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the fractures and the matrix. Several authors simplified transport in the macropore
domain, for example by ignoring diffusion and dispersion in the macropores and
considering only piston displacement there (Ahuja and Hebson
1992
; Jarvis et al.
1994
).
Mass Transfer
The transfer rate,
Γ
s
,inEq.(
18.50
) for contaminants between the mobile and immo-
bile domains in the dual-porosity models can be given as the sum of diffusive and
advective fluxes as follows:
+
w
c
∗
s
=
α
s
(
c
mo
−
c
im
)
(18.53)
where
c
∗
α
s
is the first-order
contaminant mass transfer coefficient [T
−
1
]. Notice that the advection term of Eq.
18
.
53 is equal to zero for the mobile-immobile model (Fig.
18.14b
) since the immo-
bile water content in this model is assumed to be constant. However,
is equal to
c
mo
for
Γ
w
> 0 and
c
im
for
Γ
w
< 0, and
Γ
w
may have
a nonzero value in the dual-porosity model depicted in Fig.
18.14c
.
The transfer rate,
Γ
s
,inEqs.(
18.51
) and (
18.52
) for contaminants between
the fracture and matrix regions is also usually given as the sum of diffusive and
advective fluxes as follows (e.g., Gerke and Van Genuchten
1996
):
+
w
c
∗
s
=
α
s
(1
−
w
m
)(
c
f
−
c
m
)
(18.54)
Γ
s
[T
−
1
], is of the form:
in which the mass transfer coefficient,
α
s
=
β
g
d
2
D
a
(18.55)
where
β
g
is a dimensionless geometry-dependent coefficient,
d
is the characteristic
length of the matrix structure (L) (e.g., the radius of a spherical or solid cylindrical
aggregate, or half the fracture spacing in the case of parallel rectangular voids),
D
a
is an effective diffusion coefficient [L
2
T
−
1
] representing the diffusion properties of
the fracture-matrix interface.
18.3.3.2 Chemical Nonequilibrium
Kinetic Sorption Models
A substitute to expressing sorption as an instantaneous process using algebraic equa-
tions (e.g., Eqs. (
18.38
), (
18.40
)or(
18.41
)) is to quantify the reaction kinetics based
on ordinary differential equations. A popular and simple formulation of a chemically
controlled kinetic reaction arises when first-order linear kinetics is assumed:
∂
s
t
=
α
k
(
K
d
c
−
s
)
(18.56)
∂
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