Mud Acoustics in Modern Drilling - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 285

Now, let us introduce the delta function term c 2 V s (t) (x-x s ) to the right
side of Equation 5.62, that is, add an internal source to the potential wave
equation in the form
2 (x,t)/ t 2 - c 2 2 (x,t)/ x 2 = c 2 V s (t) (x-x s ) (5.65)
Following Chapter 1, an integration through the source point x = x s shows that
the right-side term c 2 V s (t) (x-x s ) is responsible for a discontinuity in the
quantity c 2 (x,t)/ x. That is, substituting from Equation 5.63, we have a jump
in the term c 2 u t (x,t) with the strength c 2 V s (t); canceling out c 2 , we find that the
jump in the material velocity u t (x,t) is specified by assigning the velocity source
strength V s (t) in a potential function formulation. An arbitrary MWD source
can, in general, be modeled as the superposition of appropriate positive and
negative pulsers.
5.5.3.3 Mud siren sources.
A mud siren consists of a turbine-like rotor and stator pair mounted in a
low-clearance housing, through which drilling mud passes, as shown in Figure
5.6. The “rotor” rotates, while a stationary “stator” remains in place; as it
periodically interrupts the mud flow, acoustic pressure waves are created that
propagate both upwards and downwards. Sirens are only “turbine-like” to the
extent that they rotate. Their blades are chunky as shown, rather than thin and
cambered, as in aircraft turbines or turbodrills. Police sirens create high pitch
whining sounds; mud sirens, by contrast, periodically block the complete cross-
sectional area, thus creating acoustic plane waves that tend to be low-frequency
in existing designs. These plane waves, like those for poppet valves and
negative pulsers, are amenable to wave modeling.
As the siren closes, flow at the uphole side is brought to rest, and an over-
pressure relative to ambient conditions is created. At the same time, flow pulls
away from the siren at the downstream side, creating an underpressure. In this
sense, the physics of the siren acoustic source is identical to that for positive
pressure poppet pulsers: the mathematical discussion leading to Equation 5.61
applies without modification. When the siren opens, the over-pressure at the top
is relieved, but the under-pressure at the bottom increases; the above description
reverses, and Equation 5.61 again applies. Continuous mud siren rotation
creates a continuous periodic pulse, but we note that poppet valves and negative
pulsers can likewise function in continuous mode if they are so mechanically
designed. Acoustically speaking, positive pressure valves and mud sirens
function identically and the same displacement simulation models apply.
Negative pressure valves, however, require a potential function formulation; of
course, simulators for the former can be easily modified to handle the new
problem.