Mud Acoustics in Modern Drilling - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 267

We have already identified two means of characterizing acoustic waves,
namely, “acoustic pressure” and “acoustic impedance.” Other properties are
also used in the literature. The “acoustic intensity” or “I” of a sound wave is
defined as the average power transmitted per unit area in the direction of the
wave propagation, whereas the “sound energy density” is defined as the energy
per unit volume. This energy is partly kinetic, due to the motion of the fluid
elements, and partly potential, due to the displacement of the medium.
5.2 Governing Eulerian Equations
In our derivation above, we ignored any mean background velocities and
dealt with acoustic motions directly. This is justified for two reasons. If the
sound speed c far exceeds any local flow velocity V, so that the dimensionless
“Mach number” M = V/c << 1, the effects of acoustic and hydraulic flow
physically decouple and the two fields can be treated independently insofar as
wave propagation is concerned. It is important to understand that the sound
generation process does involve a complicated interaction between acoustic and
hydraulic flow elements that is difficult to model - signal strengths as functions
of fluid properties and MWD pulser geometry are more easily measured in
controlled experiments. Also, we emphasize that sound generation in mud pulse
telemetry is a purely subsonic phenomenon since M << 1.
We illustrate this decoupling with simple examples. The sound speed in
water is approximately 4,500 ft/sec and downhole fluid speeds are unlikely to
exceed 1,000 ft/sec, anywhere. The corresponding Mach number is less than
0.25. In classical applications, e.g., high speed aerodynamics, the dynamic
coupling is ignored in all applications where M < 0.3 (actual corrections to the
“potential equation” derived later are on the order of O(M 2 )). This is analogous
to ignoring the effects of flow on human speech (and vice-versa) on windy street
corners: flow effects are insignificant, except to the extent that a simple
kinematic Doppler correction may be required. Calculated pressures are linearly
additive, following Equation 5.36 below. From a mathematical perspective, the
effect of a uniform background flow is dynamically insignificant, since a
Galilean transformation with speed V transforms ( u / t + V u / x) = - p/ x
into u / t = - p/ x, which applies in the no-flow case. In other words,
constant background speeds are not consequential to the dynamics.
Flow effects are negligible in sound propagation, but they will have some
effect on attenuation, because the effects of background viscous stresses due to
flow cannot be completely ignored. And again, flow effects are important to
sound generation or signal creation process at MWD mud pulsers. In general,
this mechanism is three-dimensional: it occurs over regions very small
compared to a wavelength, and it naturally involves significant acoustic-
hydraulic flow interaction. These subtleties plus examples are discussed later.