8.3 Large unsaturated zone
In order to test the variable substitution strategies, the pro- blem shown in Fig. 16 was used. Note that this problem has
a large unsaturated zone, with hydrostatic pressure con- ditions on the left and right hand boundaries location of
water table as indicated in Fig. 16. This is a cross-section, with thickness 0.25 m in the direction normal to the cross-
section. The domain was discretized using a 16 3 11 grid, Dx ¼ Dz ¼ 0.25 m.
The data in Table 3, Table 5 and Table 6 were used. In addition, another set of relative permeability and capillary
pressure data was tested Table 7 and Table 8. The data in these tables is representative of a Borden sand
47
. In parti- cular, the capillary pressure data is based on the dimension-
less capillary pressure curves in Ref.
48
. A number of cases were tested, using both LNAPL and
DNAPL data Table 4, and both equilibrium eqn 2 and nonequilibrium eqn 4, eqn 5, eqn 7 assumptions. The
problem was initially fully saturated, and was run to 7 days with no NAPL injection to equilibrate the unsaturated flow
model. From 7.0 to 7.1 days, a NAPL contaminant was injected at a rage of 0.153 m
3
day. Injection then ceased, and the simulation was run to 500 days.
9 TEST RESULTS 9.1 LIQUID_STATE variable substitution
As discussed
previously, variable
substitution for
LIQUID_STATE primary variables is not actually required since P
n
can be used as a primary variable without switching to S
w
. However, in
26
, it was demonstrated that this type of variable switching speeded up the convergence of the
Newton iteration for unsaturated flow problems under very dry conditions. Sometimes an order of magnitude
improvement in
number of
Newton iterations
was observed
26
. However, this improvement was only seen for very dry conditions, where the capillary pressure–
saturation curve had a very large derivative. We did not anticipate this type of problem for multi-phase
passive air phase simulations, since we were mainly inter- ested in problems near and below the water table, which
have water saturations well above the critical value, due to the capillary fringe. However, rather to our surprise, we
found that variable substitution had a large effect for simu- lations involving a NAPL phase.
As a first example, consider the LNAPL Problem Section 8.1. Two runs were carried out. The first run disabled the
LIQUID_STATE variable substitution eqn 27 by setting tol
f
¼ tol
b
¼ ¹` see eqn 27, in which case P
n
was always the primary variable. The second run used the vari-
able switching parameters tol
f
¼ 0.99 and tol
b
¼ 0.9. This
problem was run to a stopping time of 11 days. Full Newton iteration was used with a van Leer flux limiter. There was
virtually no difference between these two methods during the initial part of the run to 7 days, which was simply an
unsaturated flow situation no NAPL. The run using LIQUID_STATE switching used 72 Newton iterations,
compared with 80 Newton iterations for the run without LIQUID_STATE switching, to complete the initial 7 days.
This indicates that the initial unsaturated flow part of this scenario is not particularly difficult, in contrast to the situa-
tion in
26
. This is to be expected since the capillary pressures do not have the large derivatives with respect to saturation,
Fig. 16. Computational domain for test of variable substitution.
Table 7. NAPL: water data: data set 2
S
w
k
rw
k
rn
P
cnw
kpa 0.074
0.0 0.45
9.4 0.1
0.001 0.42
7.0 0.2
0.005 0.34
4.2 0.3
0.025 0.28
3.3 0.4
0.05 0.22
2.8 0.5
0.08 0.17
2.5 0.6
0.14 0.115
2.2 0.7
0.22 0.065
2.0 0.8
0.36 0.025
1.9 0.85
0.46 0.01
1.85 0.9
0.57 0.0
1.8 1.0
1.0 0.0
0.0
Table 8. Liquid–gas data: data set 2
S
w
þ S
n
k
ra
k
rn
P
can
kpa P
caw
kpa 0.074
0.65 0.0
9.0 11.8
0.1 0.6
0.0 5.0
6.1 0.2
0.45 0.02
3.1 4.5
0.3 0.34
0.005 2.4
3.8 0.4
0.25 0.09
2.0 3.4
0.5 0.18
0.13 1.75
3.0 0.6
0.12 0.17
1.5 2.8
0.7 0.07
0.22 1.25
2.6 0.8
0.03 0.28
1.0 2.5
0.9 0.0033
0.35 0.9
2.4 0.92
0.0012 0.37
0.88 2.38
0.95 0.0
0.4 0.85
2.35 1.0
0.0 0.45
0.0 0.0
S
n
¼ 0.1.
Nonlinear multiphase flow 443
as in
26
. The run statistics for the complete run to 11 days are shown in Table 9. Clearly, the use of LIQUID_STATE
variable substitution has a very large effect on performance after the NAPL is present in the system.
At this point, we changed the input data to try to deter- mine the cause of this problem. Problems disappeared, for
example, when P
cnw
¼ 0.0, and when the P
caw
and P
can
were changed to straight line curves. Consequently, this problem
definitely appears to be due to the non-linearities in the capillary pressure curves. Adjusting the value of S
n
in eqn 11 occasionally caused improvement, but sometimes
made matters worse. It would therefore appear that the switching between P
can
and P
caw
as in eqn 9 was not the primary source of the problem. Note that other models
of the two-phase to three-phase capillary pressure transition are similar in spirit. For example in
11
a P
caw
curve is used when no NAPL is present, and the three-phase P
cnw
curve is used for situations where free NAPL is present. For cases
where the NAPL is trapped i.e. below residual, a weighted average of water- and gas-phase pressures is used for P
n
. An extensive series of tests was carried out to determine
the range of liquid saturations where the LIQUID_STATE switching was effective, and where it ceased to be effective.
The results are summarized in Table 10. Rather surprisingly, there was little effect on the number of Newton iterations for
the parameters eqn 27, until an increase was observed at a value of tol
f
, tol
b
¼ 0.4, 0.3. Lowering tol
f
, tol
b
any further caused the number of Newton iterations to increase
to the number observed when the LIQUID_STATE switch- ing was entirely disabled.
Consequently, it would appear that it is necessary to use saturations as primary variables until the liquid saturation is
above 0.5. Since the problem in convergence seems to occur near the water table, this suggests that convergence difficul-
ties are encountered as the NAPL phase becomes mobile at a given node, but this is not entirely clear. The safest choice
of parameters appears to be use a value of tol
f
which is as large as possible. In the following, we will use tol
f
¼ 0.99
and tol
b
¼ 0.9.
To study the effect of LIQUID_STATE switching in more detail, a comprehensive series of tests was carried
out using the Large Unsaturated Zone problem see Section 8.3. If the LIQUID_STATE switching mechanism was
enabled, we used the following parameters for tol
f
and tol
b
in eqn 27. tol
f
¼ 0:99 tol
b
¼ 0:9:
33 Full Newton iteration was used with a van Leer flux limiter.
A number of cases were tested, using both LNAPL and DNAPL data Table 4, and both equilibrium eqn 2
and nonequilibrium eqn 4, eqn 5, eqn 7 assumptions. In addition, another set of relative permeability and capil-
lary pressure data was tested Tables 7 and 8. The total Newton iteration counts for these tests are given in Table 11
CPU time roughly proportional to Newton iteration count. The total iteration counts are given for the complete run to
500 days.
These tests indicate that LIQUID_STATE variable sub- stitution is always faster in terms of Newton iteration
count than not using LIQUID_STATE switching. The decrease in Newton iteration count varies from more than
a factor of 10 to less than a factor of 2. The increase in efficiency obtained using LIQUID_STATE switching is
clearly highly dependent on the physical properties data, and on the modelling assumptions i.e. equilibrium, none-
quilibrium. However, in all cases involving LNAPLs, dis- abling the LIQUID_STATE state switching increased the
number of Newton iterations by more than a factor of 5.
We have seen the same results in many tests. LIQUID_STATE state variable substitution is never signif-
icantly slower than using P
n
as a primary variable, and is sometimes an order to magnitude faster. We emphasize that
this is a different effect form that in
26
, since the work in
26
considered unsaturated flow in very dry soils. In the problems in this work, the capillary pressure data for two-phase
Table 9. Performance of variable switching: LNAPL example, nonequlibrium model, stopping time 11 days
Variable substitution method
Total Newton iterations
Normalized CPU time
LIQUID_STATE switching
405 1.0
No LIQUID_STATE switching
1298 2.96
Table 10. Performance of variable switching: effect of switch- ing parameters: CPU time proportional to Newton iteration
count
tol
f
, tol
b
Total Newton iterations 0.9, 0.8
433 0.8, 0.7
409 0.7, 0.6
405 0.6, 0.5
489 0.5, 0.4
483 0.4, 0.3
1118
Table 11.
Total Newton
iteration count
for test
of LIQUID_STATE switching Fig. 16: CPU time proportional
to Newton iteration count
Scenario Switching
No switching Data in Tables 5 and 6
Equilibrium LNAPL
160 2506
DNAPL 163
2703 Nonequilibrium
LNAPL 205
1199 DNAPL
142 1472
Data in Tables 7 and 8 Equilibrium
LNAPL 205
1199 DNAPL
228 707
Nonequilibrium LNAPL
169 1064
DNAPL 177
332
444 P. A. Forsyth et al.
air–water flow does not give rise to convergence pro- blems, since the number of Newton iterations is virtually
the same for both LIQUID_STATE switching and no LIQUID_STATE switching runs, before the NAPL is
injected. Convergence difficulties arise only when the NAPL is injected. Although the cure for convergence
problems is similar to that used in
26
, the cause of these problems appears to be different.
In the following examples, we will always use LIQUID_STATE variable substitution.
9.2 Jacobian selection for the flux limiter: experiments
Kategori