Geology Reference
In-Depth Information
Transmission efficiency is measured by the “power transmission
coefficient”
P
t
, formed by the ratio of the transmitted power to the incident
power. Some algebra shows that, in general,
P
t
satisfies
P
t
= 4 / { ( A
3
/A
1
+1 )
2
cos
2
kL + (A
2
/A
1
+A
3
/A
2
)
2
sin
2
kL} (5.30a)
Let us consider an MWD application, and set A
1
= A
3
= A
c
and A
2
= A
mwd
,
where A
c
is the cross-sectional area of the drill collar containing a conventional
area-blocking MWD valve, and A
mwd
is the restricted area at that point. In this
limit, we have
P
t
= 4 / { 4 c os
2
kL + (A
mwd
/A
c
+A
c
/A
mwd
)
2
sin
2
kL} (5.30b)
For any fixed value A
mwd
/A
c
, it is clear that in the limit as kL 0, complete
wave passage is achieved with
P
t
1. In other words, MWD generated signals
and all reverberant signals reflecting at the drill bit and pipe-collar interfaces
pass straight through the valve unimpeded. In practice, we might that k = 2 /
6.28/500 ft, L 1 ft, so that kL 0.0l.
5.1.9 Example 5-5. Transmission through contrasting media.
Let us now refer to Figure 5.1 again, this time allowing different materials
in Sections 1 and 2. Here, the matching of pressures and volume velocities
leads to
P
t
= 4 A
1 1
c
1
A
2 2
c
2
/{A
1 2
c
2
+ A
2 1
c
1
}
2
(5.30c)
We emphasize that our matching conditions for continuous pressure and volume
velocity are not as general as they appear, e.g., they do not allow nonequilibrium
elastic delays that may occur at unusual interfaces (e.g., refer to our discussion
in Chapter 1 on elastic boundary conditions). Such problems may arise in
transducer design when the pressure elements are isolated from the propagation
medium by rubber diaphragm partitions that separate other fluids and gases
having different properties. They also arise when mud waves impinge upon
rubber elements of mud motors. The foregoing formula indicates the possibility
of 100% power transmission
if
A
1 2
c
2
= A
2 1
c
1
even if
A
1
A
2
. However, if
A
1
= A
2
, then P
t
= 4
1
c
1 2
c
2
/{
2
c
2
+
1
c
1
}
2
. The so-called “acoustic
impedance” is simply the product “ c” between fluid density and sound speed in
a pipe of uniform cross-section. It turns out, in modeling plane wave
transmission at normal incidence from one media to another, that partial
reflections and transmissions are absent whenever the acoustic impedances so
defined are continuous. That is, when the value c remains unchanged in the
direction of wave motion, complete energy transmission is assured. In Example
5.3, we implicitly assumed that the impedance contrast at a closed “fixed” end
was infinite, so that perfect reflection occurs.
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