Page 23 - Transmissibility Corrections and Grid Control for Shale Gas Numerical Simulation
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VA - DF: Transmissibility Corrections and Grid Control for Shale Gas Numerical Simulation
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Appendix A. Unitary solutions
3D Segment source solution with linearly distributed source:
Considering a source distribution of the form
| |
)(
s
s
, the potential due to a segment
source of length 2L is:
L
L
ds
r s z
s
zr
2 2
)
(
| |
) ,(
After integration we get:
)
( )
(
)
( )
(
)
( )
(
)
( )
(
2 )
(
)
(
) ,(
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
L z
L z r
L z
L z r
Log
z z r
z L z L r z L z L r
Log z
z r
z L r
z L r
zr
Further considering uniform source distribution,
1 )(
s
, the solution simplifies to
)
( )
(
)
( )
(
) , , , , (
2
2
2
2
L z
L z r
L z
L z r
Log
LFM
Seg
u
In this case one can also express analytically the gradient
) , , , , (
LFM
Seg
u
.
3D panel source solution with linearly distributed source:
Considering a source distribution along the panel length of the form
| |
)(
s
s
, the
potential due to a rectangular source of length 2L and height 2H is:
)0, , ,( )1, , ,( ) , ,(
hHzr
hHzr
Hzr
)
( tan
2 )
( tan )
( )
( tan )
(
)
(
2 )
(
)
( )
(
)
(
)
( ))
(2(
)
(
)
(
2
))
(2(
)
(
)
(
2
))
(2(
) 2
(
2
) , , ,(
1
2
2
1
1
1
1
2 2
1 1
2
2
2
2
2
2
2
1
2
1
2
1
2
1
z
z
z
z
gR
zh
z
gR
bh
z
gR
bh
z
R
z g Log z
b g Log
z
b g Log
z
h
z R h g Log
z
b b R h g Log
z
b b R h g Log
g g g
h
hHzr
