The Effect of Fractures on the Reclamation ofNAPL
63
using a simplified model incorporating analytical and numerical calculations. In
this study, NAPL dissolution and constituent transport is considered only in the
portion of the domain contaminated with entrapped NAPL.
As shown in Fig. 3, nodal points in the vertical direction are denoted by i (i =
1 ... ima,), and nodal points in the longitudinal direction are denoted by j. The
portion of the domain containing entrapped NAPL extends to the longitudinal
nodal points given by ic = [nc (imax - 1) + 1]. The fracture network is divided to
consecutive segments. Each one of the domain sections and fracture segments is
enumerated with an ifr number. Each block of the domain is enumerated with an
ibl number. Nodal points along the fracture segment are denoted by k (k =
1 ... imax).
The finite difference approximation of Eq. (6), applied to the calculation of Cat
the k+ 1 nodal point of the ifr fracture segment at the m time step, is given by
(Rubin et al. 1997):
C;+I[l+~ NM ]=C;[l-~ NM ]+~ NM[c:k+1 +c:J,
(20)
where C bk and C bk + 1 are values of Cb entering the nodal points k and k+1 of
the fracture segment, respectively. The finite difference approximation of Eq. (14)
is given by:
1 C m+1
-
b' '+1
[
]
___ '.:::,j_=exp -K f
m . 1/2,1tl .
l-Cbm.
l,j+
l,j
(21)
A finite difference approximation of Eq. (16) is used to evaluate the entrapped
NAPL saturation at the midpoint of the longitudinal interval, as follows:
S m+1
S m
C ~C m+1 C m ~t
n' '+1/2 = n' '+112 - nv b' '+1 - b"
.
l,j
l,j
l,j
l,j L1x
(22)
The flux average concentration at the cross-section perpendicular to the flow
direction and represented by the longitudinal nodal points ia is evaluated with a
finite difference approximation of Eq. (19), trapezoidal numerical integration, and
appropriate reference to the crossed fracture segment.
Consider an integer number 13 satisfying the following expression
f3Vrnax -1)+ 1 < ia < (13 + l)(jrnax -1)+ 1.
(23)
Then the cross section chosen for the calculation of Cay crosses the fracture
segment at the nodal point k{Jo, which is given by
k{3 = ia - f3Vrnax -1)-1,
(24)
63
using a simplified model incorporating analytical and numerical calculations. In
this study, NAPL dissolution and constituent transport is considered only in the
portion of the domain contaminated with entrapped NAPL.
As shown in Fig. 3, nodal points in the vertical direction are denoted by i (i =
1 ... ima,), and nodal points in the longitudinal direction are denoted by j. The
portion of the domain containing entrapped NAPL extends to the longitudinal
nodal points given by ic = [nc (imax - 1) + 1]. The fracture network is divided to
consecutive segments. Each one of the domain sections and fracture segments is
enumerated with an ifr number. Each block of the domain is enumerated with an
ibl number. Nodal points along the fracture segment are denoted by k (k =
1 ... imax).
The finite difference approximation of Eq. (6), applied to the calculation of Cat
the k+ 1 nodal point of the ifr fracture segment at the m time step, is given by
(Rubin et al. 1997):
C;+I[l+~ NM ]=C;[l-~ NM ]+~ NM[c:k+1 +c:J,
(20)
where C bk and C bk + 1 are values of Cb entering the nodal points k and k+1 of
the fracture segment, respectively. The finite difference approximation of Eq. (14)
is given by:
1 C m+1
-
b' '+1
[
]
___ '.:::,j_=exp -K f
m . 1/2,1tl .
l-Cbm.
l,j+
l,j
(21)
A finite difference approximation of Eq. (16) is used to evaluate the entrapped
NAPL saturation at the midpoint of the longitudinal interval, as follows:
S m+1
S m
C ~C m+1 C m ~t
n' '+1/2 = n' '+112 - nv b' '+1 - b"
.
l,j
l,j
l,j
l,j L1x
(22)
The flux average concentration at the cross-section perpendicular to the flow
direction and represented by the longitudinal nodal points ia is evaluated with a
finite difference approximation of Eq. (19), trapezoidal numerical integration, and
appropriate reference to the crossed fracture segment.
Consider an integer number 13 satisfying the following expression
f3Vrnax -1)+ 1 < ia < (13 + l)(jrnax -1)+ 1.
(23)
Then the cross section chosen for the calculation of Cay crosses the fracture
segment at the nodal point k{Jo, which is given by
k{3 = ia - f3Vrnax -1)-1,
(24)