Mud Acoustics in Modern Drilling - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 269

In unsteady incompressible flow, the transient term u / t remains. When
the flow is “irrotational” we can write the velocity u (x,t) in terms of a potential
function (x,t) using u (x,t) = (x,t)/ x (see Equation 5.47). Since u / t =
2 / t x = ( / t)/ x, Equation 5.31 can be rewritten in the form ( ( / t)/ x
+ u u / x) = - p/ x, or { / t + p + 1 / 2 u 2 }/ x = 0, so that / t + p(x) +
1/2 u 2 (x) = f(t), where f(t) is an unsteady function of integration. This
unsteady Bernoulli equation was used earlier in Chapter 3 in our study of water
waves (see Equations 3.69 to 3.71), and we will again encounter it again in our
study of ocean waves and currents.
5.2.2 Separating hydraulic and acoustic effects.
Earlier we introduced the term “hydraulic,” which is used synonymously
with constant density and incompressible . “Acoustic,” again, always refers to
transient compressible fluid motions satisfying wave-like equations. These
distinctions must be made, especially in the study of mud pulse telemetry, to
emphasize the coexistence of two contrasting physical phenomena. Equation
5.32 is the constant density version of Bernoulli's equation (a compressible
version exists, but is not treated in this topic). It does not describe wave motion;
because the fluid is assumed to be incompressible; disturbances in this limit
travel with infinite speed. Also, it does not describe viscous flows; Bernoulli's
equation assumes inviscid (that is, frictionless) flow, since viscous shear stresses
are not included in the momentum balance underlying Equation 5.31. However,
Equation 5.32 can be used to relate pressures between two points “1” and “2” in
a variable area duct containing viscous flow if the separation distance is small.
For such applications, we have the governing equations p 1 (x) + 1/2 u 1 2 (x) =
p 2 (x) + 1/2 u 2 2 (x) and A 1 u 1 = A 2 u 2 .
We all know from experience that acoustic wave motions can “ride” on
hydraulic ones. For example, we can speak and be heard, whether wind is
blowing or not, assuming that there is no noise due to turbulence. Very often, it
is desirable to separate hydraulic from acoustic effects, since these satisfy very
different sets of physical laws. This separation helps us understand and better
appreciate the actions of different types of forces that can act on the surfaces of
or through boreholes and drillpipes. Now, the one-dimensional equation
describing mass conservation takes the form
/ t + ( u )/ x = 0 (5.33)
Equation 5.33 indicates that time-wise increases or decreases in the mass density
within a control volume must be balanced by the flux of mass. This equation
applies to combined hydraulic and acoustic effects. In order to separate the two
effects, we write the density, velocity and the pressure as the sum of mean ( m )
and acoustic disturbance ( primed ) parts, namely,
(x,t) = m (x,t) + '(x,t) (5.34)