Signal Processor Design and Additional Noise Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 157

The two pressure spikes, which possess opposite signs, are characteristic of
dipole sources; and since they are separated in time, the dipole source shown
applies to positive pulser valves - which are also known for large pressure
amplitudes. Physically, the acoustic force exerted on the desurger membrane is
“large” and acts over “long enough” times to be dynamically significant (these
are characterized dimensionlessly later). If the wave equation for displacement
is solved subject to the above spring boundary condition, we can derive the
following exact solution for desurger response to the rectangular wave, namely,
f(ct-x) = g(ct-x) - 2[1-exp{-k(ct-x-L)/T}] H(ct-x-L)
+ 2[1-exp{-k(ct-x-L-a)/T}] H(ct-x-L-a) (6.1a)
where H is the Heaviside step function. Note that unlike the simple piston
solution previously obtained for mudpump piston reflections in which f = - g
without changes in shape, Equation 6.1a is complicated and contains exponential
distortions to the rectangular pulse. This is the shape distortion observed for
positive pulsers associated with longer duration with high pressure amplitude.
An expression not too different from Equation 6.1a should apply to negative
pulsers. Unless their downward motions are filtered by the multiple transducer
techniques of Chapter 4, their presence will lead to serious surface signal
processing and interpretation problems (these effects are also likely to be found
with negative pressure MWD signals).
6.1.2 Higher frequency mud sirens.
Figure 6.1c does not apply to higher frequency mud siren operations,
which employ periodic frequencies, plus signal amplitudes that are typically
much lower than those of positive pulsers. Changes in phase or frequency are
used to telemeter 0's and 1's to the surface. When frequencies are high, the
wave literally “does not have time” to act on the desurger membrane, and vice-
versa - this inaction is, to be sure, more so, given the lower pressure amplitudes
involved. Thus, one does not expect desurgers to affect signals adversely. For
higher frequency mud siren applications, desurgers will do what they are
suppose to: remove dangerous transients and eliminate mudpump noise. These
statements can be demonstrated mathematically.
Again, the partial differential equation governing mud pulse acoustics is
the classical wave equation U mud u tt - Bu xx = 0, and its general solution is u(x,t)
= f(ct-x) + g(ct+x), where f and g represent right and left-going waves
respectively. This solution need not include damping because only the desurger
nearfield is considered. Over short distances, damping in the pipeline itself is
unimportant. Our objective is simple: what happens to a wave of a given shape
upon reflection? In this acoustic study, we will assume that the functional form
of the incident wave g(ct+x) transmitted by the pulser and impinging at the
desurger x = 0 is given. The problem consists in solving for the reflected wave
f(ct-x), and then, the complete superposition solution u(x,t), so that the acoustic
response is fully determined everywhere and at all instances in time.