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

PII: S 0 3 0 9 - 1 7 0 8 ( 9 8 ) 0 0 0 3 6 - 0

Advances in Water Resources Vol. 22, No. 6, pp 597±609, 1999
Ó 1999 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0309-1708/99/$ ± see front matter

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 nonuniform ¯ow-®elds are solved to demonstrate the exact mass conservation
property. In¯ow and out¯ow boundaries do not cause any problems in the implementation. In all cases, the characteristic-conservative method obtains velocities 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.

C~ is the injection concentration for the source and for
the sink, it is equal to C. Here we take k h to be at most a
diagonal tensor if not isotropic. If the solute undergoes
an instantaneous adsorption governed by a linear isotherm, / 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 Hirt12 et al.
for computing the dynamics of an incompressible Navier-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 contaminants in groundwater. Schemes based on the

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. If C is the concentration [M=L3 ],
q is the source/sink strength [1/T, positive for a sink and
~ is the Darcy velocity ®eld [L/T],
negative for a source], V
w is the potential [L], / is the porosity, k h is the hydraulic conductivity [L/T], then the equations describing
the advection of solute C can be written as (Bear2),

1

~ ! Darcy0 s Law;
V~ ˆ ÿk h rw;

…1a†

~
V
~ ˆ ÿq;
r
oC ~
~
~† ˆ ÿqC:
‡ r
…C V
/
ot

…1b†
…1c†

E-mail: a_chilakapati@pnl.gov.
597

598

A. Chilakapati

method of characteristics for the advection equation
have been successfully applied in the past to the contaminant 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 oscillations and enable the use of large time step sizes, characteristics 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 boundary ¯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 † and X…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

Fig. 1. (A) Backtracking. X…t2 † is the regular grid cell. Its
traceback-region, the twisted cell X…t1 † is identi®ed by ``backtracking'' the streamlines originating on the surface of X…t2 †
for the duration …t2 ÿ t1 †. (B) Forward-tracking. X…t1 † is the
regular grid cell. Its trace-forward-region, the twisted cell X…t2 †
is identi®ed by ``forward-tracking'' the streamlines originating
on the surface of X…t1 † for the duration …t2 ÿ t1 †.

solute at time t1 advects into the regular grid cell X…t2 † at
time t2 , during a time step of length t2 ÿ t1 . In Fig. 1(A),
X…t1 † is approximately identi®ed by backtracking the
streamlines from the surface of the grid cell X…t2 †, for the
duration of t2 ÿ t1 , and then joining the end points. The
volume occupied by X…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 inside X…t1 † will
have to be computed by explicitly evaluating the contribution from these di€erent pieces of the regular grid
cells that make up X…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 balance 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 following.
1. An exact evaluation (as de®ned above) of the mass
storage integral is usually expensive but a numerical approximation will clearly allow errors in the
mass of solute placed in a grid cell, thus losing local mass balance (Chilakapati6, Healy and Russell10).
2. While the identi®cation of X…t1 † associated with a
given grid cell will always be approximate (since
the ¯ow-®eld is usually numerically approximated), the X…t1 † associated with adjacent regular grid
cells should neither overlap nor leave a gap between them. While this is not a major issue in
two dimensions, a careful identi®cation of volumes
would be necessary in three dimensions. Otherwise, there would be neither local nor global mass
balance.

3. Even when both the items above have been satisfactorily addressed, there is still the need for volume-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 solute would advect into the regular grid cell, depending on whether the volume of X…t1 † is greater or
smaller than the volume of X…t2 †. This will destroy
local mass balance (that can cause overshoot/undershoot 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 time t2 . This is shown
in Fig. 1(B), where X…t1 † is now a regular grid cell and
X…t2 † is the irregular region in space. Fluid from X…t1 † at
time t1 will end up in X…t2 † after a duration of t2 ÿ t1 .
The forward-tracking approach clearly works with all

A characteristic-conservative model for Darcian advection
the mass at time t1 , 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 backtracking approach. The ®rst item above is now replaced
by the following question. What fraction of the mass in
the regular grid cell X…t1 † at time t1 will end up in another regular grid cell at time t2 ? This will require a
mapping of the irregular region X…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 approximation 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 balance 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, enabling the method to obtain volume preserving velocities
and mass preserving concentrations. The method models ¯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 computation of the ¯ow-®eld.
(a) On discretization, the linear ¯ow problem is rendered nonlinear as 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 characteristics to advance the solution of eqn (1c) in time. Section 2 describes brie¯y the characteristic-conservative
method for the combined solution of the ¯ow and advection problem. A backtracking procedure is used to
identify the deformed ¯uid element X…t1 † at time t1 . The
backtracking procedure, the identi®cation of deformed
¯uid elements X…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 applies when a forward-tracking approach is used to identify 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.

599

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 approximation 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. If X…t1 † and X…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 condition C…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,
R
C
/ dX
X…t1 †
:
…2b†
C…t2 † ˆ R
/ dX
X…t2 †

Clearly, the analytic solution C…t2 †  C  can be achieved
if and only if
Z
Z
/ dX ˆ
/ dX:
…2c†
X…t1 †

X…t2 †

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, Y and Z to form a tensor
grid. Physical properties like porosity /, and hydraulic
conductivity k h are assigned to each cell. / is taken to be
uniform throughout this paper and the case of nonuniform / is brie¯y discussed. Each grid cell represents a
¯uid element X…t2 † at time t2 , the current time. The
volume X…t1 † occupied by this ¯uid element at an earlier
time t1 is identi®ed by a ``traceback''. That is, following
the streamlines originating on the surface of the regular

600

A. Chilakapati

grid cell back in time for the duration of the time step
…t2 ÿ t1 † 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 identical, 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 velocities 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 potentials is obtained. Such an equation can be written for
each of the regular grid cells to obtain as many equations 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,
d~
r ~
ˆ V …~
r…t††=/; t1 6 t 6 t2 ;
dt
Zt2 ~
V …~
r†
~
r…t2 † ÿ
dt:
r…t1 † ˆ ~
/

…2d†
…2e†

t1

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 guarantee 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.

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

3 VOLUME OF THE TWISTED CELL
Let X…t† refer to a ¯uid element at time t. At the current
time t2 it coincides with a regular grid cell X…t2 † and at a
previous time t1 it occupies the volume X…t1 †, the twisted
cell. Its volume at any time is computed as,
Z
/ dX:
…3a†
X…t†

So the principle of volume balance (eqn (2c)) is,
Z
Z
/ dX ˆ
/ dX  /Dx Dy Dz;
X…t1 †

…3b†

X…t2 †

where Dx; Dy; and Dz are the dimensions of the regular
cell X…t2 †.
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 diagonal 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®ne oddp as,

A characteristic-conservative model for Darcian advection
oddp ˆ mod…i ‡ j ‡ k; 2†:

…3c†

Clearly any two adjacent cells will have di€erent oddp
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 (or acge) and their
decomposition into tetrahedra. Fig. 3(B) shows the
twisted forms of these two adjacent cells now sharing the
twisted face 20 40 80 60 (or a0 c0 g0 e0 ). It shows that the decomposition 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,
{10 20 30 40 50 60 70 8} and {a0 b0 c0 d0 e0 f0 g0 h}, are obtained by
tracing back the eight corners of the corresponding adjacent regular grid cells, {12345678} and {abcdefgh}.
Now the volume of the twisted cells 10 20 30 40 50 60 70 8 and
0 0 0 0 0 0 0 0
a b c d e f g h can be written as,

601

volume…10 20 30 40 50 60 70 80 †

ˆ / ÿ vol…10 20 30 50 †ÿvol…80 20 60 50 † ÿ vol…80 20 30 40 †

ÿ vol…80 70 30 50 † ‡ vol…80 20 30 50 †  for oddp ˆ 0; …3d†

volume…a0 b0 c0 d 0 e0 f 0 g0 h0 †

ˆ / vol…h0 g0 f 0 d 0 †‡vol…a0 g0 c0 d 0 † ‡ vol…a0 g0 f 0 e0 †

‡ vol…a0 b0 f 0 d 0 † ÿ vol…a0 g0 f 0 d 0 †  for oddp ˆ 1: …3e†

In two dimensions oddp is 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 †.


 …xq ÿ xp † …yq ÿ yp † …zq ÿ zp † 


1

vol…pqrs† ˆ  …xr ÿ xp † …yr ÿ yp † …zr ÿ zp † : …3f†

6
 …xs ÿ xp † …ys ÿ yp † …zs ÿ zp † 

Fig. 3. (A) The volumes of two adjacent regular cells. The face 2486 (or acge) is shared by these two cells. (B) The volumes of the two
adjacent twisted cells. The face 20 40 80 60 (or a0 c0 g0 e0 ) is shared by these two cells.

602

A. Chilakapati

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 size Dt is small, a one-step forward
Euler scheme can be used to integrate eqn (2e).
~
r…t2 † ÿ V~…~
r…t2 ††…t2 ÿ t1 †=/:
r…t1 † ˆ ~

…3g†

This means that the streamline between the two locar…t1 † has been approximated as a straight
tions ~
r…t2 † and ~
r…t1 † j is not
line, which is reasonable whenever j ~
r…t2 † ÿ~
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 †, …t2 ÿ t1 †; / and
V~…~
r…t2 ††. This volume is,
 2
 3
Dt
Dt
‡ g3
; …3h†
Voltwisted ˆ g0 / ‡ g1 Dt ‡ g2
/
/2
where Dt is the time step size t2 ÿ t1 , g0 is a constant
equal to the bulk volume of the regular grid cell, g1 is a
linear function of the velocities at the eight corners, g2
has terms with products of two velocities and, g3 involves 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,
 2
Dt
Dt
g1 ‡ g 2 ‡ g 3
ˆ 0:
…3i†
/
/
The x; y and z velocities 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 approximated from eqn (1a) as,
vxi‡1=2;j;k ˆ

…Dxi ‡ Dxi‡1 † wi;j;k ÿ wi‡1;j;k
;
…kxDxi ‡ kxDxi‡1 † …Dxi ‡ Dxi‡1 †=2
i;j;k

…3j†

i‡1;j;k

where kxi;j;k is the hydraulic conductivity in the X-direction in the grid block with the grid indices fi; j; kg.
Similarly the Y and Z-velocities are approximated. wi;j;k
is the potential in the grid block fi; j; kg. eqn (3j) turns
g1 into a linear, g2 into a quadratic and, g3 into a cubic
function of w.
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 terms g2 and g3 . It
follows from eqn (3h) that the contribution of the
nonlinear terms at iteration m in eqn (3i) is given by,
 2
 2
m Dt
mÿ1 Dt
m Dt
mÿ1 Dt
‡ g3
‡ g3
 g2
g2
/
/
/
/
mÿ1
ˆ …Volmÿ1
:
twisted ÿ g0 /†=Dt ÿ g1

So the equation solved for w at iteration m is,

…3k†



g1m ˆ g1mÿ1 ‡ g0 / ÿ Volmÿ1
twisted =Dt:

…3l†

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 matrix G is
formed. G has 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 ORTHOMIN 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 approximation for streamlines then eqn (3l) may be extended
for this case as,
2
3
,
Z
Z
6
7
m
mÿ1
/ dX5 Dt:
g1 ˆ g1 ‡ 4 / dX ÿ
…3m†
X…t2 †

X…t1 †mÿ1

R
But the exact evaluation of 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 intersected 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,



vx2 ‡ vx4 vx1 ‡ vx3
vy3 ‡ vy4
Dy
‡ Dx
ÿ
2
2
2

vy1 ‡ vy2
:
…3n†
ÿ
2
Here vx1 refers to the x-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 k h and a uniform, square grid, the
discretization of g1 according 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 Laplace 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
only the 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 if g1 involves velocities not only at the corners
but also at other points on the surface of the cell. What

A characteristic-conservative model for Darcian advection

603

Fig. 4. (A) The construction of the twisted cell 10 20 40 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 dimensions when both the corners and the centers of the edges are traced back. (D) A stable ®nite di€erence operator results from the
added degrees of freedom to the twisted cell in (C).

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.

Equating this to the area of the regular grid cell
area…1397†, the new expression for g1 in place of
eqn (3n) is,
Dy ‰ÿvx1 ÿ 2vx4 ÿ vx7 ‡ vx3 ‡ 2vx6 ‡ vx9Š=4
‡ Dx‰ÿvy1 ÿ 2vy2 ÿ vy3 ‡ vy7 ‡ 2vy8 ‡ vy9Š=4:
…3o†

3.2 Re®ned volume computation
Following the above discussion we allow the regular
grid cell to twist into more complex shapes. For example, 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…10 20 50 40 † ‡ area…20 30 60 50 † ‡ area…40 50 80 70 †
‡ area…50 60 90 80 †:

With this g1 the ®nite di€erence operator has the weights
shown in Fig. 4(D) leading to a stable and more accurate (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
/),

604

A. Chilakapati

Fig. 5. Locations of the 26 degrees of freedom for the grid cell
in three dimensions.

/‰volume …10 20 40 50 100 110 130 140 †
‡ volume …50 60 80 90 140 150 170 180 †
volume …110 120 140 150 200 210 230 240 †
‡ volume …130 140 160 170 220 230 250 260 †Š
/‰volume …20 30 50 60 110 120 140 150 †
‡ volume …40 50 70 80 130 140 160 170 †
volume …100 110 130 140 190 200 220 230 †
‡ volume …140 150 170 180 230 240 260 270 †Š

…3p†

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 oddp ˆ 1. 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 oddp ˆ 0. 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 approximation to Darcy's law. The matrix G is usually
nonsymmetric and has the structure of a 27-point operator. 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.
…3q†
Voltwisted ˆ Volregular ‡ q^w Dt;
where q^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 contained 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 size Dt should be small enough so that
the Voltwisted for the injection well-block does not become negative. That is, we need the volume of the ¯uid
entering the injection well-block during the time Dt to be
smaller than the volume of that well-block. For a production 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 wellblock. 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 summarize,
 i j k 
Dxw Dyw Dzw /
;
…3r†
Dt 6 min
…i;j;k;w†
j q^w j
where q^w is the ¯ow-rate (L3 =T ) into the well-block located at …i; j; k† and Dxiw is the x-width of that wellblock. Dywj and Dzkw are similarly de®ned. This is similar
to a CFL constraint on Dt. Fig. 6 shows the shape of a
twisted injection well-block. This is obtained by placing
a unit strength (q ˆ )1.0) source at the center of 3  3 
3 m3 . A 3  3  3 grid is used. The source is placed in
the cell (2,2). Constant ( 0.0) potential conditions are
used on all the six faces. k h  1.0, /  1.0, and Dt ˆ 0.8
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 time t1 to reach the boundary at time t2 , can
be identi®ed. If the ¯ow enters, then a part of the corresponding 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.

A characteristic-conservative model for Darcian advection
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 volume 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 entering a grid cell from outside is that portion of the
volume of the corresponding twisted cell that is outside.
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 according 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

605

out¯ow boundaries, all the twisted cells are inside the
problem domain. The volume of the problem domain
that is not part of any twisted cell, is the volume of ¯uid
that leaves through the out¯ow boundary. In the absence 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 (Y ˆ 0, Y ˆ Ly ). A 7  7 uniform grid is used on a
domain with Lx ˆ Ly ˆ 1:0 m. The nonuniform but
isotropic hydraulic conductivity (meters) for each cell is

Fig. 7. (A) The 7  7 grid and the value of k in 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
lines, in the same order as in the regular cell.

606

A. Chilakapati

shown in Fig. 7(A). A uniform porosity / ˆ 1.0, and a
time step size Dt ˆ 0.04 days are used. The convergent/
divergent ¯ow-®eld obtained by the characteristic-conservative 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 at X ˆ 0:0, parts of the
twisted cells at the inlet boundary lie outside the problem domain. All the other twisted cells lie inside the
problem domain. The maximum relative error in volumes obtained by the characteristic-conservative method 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€erence 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
x ˆ 0 boundary in one time step, i.e. 1:0  Dt ˆ 0:04 m3
here. Since the Y-boundaries have a no¯ow condition,
the out¯ow rate at x ˆ 1:0 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 1:0  Dt ˆ
0:04 m3 , which is the amount of ¯uid that leaves the x ˆ
1:0 boundary in a single time step. The characteristicconservative 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 ®nitedi€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 (q ˆ )1.0/day) at the grid location (2,2) and a
sink of unit rate (q ˆ 1.0/day) at the grid location (6,6).
Constant potential ( ˆ 0.0 m) conditions are used on the
boundaries x ˆ Lx ; y ˆ Ly . No¯ow conditions are used
on the boundaries x ˆ 0; y ˆ 0. Uniform conductivity k
ˆ 1.0, porosity / ˆ 1.0, and a time step size of Dt ˆ 0.02
days are used. It follows from eqn (3r) that the maximum 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† ˆ 0:020408. 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^w Dt ˆ …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 ‡ 0:02  0:040408.
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=…7:0  7:0†† in excess.
E€ect of Dt. The velocity ®eld obtained here is dependent 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 on Dt 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 traceback-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 appropriate correction to the ¯ow-®eld so
that the characteristic-conservative method preserves
mass balance. This is made clear in the following example. 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 the X-direction and a no¯ow condition on the transverse faces. A 33  33 uniform 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 at X ˆ 0.0
to 0.0 at X ˆ 1.0, the e€ect of Dt on 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.
1. The characteristic-conservative method has been
implemented here by approximating the streamlines to be straight. This allows the formulation
of volume invariance condition in terms of cellcentered potentials, thus obtaining a velocity ®eld
that preserves ¯uid volume and local, global mass
balance. When the error associated with the
straight line approximation for the streamlines is

A characteristic-conservative model for Darcian advection

607

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
not preserve volumes.

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 solution to a linear problem increases the computational e€ort here compared to a cell-centered
scheme for velocities. In particular, this introduces
inner and outer iteration loops since the inner linear problem also is solved iteratively with
ORTHOMIN. Because of the non-symmetric 27point 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 recommended if the Darcy ¯ow problem alone needs to
be solved. The bene®ts of this volume-balance approach for velocities are realized only when it is
used in conjunction with a method of characteristics 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 number 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 transport 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 advection problem, while using a characteristics
based method, is a de®nite drawback with the
characteristic-conservative method. To some extent it defeats the advantage o€ered by the characteristics based method. However for many reactive
transport problems, the time step size may have to
be anyway limited, either because reactions have

608

A. Chilakapati

been time-split from advection, or simply because
of nonconvergence owing to large changes in reaction terms. Also the time step constraint that prevents 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 integral. An exact integration over X…t1 †, will help
preserve local and global mass balance but will
also require a larger CPU, compared to an approximate numerical integration by quadrature
rules. An increased accuracy in the numerical integration 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 integration or by an accurate numerical integration, has
to be justi®ed by the need for good local mass balance.
6. The number of species, and the complexity of the
reaction mechanisms (if reactions are present) do
not have any special consequences to the characteristic-conservative method. When there are multiple 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 modeling of the adsorbed species under equilibrium
with its counterpart in solution, and time-splitting
reactions, can avoid the retardation factor approach. This will cause all the mobile species to
move with the same speed, thus requiring the mapping of X…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 incompressible ¯ow and transport problem is presented and
implemented in multiple dimensions. Velocities are obtained 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 for g1 in eqn (3l) or eqn (3m) is
given by the following.
g1 ˆ 1=6…ÿ1=2vy10 ÿ 1=2vy21 ÿ 2vy11
ÿ 1=2vy2 ÿ 1=2vy20 ÿ 1=2vy19 ‡ 1=2vy27
‡ 1=2vy8 ‡ 1=2vy9 ‡ 1=2vy16 ‡ 2vy17
‡ 1=2vy26 ÿ 1=2vy12 ‡ 1=2vy18 ÿ 1=2vy1
‡ 1=2vy7 ÿ 1=2vy3 ‡ 1=2vy25†DxDz
‡ 1=6…2vz23 ÿ 2vz5 ‡ 1=2vz19 ‡ 1=2vz26
ÿ 1=2vz4 ÿ 1=2vz6 ‡ 1=2vz27 ‡ 1=2vz22
ÿ 1=2vz2 ÿ 1=2vz8 ‡ 1=2vz21 ÿ 1=2vz1
‡ 1=2vz24 ‡ 1=2vz20 ÿ 1=2vz3 ÿ 1=2vz7
‡ 1=2vz25 ÿ 1=2vz9†DxDy
‡ 1=6…1=2vx21
ÿ 1=2vx7 ‡ 1=2vx9 ÿ 2vx13 ÿ 1=2vx19
‡ 1=2vx18 ÿ 1=2vx10 ÿ 1=2vx16 ÿ 1=2vx4
‡ 1=2vx27 ÿ 1=2vx1 ‡ 1=2vx12 ÿ 1=2vx22
ÿ 1=2vx25 ‡ 1=2vx6 ‡ 1=2vx3 ‡ 1=2vx24
‡ 2vx15†DzDy
vx1 . . . vx27; vy1 . . . vy27 and vz1 . . . vz27 are the x, y and
z velocities at the locations 1 . . . 27 on the regular grid
block (Fig. 5). Dx; Dy and Dz are the widths of the grid
block in the x, y and z directions.

REFERENCES
1. Arbogast, T. and Wheeler, M. F. A characteristics-mixed
®nite element method for advection dominated transport
problems. SIAM J. Numer. Anal., 1995, 32, 404±425.
2. Bear, J. Dynamics of Fluids in Porous Media. Dover, New
York, 1972.
3. Binning, P. and Celia, M. A. A ®nite volume EulerianLagrangian adjoint method for solution of the contaminant transport equations in two-dimensional multiphase
¯ow systems. Water Resources Research, 1996, 32(1), 103±
114.
4. Celia, M. A., Russell, T. F., Herrera, I. and Ewing, R.E.
An Eulerian-Lagrangian localized adjoint method for the
advection-di€usion equation. Advances in Water Resources, 1990, 13(4), 187±206.
5. Chiang, C. Y., Wheeler, M. F. and Bedient, P. B. A
modi®ed method of characteristics technique and a mixed

A characteristic-conservative model for Darcian advection

6.
7.

8.

9.

10.

11.

®nite element method for simulation of groundwater
solute transport. Water Resources Research 1989, 25,
1541±1549.
Chilakapati, A. Numerical Simulation of Reactive Flow
and Transport Through the Subsurface. Ph.D. thesis, Rice
University, Houston, TX, 1993.
Douglas, J. and Russell, T. F. Numerical methods for
convection-dominated di€usion problems based on combining the method of characteristics with ®nite element or
®nite di€erence procedures. SIAM Journal of Numerical
Analysis, 1982, 19, 871±885.
Ewing, R. E., Russell, T. F. and Wheeler, M. F. Simulation of miscible displacement using mixed methods and a
modi®ed method of characteristics. In Proceedings of the
Seventh SPE symposium on Reservoir Simulation, SPE
12241, November 1983.
Healy, R. W. and Russell, T. F. A ®nite-volume EulerianLagrangian localized adjoint method for solution of the
advection-dispersion equation. Water Resources Research,
1993, 29(7), 2399±2413.
Healy, R. W. and Russell, T. F. Solution of the advectiondispersion equation in two dimensions by a ®nite-volume
Eulerian-Lagrangian localized adjoint method. Advances
in Water Resources, 1998, 21(1), 11±26.
Hirsch, C. Numerical Computation of Internal and External Flows, Vol. 1. Wiley, New York, 1990.

609

12. Hirt, C. W., Cook, J.L. and Butler, T.D. A Lagrangian
method for calculating the dynamics of an incompressible
¯uid with free surface. Journal of Computational Physics,
1970, 5, 103±124.
13. Huang, K., Zhang, R. and van Genuchten, M. T. An
Eulerian-Lagrangian approach with an adaptively corrected method of characteristics to simulate variably saturated
water ¯ow. Water Resources Research, 1994, 30(2), 499±
507.
14. Moore, D. Understanding Simploids. In David Kirk,
editor, Graphics Gems III, ch. 5, Academic Press, New
York, 1992, pp.250±255.
15. Oppe, T. C., Joubert, W. D. and Kincaid, D. R. NSPCG
User's Guide, Version 1.0, April 1988.
16. Russell, T. F. Finite Elements with Characteristics for
Two-Component Incompressible Miscible Displacement.
In Proceedings of the Sixth Symposium on Reservoir
Simulation of the SPE, SPE 10500, February 1982.
17. Russell T. F. and Trujillo, R.V. Eulerian-Lagrangian
localized adjoint methods with variable coecients in
multiple dimensions. In G. Gambolati et al., editor,
Computational Methods in Subsurface Hydrology: Proceedings of the Eighth International Conference o