The physical properties data are given in Table 3 and Table 4. The longitudinal dispersivity is typical of the
Borden sand, based on matching field scale experiments
46
. The value of b in Table 3 b ¼ 1.0 is typical of that obtained by matching field-scale experiments
13
. Laboratory experiments typically have b . 0.5–0.75
16
. Note that in the following we assume that the permeability tensor is
diagonal, with diagonal entries in the x and z directions k
x
, k
z
: K ¼
k
x
k
z
31 The Courant number for this problem, based on the max-
imum timestep of 10 days was . 10. The Peclet number is about . 2, for the dispersion parameters in Table 3. The
LNAPL data in Table 4 was used for this example. The relative permeability and capillary pressure data are given
in Table 5 and Table 6. Hysteresis effects are ignored in all simulations reported in this paper.
This problem is initially fully saturated S
w
¼ 1 every-
where, and was run to 7 days with no NAPL injection to equilibrate the water table. LNAPL was then injected a rate
1.6 m
3
day for 0.5 days at the point shown in Fig. 2. Injec- tion of LNAPL then ceased, and the problem was run to
completion. Fig. 3 shows the Log
10
normalized mole frac- tion X
n
X
n
at 11 days for this run, using a nonequilibrium model eqn 1, eqn 4, eqn 5 and eqn 7.
Fig. 4 shows the saturation contours for this problem, but using the equilibrium model. Comparing the contours at
23.5 and 365 days, it is clear that a large portion of the LNAPL has been dissolved by the flowing water phase.
In contrast, Fig. 5 shows the LNAPL saturation contours at 365 days using the nonequilibrium model. Comparing
Fig. 5 with Fig. 4 365 days, we can see that the extent of the NAPL phase is much greater for the nonequilibrium
model than for the equilibrium model. Of course, this is to be expected since the nonequilibrium model will tend to
dissolve the NAPL phase slower than the equilibrium model.
8.2 DNAPL problem
Fig. 6 shows the computational domain for these tests, which consists of a 50 3 40 grid, with Dx ¼ 1.0 m, Dz ¼
0.1 m. A fine grid 99 3 79 was also used for this problem, with all node spacing halved. A unit thickness was specified
in the y direction normal to the cross-section. Hydrostatic pressure boundary conditions are imposed with the water
table elevations as shown in Fig. 6.
All the physical data used for this example was as in Table 3 and Table 4. The Courant number for this problem,
Table 3. Physical properties data
k
x
¼ k
z
1.0 3 10
¹ 11
m
2
X
n
0.00285 M
n
0.74 cp a
L
0.50 m a
TH
0.03 m a
TV
0.001 m d
w
0.0 m
2
day l
R
2.0 day
¹ 1
b 1.0
r
b
K
d
0.03 l
A
1.0 day
¹ 1
f 0.3
Table 4. Density data for the examples
LNAPL data M
n
9.0 3 10
3
molm
3
r
n
100.1 3 10
¹ 3
kgmol DNAPL data
M
n
11.4 3 10
3
molm
3
r
n
131.1 3 10
¹ 3
kgmol Water data
M
w
55.5 3 10
3
molm
3
r
w
18.02 3 10
¹ 3
kgmol
Table 5. NAPL: water data: data set 1
S
w
k
rw
k
rn
P
cnw
kpa 0.20
0.0 0.68
9.0 0.30
0.04 0.55
5.4 0.40
0.10 0.43
3.9 0.50
0.18 0.31
3.3 0.60
0.30 0.20
3.0 0.70
0.44 0.12
2.7 0.80
0.60 0.05
2.4 0.90
0.80 0.0
1.5 1.0
1.0 0.0
0.0
Table 6. Liquid–gas data: data set 1
S
w
þ S
n
k
ra
k
rn
P
can
kpa P
caw
kpa 0.20
0.64 0.0
9.0 6.0
0.32 0.46
0.00 3.0
4.5 0.40
0.36 0.0009
2.4 3.9
0.50 0.25
0.045 2.1
3.6 0.60
0.16 0.116
1.8 3.3
0.70 0.09
0.21 1.5
3.0 0.80
0.04 0.34
1.2 2.0
0.90 0.01
0.49 0.9
1.0 0.95
0.00 0.58
0.5 0.5
1.0 0.0
0.68 0.0
0.0 S
n
¼ 0.2.
Fig. 3. Log
10
normalized mole fraction, 11 days, LNAPL problem, nonequilibrium model.
440 P. A. Forsyth et al.
based on the maximum timestep of 10 days was . 20. The Peclet number is about . 2, for the dispersion parameters in
Table 3. A few results will also be reported for a case where all dispersion constants are decreased by a factor of 10
compared to the base case in Table 3, which results in convection-dominated flow with a Peclet number of . 20.
Two variations of this scenario were tested.
• Homogeneous: a constant absolute permeability
was used see Table 3. •
Heterogeneous: the second case used highly hetero- geneous absolute permeability data, with perme-
abilities ranging from 10
¹ 10
to 10
¹ 15
m
2
. The exponent in the mass transfer eqn 7 for the hetero-
geneous problem was altered to b ¼ 1.5, while the time constant was l
R
¼ 2.0 days
¹ 1
. This data is typical of field-scale experiments
13
. The Hetero- geneous absolute permeability data was all perme-
ability units m
2
, and x, z units are m: k
x
¼ k
z
¼ 10
¹ 15
13 x 16 ,
2:95 z 3:05 32
k
x
¼ k
z
¼ 10
¹ 15
10 x 14 ,
2:75 z 2:85 k
x
¼ k
z
¼ 10
¹ 15
15 x 17 ,
2:55 z 2:65 k
x
¼ k
z
¼ 10
¹ 13
29 x 34 ,
1:55 z 2:05 k
x
¼ k
z
¼ 10
¹ 14
19 x 29 ,
2:25 z 2:55 k
x
¼ k
z
¼ 10
¹ 10
9 x 39 ,
0:65 z 1:15 k
x
¼ k
z
¼ 10
¹ 11
everywhere else Initially, these problems were fully saturated S
w
¼ 1
everywhere, and were first run for 7 days to equilibrate the water table no NAPL injected, and then the DNAPL
was injected for 0.1 days at the point shown in Fig. 6 at a rate of 0.8 m
3
day. DNAPL injection then stopped, and the problem was run to completion.
Fig. 4. NAPL saturation contours, homogeneous LNAPL problem,
equilibrium model.
Fig. 5. NAPL saturation contours, 365 days, homogeneous LNAPL problem, nonequilibrium model. Compare with Fig. 4,
365 days.
Fig. 6. Two-dimensional cross-section computational domain for
the DNAPL scenarios.
Fig. 7. Log
10
normalized mole fraction, 11 days, van Leer flux limiter, homogeneous DNAPL problem, equilibrium model,
coarse grid.
Fig. 8. Log
10
normalized mole fraction, 11 days, MUSCL flux limiter
21
, homogeneous DNAPL problem, equilibrium model, coarse grid.
Fig. 9. Log
10
normalized mole fraction, 11 days, upstream weight- ing, homogeneous DNAPL problem, equilibrium model, coarse
grid.
Nonlinear multiphase flow 441
In the following, we will be testing the effect of various nonlinear iteration strategies for use in conjunction with
MUSCL and van Leer flux limiters. It is therefore, of inter- est to see what effect use of these different limiters has on
the computed solution. To illustrate the effects of these different methods for discretization, we show some results
for the homogeneous DNAPL problem, equilibrium model, using the coarse grid 50 3 40. Figs 7–9 show the log
10
normalized mole fraction X
n
X
n
contours at 11 days, using van Leer eqns 23–25, MUSCL eqn 26 and upstream
methods. Figs 10–12 show the same results using the fine 99 3 79 grid.
Note than Van Leer and MUSCL weighting give similar results at both coarse and fine grids. Upstream weighting
appears to be very slowly converging, and shows considerable numerical dispersion compared to the flux limiter methods.
A complete comparison of grid refinement studies showing that the van Leer limiter converges more quickly than
upstream weighting as the grid is refined is given in
20
, and will not be repeated here.
We will also be using the heterogeneous version of the DNAPL problem in the following. Figs 13 and 14 show the
log normalized NAPL contaminant mole fraction contours at 11 days, for both equilibrium and nonequilibrium models.
Note that the plume for the nonequilibrium model has not spread as far as the equilibrium model, which is to be expected.
As discussed in
20
, the use of the maximum potential method
for determining
the second
upstream point
would seem to be more physically reasonable compared to the geometric method, especially for heterogeneous problems.
Fig. 15 compares the normalized contaminant mole frac- tion contours at 100 days, for the heterogeneous DNAPL
problem, nonequilibrium model, using both the geometric and maximum potential methods for location of the second
upstream point. Van Leer weighting was used for these examples. Note that there appears to be little difference
between using either geometric and maximum potential methods, consistent with the results in
20
. However, as will be shown, there is a considerable difference in the
numerical performance in terms of CPU cost of these two methods.
Fig. 10. Log
10
normalized mole fraction, 11 days, Van Leer flux limiter, equilibrium model, fine grid.
Fig. 11. Log
10
normalized mole fraction, 11 days, MUSCL flux limiter, equilibrium model, fine grid.
Fig. 12. Log
10
normalized mole fraction, 11 days, upstream weighting, equilibrium model, fine grid.
Fig. 13. Log
10
normalized mole fraction, 11 days, heterogeneous DNAPL problem, equilibrium model, coarse grid.
Fig. 14. Log
10
normalized mole fraction, 11 days, heterogeneous DNAPL problem, nonequilibrium.
Fig. 15. Log
10
normalized mole fraction, 100 days, heterogeneous DNAPL, problem, nonequilibrium model coarse grid. Comparison
of maximum potential and geometric second upstream point method.
442 P. A. Forsyth et al.
8.3 Large unsaturated zone
