Fig. 8(A) shows the

streamlines diverging from the source and converging at the sink. Fig. 8(B) shows the mapping between the twisted and regular cells. The twisted cell volume for the regular cell with the source (at (2,2)) is,

DxDy/‡q^_wDtˆ …1:0=7:0†…1:0=7:0†…1:0† ‡ …ÿ1:0†…0:02†

0:000408:

Similarly the twisted cell volume for the regular cell with

the sink (at (6,6)) is 1:0=49:0‡0:020:040408.

Fig. 8(C) shows the mapping of the material volume if the conventional cell-centered, ®nite-di€erence scheme is used for the ¯ow-®eld. The predicted volume of the twisted injection well-block turns out to be 0.007078 instead of 0.000408. This means that after a single time step the mass in the injection well-block would be 32%

……0:007078‡0:02† 100=…7:07:0††in excess.

E€ect of Dt. The velocity ®eld obtained here is de-pendent on the time step size. The dependence of the

incompressible ¯ow-®eld on Dt may seem unphysical

since eqns (1a) and (1b) do not feature time. The reason

for its dependence onDt of course is the need to follow

the streamlines for the duration of Dt to locate and

identify the material volume, so that its volume may be computed. The location and identi®cation of the trace-back-region (material volume at an earlier time) will always be approximate in any numerical procedure and so it serves to make an approximation that will preserve

mass balance. The dependence of ¯ow-®eld on_Dt here

provides the appropriatecorrection to the ¯ow-®eld so

that the characteristic-conservative method preserves mass balance. This is made clear in the following ex-ample. To test the dependence on the time step size, simulations with di€erent time step sizes (less than the maximum allowable value to prevent excessive twisting) are carried out for a problem with a nonuniform ¯ow pattern. The problem is similar to the one in Fig. 7(A)

with a unit head drop along theX-direction and a

no-¯ow condition on the transverse faces. A 3333

uni-form grid is used. The potential at the grid location

(17,17) with a Dt of 0.0, 0.000625, 0.00125, 0.0025,

0.005, 0.01, days is computed to be 0.62075, 0.62073, 0.62070, 0.62065, 0.62054 and 0.62027 m respectively.

Considering that the potential varies from 1.0 atXˆ0.0

to 0.0 atXˆ1.0, the e€ect of_Dton potential is seen to

be small.

Implementation/CPU considerations. Some of the important factors that impact the solution eciency and accuracy of the characteristic-conservative scheme are summarized here.

a concern, grid re®nement and/or the use of small enough time step sizes seem to be the only practical means to reduce it.

2. To have to solve a nonlinear problem to get a so-lution to a linear problem increases the computa-tional e€ort here compared to a cell-centered scheme for velocities. In particular, this introduces inner and outer iteration loops since the inner lin-ear problem also is solved iteratively with ORTHOMIN. Because of the non-symmetric 27-point stencil, and the need for iterating over the

nonlinearities, the characteristic-conservative

method is always slower than the cell-centered scheme which has a symmetric 7-point stencil and no nonlinearities. So this method is not recom-mended if the Darcy ¯ow problem alone needs to be solved. The bene®ts of this volume-balance ap-proach for velocities are realized only when it is used in conjunction with a method of characteris-tics scheme for solute transport. In that case the extra e€ort expended in obtaining the velocities,

pays itself o€ in terms of exact local and global mass balance.

3. For the velocity problem, grid re®nement increases the time for solving the linear system but the num-ber of outer iterations do not increase signi®cantly. The reason of course is that the appropriate time step size that avoids excessive twisting (Fig. 2) on

a ®ner grid is smaller, and, a smaller _Dt reduces

the amount of nonlinearity to be resolved in the outer loop (eqn (3i)). But the time for the trans-port problem increases due to both the increased number of grid cells, and the larger number of time steps that would be needed.

4. To have to obey a time step constraint for the ad-vection problem, while using a characteristics based method, is a de®nite drawback with the characteristic-conservative method. To some ex-tent it defeats the advantage o€ered by the charac-teristics based method. However for many reactive transport problems, the time step size may have to be anyway limited, either because reactions have

Fig. 8.Problem with a source and a sink. (A) The magnitude and direction of the ¯ow is shown by the arrows. (B) The regular cells (- - - edges) and their twisted counterparts (± edges). Points (*) are traceback locations. (C) As in B, but with a ¯ow-®eld that does

been time-split from advection, or simply because of nonconvergence owing to large changes in reac-tion terms. Also the time step constraint that pre-vents excessive twisting is not usually severe in the absence of sinks and sources.

5. Since the velocities are obtained only once, the CPU requirement for the velocity calculation is a small fraction of the total CPU needed to obtain a solution to the transport problem. The most CPU intensive part in any characteristics based scheme is the evaluation of the mass storage

inte-gral. An exact integration over X…t1†, will help

preserve local and global mass balance but will also require a larger CPU, compared to an ap-proximate numerical integration by quadrature rules. An increased accuracy in the numerical inte-gration may be achieved by using a large number of integration points, which will in turn increase

the computational e€ort (Healy and Russell 10).

So the extra CPU consumed by an exact integra-tion or by an accurate numerical integraintegra-tion, has to be justi®ed by the need for good local mass bal-ance.

6. The number of species, and the complexity of the reaction mechanisms (if reactions are present) do not have any special consequences to the charac-teristic-conservative method. When there are mul-tiple solutes and some of them have been modeled with a retardation factor, then those solutes move with a di€erent speed. For each solute that moves

with a di€erent speed,X…t1†is di€erent, requiring

additional integration e€ort. But an explicit mod-eling of the adsorbed species under equilibrium with its counterpart in solution, and time-splitting reactions, can avoid the retardation factor ap-proach. This will cause all the mobile species to move with the same speed, thus requiring the

map-ping ofX…t1†onto the regular grid to be done only

once.

5 CONCLUSIONS

A characteristic-conservative method based on the method of characteristics for the combined incom-pressible ¯ow and transport problem is presented and implemented in multiple dimensions. Velocities are ob-tained by requiring that the volume of the ¯uid elements be invariant as they deform under advection. The use of the same characteristics to solve for both velocities and concentrations allows the characteristic-conservative method to achieve exact local and global mass balance. But this bene®t has to be weighed against its generally larger CPU requirements compared to conventional ®-nite-di€erence or ®nite-element techniques to the solute transport problem.

ACKNOWLEDGEMENTS

The author wishes to thank Todd Arbogast, Mary Wheeler and Clarence Miller for the discussions that led to the development of the ideas presented here. The author is also grateful to the three anonymous reviewers for their suggestions on improving this paper.

APPENDIX A REFINED VOLUME

In Section 3.2 the volume of the twisted grid block was found as the sum of the volumes of the eight subcells

(Fig. 5). The expression forg1in eqn (3l) or eqn (3m) is

z velocities at the locations 1. . .27 on the regular grid

block (Fig. 5).Dx;Dy andDzare the widths of the grid

block in thex,yandzdirections.

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