Geology Reference
In-Depth Information
U
b
xx
(x) +
(Z
2
/c
mud
2
)U
b
=
0 (2.7d)
U
a2
xx
(x) +
(Z
2
/c
mud
2
)U
a2
=
0 (2.7e)
U
a1
xx
(x) +
(Z
2
/c
mud
2
)U
a1
=
0 (2.7f)
We observe that c
mud
appears in all of the above equations, except Equation 2.7c,
which contains c
mm
. This quantity is just the speed of sound in the mud motor
passage, which must be further obtained as a suitable weighted average of
rubber, mud, and possibly, steel rotor properties (note that use of our Helmholtz
model will preclude the modeling of signal shape distortions possible at
downhole rubber interfaces). More than likely, laboratory investigation will be
required to determine c
mm
, the corresponding bulk modulus B
mm
and the
equivalent density U
mm
(as in the case of drilling mud, the relationship of the
form c
mm
= (B
mm
/U
mm
applies). We also emphasize that the attenuative nature
of the hard rubber that makes up the stators in the mud motor is not important at
MWD frequencies less than 100 Hz and typical motor lengths less than 100 ft.
Now, the general solutions for Equations 2.7a - 2.7f can be easily written down
in closed analytical form, and in particular, are obtained as
U
p
(x)
=
C
1
exp (-iZx/c
mud
) (2.8a)
U
c
(x)
=
C
2
cos Zx/c
mud
+ C
3
sin Zx/c
mud
(2.8b)
+ 0
if
x < x
s
,
or
- {c
mud
'p/(ZB
mud
)}sin Z(x-x
s
)/c
mud
if
x > x
s
U
m
(x) =
C
4
cos Zx/c
mm
+ C
5
sin Zx/c
mm
(2.8c)
U
b
(x)
=
C
6
cos Zx/c
mud
+ C
7
sin Zx/c
mud
(2.8d)
U
a2
(x) =
C
8
cos Zx/c
mud
+ C
9
sin Zx/c
mud
(2.8e)
U
a1
(x) = C
10
exp (+iZx/c
mud
) (2.8f)
Observe that sines, cosines, and complex exponentials have been used in
the solutions given by Equations 2.8a - 2.8f. Let us explain the motivation
behind the exact choices made. Note that the typical homogeneous differential
equation, e.g., U
xx
(x) + (Z
2
/c
mud
2
) U = 0, has two
real
linearly independent
solutions, namely, the usual sin Zx/c
mud
and cos Zx/c
mud
, but that equivalent
solutions are also given by the
complex
mathematical expressions
exp(+iZx/c
mud
) and exp(-iZx/c
mud
). Direct substitution in the governing
equation, of course, demonstrates that both of these solution pairs are valid.
The exact representation useful in any particular instance, however,
depends on the nature of the wave propagation found in the waveguide section
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