Signal Processor Design and Additional Noise Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 155

For example, engineers operating higher data rate siren tools will explain
that desurgers are never problematic, that is, desurgers never distort signals and
in fact are an essential part of the MWD surface setup. On the other hand,
operators of lower data rate tools will describe how desurgers distort signals so
badly that the original signal is sometime unidentifiable. These remarks are not
hearsay: these comments are based on the author's first-hand experiences. It
turns out that, on returning to first principles, these discrepancies can be
explained logically. Again, we turn to the modeling principles established
earlier in Chapters 2-5. The one-dimensional wave model used applies so long
as a typical wavelength greatly exceeds the cross-dimensions of the drillpipe, a
condition easily met in all mud pulse operations.
The Lagrangian displacement variable u(x,t) is the distance that a fluid
particle moves as the wave travels past a given location “x.” The corresponding
velocity is wu/wt while the local acoustic pressure p is p(x,t) = -B wu/wx, where B
is the bulk modulus. If U is the fluid density, then c 2 = B/U. The general
solution to the above wave equation is u(x,t) = f(ct-x) + g(ct+x), as is well
known from partial differential equations. The “f” represents a right-going wave
and the “g” represents a left-going wave. For our convention, let us assume that
a reflector is located at x = 0. The wave originating from downhole is “g” and
the reflected wave is “f.” If the reflector is a solid reflector like a mudpump
piston, then the fluid displacement u = 0 at x = 0. Thus, u(0,t) = f(ct) + g(ct) = 0
at x = 0 so that f = -g. If we take x-derivatives of the general solution, then
wu(x,t)/wx = -f'(ct-x) + g'(ct+x) = -f'(ct) + g'(ct) = 2g'(ct), thus proving that at
the piston, the original pressure g'(ct) doubles - an elementary derivation indeed
and a correspondingly exact result.
This result, interestingly, has been used to develop the simplest analogue
signal amplifier - the “hundred feet hose” attachment shown in Figure 6.1b and
discussed in detail in the author's U.S. Patent Nos. 5,515,336 and 5,535,177.
Assume that an amplitude “A” is measured at a pressure transducer installed on
the standpipe. If this transducer is removed and attached to one end of a long
hose, with the other end then installed to the standpipe, the measured pressure is
obtained as “2A.” This application has been validated in field tests. Simply
installing the transducer ahead of a pump piston will also double the signal.
This is not usually done because the pump is viewed as a noise source.
Figure 6.1b. “One-hundred feet hose” analogue signal amplifier.