Geoscience Reference
In-Depth Information
Figure 4.23 Volume of a 3-D cell.
(∂
y
/∂η∂
z
/∂ζ
y
/∂ζ ∂
z
/∂η)ηζ
and discretizing the coordinate derivatives at cell
center P yields
b 1 P = (
y n
y s
)(
z t
z b ) (
y t
y b )(
z n
z s
)
(4.149)
Following this procedure, one can derive the discretized equations for all b i
at cell
center P and faces w , s , and b , which are not introduced here.
The diffusion parameters are
1
j
1
j
b 1 b 1 +
b 2 b 2 ++
b 3 b 3 )
D w
= (
J
α
α
ηζ)
=
(
/
V w
w
w
w
w
2
j
2
j
b 1 b 1 +
b 2 b 2 +
b 3 b 3 )
D s
= (
J
α
α
ξζ)
=
(
/
V s
(4.150)
s
s
s
s
3
j
3
j
b 1 b 1 +
b 2 b 2 +
b 3 b 3 )
D b
= (
J
α
α
ξη)
=
(
/
V b
b
b
b
b
where
V w
= (
J
ξηζ)
= (
V W
+
V P
)/
2,
V s
= (
J
ξηζ)
=
w
s
(
2.
In addition, the values of parameters, such as velocity, density, and diffusivity, at
cell faces need to be interpolated from their values at adjacent cell centers. The often
used method is linear interpolation. For example, the quantities of
V S
+
V P
)/
2, and
V b
= (
J
ξηζ)
= (
V B
+
V P
)/
b
at faces w , s , and
b are computed by linear interpolation between the values at two adjacent cell centers
of each face as follows:
φ
φ
=
f x , P
φ
+ (
1
f x , P
w
P
W
φ s =
f y , P φ P + (
f y , P S
1
(4.151)
φ b =
f z , P
φ
+ (
1
f z , P
P
B
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