Transient Constant Area Surface and Downhole Wave Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 127

4.5.2 Physical problem.
The key physical ideas are simply stated. Downhole information, meaning
the sequence of 0's and 1's describing logging data, is directly encoded in valve
positions and thus appears directly in 'p. Recovery of 'p implies recovery of
the train of 0's and 1's. However, 'p is never directly measured. It is only
indirectly observed in measured pressures as the latter contain the effects of
reflections. Removal of surface reflections and mudpump noise solves only part
of the problem. Downhole reflections are still prevalent even after Methods 4-1
to 4-4 are applied. Again, these downhole reflections are problematic and only
clever surface processing will remove their influence.
When a positive pulse valve opens and shuts, or when a mud siren changes
rotation speed based on azimuthal location of the rotor relative to the to indicate
0's and 1's, over-pressure and under-pressure acoustic signals are created at
opposite sides of the source which have equal magnitude (this is confirmed in
numerous experiments). The downgoing signals reflect at the drillbit and
interfere with newly created upgoing signals at the source. These ghost
reflections contaminate the intended 'p signal by superposing with tail end of
the later upgoing signal. In summary, a single valve action (that is, an “open” or
a “close”) creates two upgoing pressure pulses that enter the drillpipe to travel to
the surface. At the surface, these may reflect at the desurger and at the
mudpump, thus forming even more signals that are sensed in the standpipe
pressure transducer array. Methods 4-1 to 4-4 handle surface reflections, while
Methods 4-5 and 4-6 handle 'p recovery from surface-filtered results.
In terms of the math symbols below, the p 2 L,t` remaining after Methods
4-1 to 4-4 are applied must be further processed to extract 'p(t), which is the
transient function containing the 0's and 1's that are implicit in the position-
encoding used. For Methods 4-5 and 4-6, we assume a “dipole” source. By a
“dipole source” we mean a pressure source whose created pressures are
antisymmetric with respect to source point - this section applies to dipole
sources, e.g., positive pulsers and mud sirens, but not negative pulse systems.
4.5.3 On solid and open reflectors.
It is important to clarify at this point possible confusion on the use of “zero
acoustic pressure” boundary conditions in modeling open ends. We have stated,
and proved mathematically, why solid reflectors like pump pistons double
incident wave pressures locally. This is straightforward. What happens at open
ends, however, is less clear. The results of detailed and rigorous three-
dimensional analyses show that long waves in tubes reflect at open ends with
opposite pressure polarity and almost identical magnitude. Hence, “zero
acoustic pressure” is found at open ends. Of course, to the lab technician
working outside the wind tunnel, the acoustic signature heard is far from zero
compared to ambient room conditions. We emphasize, however, that the
pressure is “zero” relative to sound pressures within the pipe.