Environmental Engineering Reference
In-Depth Information
power law of the length travelled, i.e.,
D
L
x
L
−
ʸ
D
(
x
)
=
.
(6)
In the above equation
D
L
is the dispersion coefficient at the extraction length (
x
=
L
)
and
is the connectivity index associated to the hydrodynamic dispersion.
There are several parameters involved in the model, specifically in Eqs. (
4
) and
(
6
) appear
ʸ
ˆ
,
A
cs
,
D
L
and
ʸ
. The tracer flux for constant fluid density is given by
M
F
A
cs
ˆˁ
0
C
D
L
L
ʸ
x
−
ʸ
∂
C
(
x
,
t
)
J
(
x
,
t
)
=
(
x
,
t
)
−
,
(7)
∂
x
where the first term describes advection, and the second one depicts dispersion.
Accordingly, the tracer conservation equation for uniformflow is (Herrera-Hernández
et al.
2013
)
∂
C
(
x
,
t
)
+
ʓ(α)
x
α
−
1
∂
J
(
x
,
t
)
=
S
T
.
(8)
∂
t
∂
x
=
/
After substituting Eq. (
7
) into Eq. (
8
), using dimensionless variables,
x
D
x
L
,
C
D
=
/
C
Max
and
t
D
=
/ʔ
C
t
t
, and assuming that no source is present, we obtain
x
−
D
∂
M
F
ʔ
∂
C
D
∂
t
D
+
ʓ(α)
t
A
cs
ˆˁ
0
∂
C
D
∂
x
D
−
ʓ(α)
D
L
ʔ
t
∂
C
D
∂
x
1
−
α
D
x
1
−
α
D
=
0
.
(9)
L
α
L
α
−
1
L
2
∂
x
D
x
D
ʔ
t
is an arbitrary reference time and
C
Max
is a reference tracer concentration.
The mathematical model given by Eq. (
9
) contains various parameters that can be
grouped as
M
F
ʔ
U
ad
=
ʓ(α)
L
α
t
A
cs
ˆˁ
0
,
(10)
D
ad
=
ʓ(α)
D
L
ʔ
t
.
(11)
L
α
−
1
L
2
Rewriting Eq. (
9
) in terms of these parameters, it becomes
x
−
D
∂
∂
C
D
∂
∂
C
D
∂
∂
C
D
∂
U
ad
x
1
−
α
D
ad
x
1
−
α
t
D
+
x
D
−
=
0
.
(12)
D
D
∂
x
D
x
D
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