Geology Reference
In-Depth Information
In these solutions, we gave the eigenfunction and eigenvalues only, which are
determined from boundary conditions. The full solution, to include initial
conditions, can be obtained using the Fourier integral methods in Chapter 1.
We emphasize that our solutions apply to highly deviated drillstrings and
boreholes, despite the appearance of a single “x” in
2
u/ t
2
- c
2 2
u/ x
2
= 0 . In
swab-surge and MWD applications, the wavelength = c / f (5,000 ft/sec)/(10
Hz ) 500 ft far exceeds most bend radii (at worst, 1 ft). Thus, reflections are
not incurred at bends, and one-dimensional theory applies. For that matter, our
wave
results will apply to coiled tubing wound about a spool of small radius,
although pressure gradients needed to pass
hydraulic flow
are higher than in
straight conduits due to centrifugal forces.
5.1.8 Example 5-4. Passage through area obstructions.
As indicated in the above paragraph, one-dimensional theory applies to
highly deviated geometries, and this is true for problems with multiple area
discontinuities as well, where partial transmission and reflection at transition
points exist. Consider two semi-infinite Sections “1” and “3” bounding a finite
duct of length L (no restrictions are placed on relative areas) as shown in Figure
5.2. An incident wave from the left will partially reflect and transmit at the 1-2
interface; the transmitted wave acts similarly at the 2-3 interface, but a single
wave emerges at the right. The problem of wave transmission through Section
“2” is relevant to MWD mud pulse telemetry. An acoustic source is located in
this section, which generates waves that propagate uphole and downhole. The
latter reflect at the bit and head up towards Section “2” once more. How much
of this wave passes and adds to new waves that are progressing uphole? This
problem is addressed rigorously and in greater detail in the author' s new topic
Measurement While Drilling Signal Analysis, Optimization and Design
(Chin
et
al
, 2014), but here, a simpler approach is taken.
+---------------
|
+----------+
|
---------+
A
1
A
2
A
3
---------+
|<----
L
---->
+----------+
|
+---------------
Figure 5.2.
Transmission through three-pipe system.
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