can also have a profound effect on the convergence of the Newton iteration
8,9,11,26,35
.
4.1 Equilibrium primary variables
The equilibrium equations eqns 14 and 15 are two partial differential equations and hence require two primary
variables. The state of a node is given by two state indicators {NAPL_STATE, LIQUID_STATE}. The primary variables
are given in Table 1. For example, if node i is in the state {napl_on, Sw_primary}, the primary variables are
{S
ni
, S
wi
}. The transition rules are: IF
LIQUID_STATE ¼ Sw_primary AND S
wi
tol
f
LIQUID_STATE : ¼ Pn_PRIMARY ELSEIF
LIQUID_STATE ¼ Pn_primary AND S
wi
, tol
b
LIQUID_STATE : ¼ Sw_primary ENDIF
27 IF
NAPL_STATE ¼ napl_on AND S
ni
, 0 NAPL_STATE : ¼ napl_off
S
ni
¼ ELSEIF
NAPL_STATE ¼ napl_off AND X
ni
. X
p ni
NAPL_STATE : ¼ napl_on X
ni
¼ X
p ni
ENDIF 28
Note that the LIQUID_STATE variable substitution is not strictly required, since we can always solve the system of
eqn 14 and eqn 15 if we use P
n
as a primary variable for the LIQUID_STATE. In
26
it was demonstrated that using the LIQUID_STATE variable switching resulted in a large
gain in efficiency for unsaturated flow under dry conditions compared to using P
n
as a primary variable. Somewhat surprisingly, we have found using the LIQUID_STATE
variable switching to be more efficient when solving multi-phase flow problems. We will demonstrate this
effect in some numerical examples. This phenomena has also been observed previously
36
.
4.2 Nonequilibrium primary variables
In the case of nonequlibrium between phases eqns 14, 18 and 19 there are three primary variables required. In
this case, there is only one state indicator {LIQUID_ STATE}. The primary variables are described in Table 2, and the
transition rule is given in eqn 27.
5 NONLINEAR ITERATION METHODS
As discussed in Section 1, we will consider fully implicit methods in the following. In our experience, it is essential to
solve for the saturation in a fully coupled, fully implicit manner for reliable, robust simulations. It is possible that
adaptive implicit methods
3
may prove to be useful in some situations. As well, we will restrict attention to a single
species NAPL contaminant. If there are a large number of reacting chemical components, it may be more attractive to
use some form of operator splitting.
5.1 Jacobian selection for Newton iteration
Kategori