Signal Processor Design and Additional Noise Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 156

An extension of the above analysis can be used to model desurger effects
on acoustic signal reflection. Now, the rubber membrane within a desurger
essentially acts like a large spring as opposed to a solid piston. The spring will
move with a distance “u” and the force exerted by the membrane would be
ku(0,t) where k is the spring constant. This force must equal that exerted by the
acoustic wave. If the acoustic pressure is -B wu/wx and SD 2 /4 is the area of the
membrane, D being the diameter of the circular membrane, then the boundary
condition is ku(0,t) = T wu(0,t)/wx where T = BSD 2 /4. Note the following two
physical limits: the mud pump piston possesses “infinite k” so that u = 0, while a
centrifugal pump satisfies wu(0,t)/wx = 0 instead.
6.1.1 Low-frequency positive pulsers.
Consider the situation at the top of Figure 6.1c. A rectangular pulse travels
toward the desurger location x = 0 at time t = 0 - but what does a rectangular
“u” pulse physically mean? Again, pressure is the spatial derivative or slope of
“u.” The head of the wave shows a large positive slope while the tail has a large
negative one. The flat portion of the rectangular has zero slope and is therefore
not associated with any compression or expansion. Thus, the initial t = 0
schematic in Figure 6.1c represents a sudden valve opening followed by a
sudden closure.
Figure 6.1c . MWD signal distortion for low data rate pulsers
(confirmed experimentally in mud loop tests).