Environmental Engineering Reference
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as the barycentric fluid velocity and the diffusion velocity of species
k
in the fluid
for which
n
u
k
=
0
.
(54)
k
=
1
In single-phase flow through a porous medium, the diffusive flux of a component
with respect to the barycentric fluid velocity is called
hydrodynamic dispersion
.
Each component
k
has its own mass and momentum conservation equations, which
by analogy with Eqs. (
4
) and (
10
)aregivenby
∂ (ˆˁ
k
)
∂
+∇·
(ˁ
k
v
k
)
=
I
k
,
(55)
t
and
k
μ
·
(
∇
v
=−
p
−
ˁ
g
∇
z
) ,
(56)
ˆ
where
and
k
are the porosity and absolute permeability of the porous medium.
Since the species velocities are typically inaccessible to measurement, it is more
convenient to rewrite Eq. (
55
)as
∂ (ˆ
c
k
ˁ)
∂
+∇·
(ˁ
c
k
v
)
=−∇·
d
k
+
I
k
,
(57)
t
where we have used the relations:
ˁ
k
=
c
k
ˁ
and
v
k
=
u
k
+
v
. In the above equation
d
k
=
ˁ
c
k
u
k
is the dispersive mass flux of species
k
, which we assume to obey Fick's
law, i.e.,
d
k
=−
D
·∇
c
k
,
(58)
where
D
is the diffusion-dispersion tensor given by (Peaceman
1966
)
ʶ
l
E
)
+
ʶ
t
E
↥
(
D
=
ˆˁʶ
m
I
+
ˁ
|
v
|
(
v
v
)
.
(59)
ʶ
m
is the molecular diffusivity (in units of volume rate per unit
The coefficient
length), while
ʶ
t
(having units of length) are the longitudinal a
nd transversa
l
dispersivities, respectively,
ʶ
l
and
v
x
+
|
v
|
is the Euclidean norm of
v
, i.e.,
|
v
|=
v
y
+
v
z
,
I
is the identity tensor, and the tensor
E
(
v
)
is the orthogonal projection along the
velocity
1
v
t
E
(
v
)
=
2
v
·
,
(60)
|
|
v
where the superscript
t
means transposition and
E
↥
(
. Summing up the
contributions of all species in Eq. (
58
), we obtain the further constraint
v
)
=
I
−
E
(
v
)
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