The Effect of Fractures on the Reclamation ofNAPL
65
and the finite difference approximation ofEq. (19) yields:
I
[k f3 -1
imax-l
1
+
L Chi + L Chi
1 + N M i=2
i=kf3+1
(25)
Here, subscripts refer to the vertical position of nodal points whose common
longitudinal position is ilP and all quantity values are calculated at the same time
step. Subscripts kf3. and k(3+ of Cb refer to values upstream and downstream of the
fracture segment, respectively. Eventually values of C k and C bk are identical.
P
P+
If the integer f3 satisfies the following expression
f3Vmax -1)+ I = ia ,
(26)
then the finite difference approximation ofEq. (19) yields:
I 1 [I (
) i max -I ]}
Cay =
CI +t1xNM -\Cbl +Cbi
+ L Cbi
1 + N M
2
max
i=2
(27)
Here again, subscripts refer to the vertical position of the nodal points whose
common longitudinal position is ilP and all quantity values are calculated at the
same time step.
Rubin et al. (1997) used numerical simulations to study the transient period
during which quasi-steady state conditions are established in the fractured
permeable aquifer following the startup of pump-and-treat systems. Results of
that study suggest that the initial values of Cb in the current study should be
specified by the solute concentrations obtained after the establishment of quasisteady-state conditions. However, because the transient period is short relative to
the period of pump-and-treat remediation examined in this study, it is possible to
consider that initially Cb = 0 in the entire domain, with negligible influence on the
numerical results.