Environmental Engineering Reference
In-Depth Information
2.3 Boundary and Initial Conditions
2.3.1 Boundary Conditions
As mentioned in Sect.
2.1
, the tracer and fluid are injected at
x
=
x
w
(wellbore
radius) and extracted at
x
L
. Those points (plates) constitute the boundaries of
the system where particular conditions are imposed. At the injection plate (
x
=
=
x
w
)
a zero-flux condition is imposed, this is
x
w
L
ʸ
C
D
,
M
F
L
A
cs
ˁ
0
ˆ
x
D
x
D
=
∂
C
D
∂
=
(13)
x
L
D
0
and at the extraction plate (
x
L
) zero dispersive flux is set since beyond this point
no more porous medium is present (Danckwerts
1953
)
=
x
D
x
D
=
1
=
∂
C
D
0
.
(14)
∂
2.3.2 Initial Condition
The initial condition establishes that no tracer is present inside the fractal medium,
except for a small truncated Gaussian pulse located between
x
D
min
and
x
D
max
and
centred at
x
D
in a zone close to the injection plate, this is
⊧
⊨
B
exp
L
2
(
x
D
−
x
D
)
2
−
2
2
˃
if
x
D
min
≤
x
≤
x
D
max
C
D
(
x
D
,
t
D
=
0
)
=
√
2
(15)
⊩
ˀ˃
2
0
elsewhere
,
where
B
is related to the amount of tracer injected to the system,
is the initial width
of the pulse and
x
D
is the dimensionless length as defined in Sect.
2.2
.
˃
2.4 Numerical Solution of the Advection-Dispersion Equation
We solve Eq . (
12
) numerically, and for this purpose we write it as
2
C
D
∂
∂
x
D
)
∂
x
D
)
∂
C
D
∂
C
D
∂
t
D
=
f
1
(
x
D
+
f
2
(
x
D
,
(16)
whose coefficients are
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