60
H. Rubin et al.
specific discharge of the permeable block flow, C· is the dissolved NAPL solute
concentration in the fracture flow, C; is the dissolved NAPL solute concentration
of permeable block flow entering the fracture, x' and y' are the horizontal and
vertical coordinates, respectively, and x' is the local coordinate along the fracture
centerline. Solute transport through the fracture is assumed to occur solely by
advection, neglecting diffusion, longitudinal dispersion, and mass storage changes
within the fracture. BirkhOlzer et al. (1993a) conducted an extensive evaluation of
these assumptions by employing the numerical code STRAFE (Birkholzer 1994),
in which all these effects are considered. Their tests have indicated that these
assumptions are reasonable for transport in sandstone aquifers, which are typically
characterized by significant advection through the permeable blocks. Solute
transport in the fracture network is affected by mixing between the permeable
block and fracture flow, as indicated in Fig. 2c. Employing the foregoing
assumptions, a mass balance on the organic solute in the elementary fracture
volume can be expressed as (Birkholzer et al. 1993a):
Qf a~~ dx'+qb(C* -C;}u'sin8=O.
(2)
Observing that the projection of dx' in the horizontal and vertical directions is,
respectively:
*
* .
dx =dx'cos8; dy =dx'sm8,
(3)
Equation (2) can be rewritten as:
(4)
The following dimensionless variables, space, and time coordinates may be
employed for the development of dimensionless material balance expressions
(Rubin et al. 1997):
C*
C*
C =-.; Cb =--f,
C s
C s
*
*
*v,
x=_x __ . y=_y_. t=_t_b_.
Bcos8'
Bsin8'
Bcos8 '
(5)
where C; is the equilibrium concentration ofNAPL solute in the water phase.
Variables with an asterisk superscript represent the relevant dimensional value.
Note that under the assumption that entrapped NAPL has a negligible effect on the
block permeability, the specific discharge, qb, as well as interstitial flow velocity,
Vb, in all blocks are identical. Introducing the dimensionless variables of Eq. (5)
into Eq. (4), we obtain the following differential equation for the calculation of the
dissolved solute transport in the fracture network
ac
-+NMC=NMCb ,
(6)
ax
where NM is the mobility number, defined as
NM =.!!.!L= KbBtan8
qf
M f
(7)
H. Rubin et al.
specific discharge of the permeable block flow, C· is the dissolved NAPL solute
concentration in the fracture flow, C; is the dissolved NAPL solute concentration
of permeable block flow entering the fracture, x' and y' are the horizontal and
vertical coordinates, respectively, and x' is the local coordinate along the fracture
centerline. Solute transport through the fracture is assumed to occur solely by
advection, neglecting diffusion, longitudinal dispersion, and mass storage changes
within the fracture. BirkhOlzer et al. (1993a) conducted an extensive evaluation of
these assumptions by employing the numerical code STRAFE (Birkholzer 1994),
in which all these effects are considered. Their tests have indicated that these
assumptions are reasonable for transport in sandstone aquifers, which are typically
characterized by significant advection through the permeable blocks. Solute
transport in the fracture network is affected by mixing between the permeable
block and fracture flow, as indicated in Fig. 2c. Employing the foregoing
assumptions, a mass balance on the organic solute in the elementary fracture
volume can be expressed as (Birkholzer et al. 1993a):
Qf a~~ dx'+qb(C* -C;}u'sin8=O.
(2)
Observing that the projection of dx' in the horizontal and vertical directions is,
respectively:
*
* .
dx =dx'cos8; dy =dx'sm8,
(3)
Equation (2) can be rewritten as:
(4)
The following dimensionless variables, space, and time coordinates may be
employed for the development of dimensionless material balance expressions
(Rubin et al. 1997):
C*
C*
C =-.; Cb =--f,
C s
C s
*
*
*v,
x=_x __ . y=_y_. t=_t_b_.
Bcos8'
Bsin8'
Bcos8 '
(5)
where C; is the equilibrium concentration ofNAPL solute in the water phase.
Variables with an asterisk superscript represent the relevant dimensional value.
Note that under the assumption that entrapped NAPL has a negligible effect on the
block permeability, the specific discharge, qb, as well as interstitial flow velocity,
Vb, in all blocks are identical. Introducing the dimensionless variables of Eq. (5)
into Eq. (4), we obtain the following differential equation for the calculation of the
dissolved solute transport in the fracture network
ac
-+NMC=NMCb ,
(6)
ax
where NM is the mobility number, defined as
NM =.!!.!L= KbBtan8
qf
M f
(7)