Geology Reference
In-Depth Information
because there is no valve erosion or turbine power penalty associated with this
type of signal enhancement. On the other hand, suppose that MWD signals are
desired, which are to travel up the annulus (such signals have been used for gas
influx detection, e.g., the inability to cross-correlate with drillpipe waveforms
suggests the possible existence of un-dissolved gas bubbles). For such
applications, we might ask the complementary question, “How optimal is the
created signal that is propagating up the annulus for a given pulser 'p?”
A convenient dimensionless measure, in either case, is obtained by
normalizing the acoustic pressure by the pulser 'p; the absolute value of this
ratio, we have termed the “transmission efficiency.” In designing an optimized
pulser, it is this quantity that we seek to maximize. We emphasize, however,
that high values of p/'p alone may not suffice. Since greater (thermodynamic)
attenuation is found at higher frequencies, the increase in source strength may
not always be enough to enable transmission to the surface; these increases must
offset increases in attenuation found at higher frequencies. Whereas 'p depends
largely on flow rate and pulser geometry, the transmission efficiency is
completely independent of 'p and depends on waveguide geometry, sound
speed, and pulser location and frequency only.
In an idealized situation where the dipole source resides in an infinite pipe
without areal or material discontinuities, so that reflections and impedance
mismatches are entirely ruled out, it is clear that half of this 'p signal
propagates uphole while the remaining half travels downhole. This physical fact
can be easily deduced from D'Alembert's formula in mathematics but it is also
apparent from symmetry considerations. The theoretical value for transmission
efficiency is identically 0.5; indeed, this simple value for both pipe and annular
wave solutions serves as a critical software and programming check in the said
limit. It turns out, as detailed calculations show, that at low frequencies, e.g.,
those typical of existing positive pulsers, a value of unity is approached as a
result of constructive interference reflections at the bit - we had previously
explained why this was so using purely physical arguments for dipole sources.
However, this unit value by no means represents perfection - at higher
frequencies, depending on pulser location and BHA details, transmission
efficiencies exceeding 1.0 and approaching 3.0 are possible. In Chapter 10, a
prototype design for a 10 bits/sec system is offered, in which the transmission
efficiency is 1.7 assuming typical bottomhole assemblies and muds.
The transmission efficiencies corresponding to Equations 2.17c and 2.18c
are easily determined from division by 'p and taking absolute values. Since the
absolute value of exp iZ(t ± x/c
mud
) is exactly unity, it follows that
Transmission efficiency
1,10
=|p 'p| = {ZB
mud
/( |'p|c
mud
)} |C
1,10
|
= {ZB
mud
/(|'p|c
mud
)}(C
1,10
C
1,10
*
)
(2.19)
where the asterisk denotes complex conjugates.
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