Geology Reference
In-Depth Information
Collar-Pipe Interface
:
{1 - i tan Zx
c
/c
mud
} C
1
- {A
c
/A
p
}C
2
- {(A
c
/A
p
) tan Zx
c
/c
mud
}C
3
=
- {A
c
c
mud
'p sin Z(x
c
-x
s
)/c
mud
}/{A
p
ZB
mud
cos Zx
c
/c
mud
}
(2.9a)
{-tan Zx
c
/c
mud
- i} C
1
+ {tan Zx
c
/c
mud
}C
2
- C
3
=
- {c
mud
'p cos Z(x
c
-x
s
)/c
mud
}/{ZB
mud
cos Zx
c
/c
mud
}
(2.9b)
Mud Motor-Collar Interface:
C
2
- {A
m
/A
c
}C
4
= 0 (2.10a)
C
3
- {(C
mud
B
mm
)/(C
mm
B
mud
)}C
5
= 0 (2.10b)
Bit Passage-Mud Motor Interface
:
{A
m
cos Zx
m
/c
mm
}C
4
- {A
m
sin Zx
m
/c
mm
}C
5
- {A
b
cos Zx
m
/c
mud
}C
6
+ {A
b
sin Zx
m
/c
mud
}C
7
= 0
(2.11a)
{(B
mm
/C
mm
)sin Zx
m
/c
mm
}C
4
+ {(B
mm
/C
mm
) cos Zx
m
/c
mm
}C
5
(2.11b)
- {(B
mud
/C
mud
)sinZx
m
/c
mud
}C
6
- {(B
mud
/C
mud
) cos Zx
m
/c
mud
}C
7
= 0
Annulus (2) - Bit Passage Interface:
C
6
- {tan Z(x
m
+x
b
)/c
mud
}C
7
- {A
a2
/A
b
}C
8
+ {(A
a2
/A
b
) tan Z(x
m
+x
b
)/c
mud
}C
9
= 0 (2.12a)
{tan Z(x
m
+x
b
)/c
mud
}C
6
+ C
7
- {tan Z(x
m
+x
b
)/c
mud
}C
8
- C
9
= 0 (2.12b)
Annulus (1) - Annulus (2) Interface
:
C
8
- {tan Z(x
m
+x
b
+x
a
)/c
mud
}C
9
+ (A
a1
/A
a2
){- 1 + i tan Z(x
m
+x
b
+x
a
)/c
mud
}C
10
= 0 (2.13a)
{tan Z(x
m
+x
b
+x
a
)/c
mud
}C
8
+C
9
+ {- tan Z(x
m
+x
b
+x
a
)/c
mud
- i}C
10
= 0 (2.13b)
2.4.2.5 Matrix formulation.
The foregoing equations define a 10 u 10 system of linear algebraic
equations in the
complex
unknowns C
1
, C
2
, ... , C
10
. An analytical solution is
possible with the use of algebraic manipulation software - it can, of course, be
derived more laboriously by hand. The equation system can be represented
efficiently if we rewrite our equations in a more transparent matrix form that
highlights the structure of the coefficient terms. When we do this, we
straightforwardly obtain an equation of the form [
S
] [
C
] = [
R
] which can be
easily interpreted by computer algebra algorithms, that is,
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