The Effect of Fractures on the Reclamation ofNAPL
61
Here Kb is the hydraulic conductivity of the permeable blocks and M f is the
mobility of a single fracture segment, namely the ratio of fracture flow rate to the
hydraulic gradient imposed along the fracture segment.
Representation of NAPL dissolution and solute transport in the permeable
block flow is similarly based on a material balance expression developed in
dimensionless form, using dimensionless variables defined in Eq. (5). As major
solute dispersion effects originate from mixing between the permeable and
fracture flows, solute transport in the permeable blocks incorporates mainly
dissolution of the entrapped NAPL and advection of the dissolved solute.
Therefore, the dimensionless permeable block solute transport equation is
obtained as (Rubin et al. 1997)
aCb aCb _ K (1 C)
--+--- f - b ,
at ax
(8)
where Kfis the dimensionless interphase mass transfer coefficient, given by:
K
k
Rcose
f:::: f a - - - ,
%
(9)
where k f is the mass transfer coefficient and a is the specific interfacial contact
area (area per volume of porous media) of the entrapped NAPL. An expression
for a was developed by DeZabala and Radke (1986):
a=t/>bSnf[~l'
(10)
where [~ 1 is the surface area to volume ratio of a blob, Sn is the entrapped
NAPL saturation,jis the fraction of the interphase contact area exposed to mobile
water, and t/>b is the porosity of the permeable blocks. Following the approach of
Powers et al. (1991), the entrapped NAPL is conceptualized as a fixed number of
spheres, whose radii gradually diminish throughout the dissolution process. Under
these assumptions, Eqs. (9) and (10) are used to obtain:
K f = K fOr Sn J
2
/
3
,
SnO
where KfO and SnO represent initial values of Kfand S,., respectively.
(11)
Equation (7) is simplified by adopting the Lagrangian approach of using a
moving coordinate systemxb t1
XI =x-t; tl =t.
(12)
Introducing the time and space coordinates of Eq. (12) into Eq. (7), the following
expression is obtained:
dCb =K (I-C).
dt
f
b
1
(13)
For a comparatively short time interval, Atb it is possible to assume that the
value of Kfis kept constant. Then direct integration ofEq. (13) yields:
61
Here Kb is the hydraulic conductivity of the permeable blocks and M f is the
mobility of a single fracture segment, namely the ratio of fracture flow rate to the
hydraulic gradient imposed along the fracture segment.
Representation of NAPL dissolution and solute transport in the permeable
block flow is similarly based on a material balance expression developed in
dimensionless form, using dimensionless variables defined in Eq. (5). As major
solute dispersion effects originate from mixing between the permeable and
fracture flows, solute transport in the permeable blocks incorporates mainly
dissolution of the entrapped NAPL and advection of the dissolved solute.
Therefore, the dimensionless permeable block solute transport equation is
obtained as (Rubin et al. 1997)
aCb aCb _ K (1 C)
--+--- f - b ,
at ax
(8)
where Kfis the dimensionless interphase mass transfer coefficient, given by:
K
k
Rcose
f:::: f a - - - ,
%
(9)
where k f is the mass transfer coefficient and a is the specific interfacial contact
area (area per volume of porous media) of the entrapped NAPL. An expression
for a was developed by DeZabala and Radke (1986):
a=t/>bSnf[~l'
(10)
where [~ 1 is the surface area to volume ratio of a blob, Sn is the entrapped
NAPL saturation,jis the fraction of the interphase contact area exposed to mobile
water, and t/>b is the porosity of the permeable blocks. Following the approach of
Powers et al. (1991), the entrapped NAPL is conceptualized as a fixed number of
spheres, whose radii gradually diminish throughout the dissolution process. Under
these assumptions, Eqs. (9) and (10) are used to obtain:
K f = K fOr Sn J
2
/
3
,
SnO
where KfO and SnO represent initial values of Kfand S,., respectively.
(11)
Equation (7) is simplified by adopting the Lagrangian approach of using a
moving coordinate systemxb t1
XI =x-t; tl =t.
(12)
Introducing the time and space coordinates of Eq. (12) into Eq. (7), the following
expression is obtained:
dCb =K (I-C).
dt
f
b
1
(13)
For a comparatively short time interval, Atb it is possible to assume that the
value of Kfis kept constant. Then direct integration ofEq. (13) yields: