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2 u/ t 2 - B 2 u/ x 2 = P s (t) (x-x s ) (5.61)
Equation 5.61 clearly implies a jump in the first derivative u/ x. If we recall
that the acoustic pressure is p = -B u/ x, integration across the source point
(using the delta function formalism of Chapter 1) shows that the jump in
pressure equals the “pressure source strength” P s (t). It is important, invoking
results from Chapter 1, to observe that the overpressure upstream is equal in
magnitude to the underpressure downstream (this has been observed
experimentally in the laboratory). Thus, the created signal pressure is
antisymmetric with respect to the source point. Also, in an infinite system, half
the delta-p will propagate uphole away from the source, while the remaining
half will propagate downhole. By contrast, the created acoustic velocity field is
symmetric with respect to the source; that is, upstream and downstream
velocities always possess the same signs, their directions being physically
identical. The propagating pressures, displacements and velocities in the above
paragraph will travel with a sound speed c = (B/ ) 1/2 . As the wave passes any
fixed point within the drillpipe, it momentarily induces a fluid displacement
u(x,t), associated with a kinematic strain u/ x, and a material velocity
u/ t
satisfying | u/ t| << c .
5.5.3.2 Negative pressure valves.
Negative pulsers may be conceptually visualized as corked holes in the
drill collar that periodically open and shut. When this “door” opens suddenly,
the rush of mud through the orifice creates a pressure drop or negative pressure
in the drillpipe; the pressure drop between pipe and annular flows is responsible
for signal generation. When the “door” closes, an over-pressure is created, that
propagates up and downhole. A negative pulser is schematically shown in
Figure 5.5. In either case, the physical corking nature of the signal generation
implies that created signal pressures , by contrast to those for positive pressure
valves, are symmetric with respect to the source point , the signs in pressures
being the same immediately above and below the pulser. Thus, negative
pressure valves create MWD signals detectable at the surface, but they are not
associated with nonzero delta-p' s. Instead, they are associated with nonzero
delta-velocities (whether the cork is plugging or unplugging, the velocities at the
sides of the valve are always opposite in direction). The conventional maxim
that nonzero delta-p' s are always required to transmit information is therefore
incorrect. The physical picture described above applies, as well, to a balloon
that is “popped” in a long conduit. Again, because the acoustic pressure
distribution is symmetric about the source point, there can be no pressure
discontinuity, but the existence of equal and opposite velocities at all times leads
to jumps in velocity or nonzero delta-velocities . How are these modeled?
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