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1
2
Ca vi
′ =
β−β−
v
(1
a
)
(B6)
2
2
1
2
(
)
j
j
j
j
Ec
′ =− −
avic
β −+
+
2
cc
(B7)
i
2
i
1
i
i
−
1
where β is defined as the ratio of time increment to space increment as above.
4.9 Appendix C: Derivations of Finite Difference Equations
for CDE with a Nonlinearly Time-Dependent Dispersivity
For a power law dispersivity-time model, the governing equation is as fol-
lows (Equation 4.23):
2
∂
∂
=−
∂
c
t
c
x
∂
∂
c
DD
fr
−
1
(C1)
v
+
av
t
fr
3
2
∂
x
Approximations of first partial derivatives are given in Equations A2 and
A3. The second derivative with respect to
x
is given by:
j
+
1
j
+
1
j
+
1
j
j
j
2
∂
∂
c
x
1
2
c
−
2
c
+
c
c
−+
2
c
c
D
−
1
D
−
1
D
−
1
D
−
1
i
+
1
i
i
−
1
i
+
1
i
i
−
1
t
fr
=
( )
j
+
fr
t
fr
+
(
jt
)
fr
2
2
2
x
x
(C2)
For convenience, we let
D
fr
t
x
t
x
β=
and
γ=
2
Notice that
2
γ=β
for
D
fr
=
. Substituting Equation C2 together with
Equations A2 and A3 into Equation C1 and rearranging yields:
j
+
1
j
+
1
j
+
1
Ac
′′
+ ′′
Bc
+ ′′
Cc
= ′′
E
(C3)
i
−
1
i
i
+
1
where
1
2
D
D
−
1
Aa vj
′′ =
fr
( )
+
fr
γ
(C4)
3
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