Mud Acoustics in Modern Drilling - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 260

dimensional wave modeling. These led to the difference approximation A (p) E (p)
(u i*+1,n - u i*+,n ) x = A (c) E (c) (u i*-,n - u i*-1,n ) x derived earlier in Equation
4.2.103, which supplements the use of separate differential equations for the
pipe and collar.
In mud column acoustics, however, we recognize that fluids physically do
not support internal stress discontinuities. Thus, acoustic pressure itself is
assumed to be continuous through the area discontinuity. Since p = - B u/ x,
and B is constant throughout the entire system, we obtain u (p) / x = u (c) / x.
Continuity of volume velocity A (p) u (p) / t = A (c) u (c) / t furnishes another
matching condition implies that displacement is not continuous. These
conditions are mathematically different from those for axial vibrations. This
means that a solution developed for jackhammer designer cannot be used for
water hammer unless, or course, both waveguides are uniform in space. It is not
obvious that acoustic pressure must be constant through an area change. In
constant density, incompressible flow , pressures do change. The exact level is
determined by mass continuity, Bernoulli's equation and the area ratio;
differences between “pressure area” from one side to the other are assumed to
be balanced by stresses acting on the metal structure. That (acoustic) pressure
continuity applies turns out to be realized by long waves only, following
rigorous three-dimensional arguments that are beyond the scope of this topic.
Here, we will summarize the relevant results only.
Consider a duct with a sudden areal change, that is, a short transition zone
separating two regions, each of which support long plane waves. In general, the
acoustic pressures in both regions differ due to evanescent modes (e.g., see
Chapter 3) excited at the transition, but this difference is small. For circular
ducts with radii r 1 >> r 2 , this difference is not significant unless the frequency f
= /2 is comparable to or larger than the critical value f crit = ( c r 2 2 /r 1 3 )/2 . Let
us examine r 2 = 0.25 ft, r 1 = 0.5 ft, and c water = 4500 ft/sec as representative
order-of-magnitude numbers; for this example, we have a high f crit = 358 Hz.
Thus, for slower pipe movements in swab-surge applications, or for typical
MWD mud pulse transmission data rates, the approximation p 1 p 2 suffices at
area discontinuities. Detailed mathematical discussions are offered in Miles
(1944), Miles (1946), Morse and Ingard (1968), and Pierce (1981). Nor is the
continuity of volume velocity (that is, v = u/ t) assumed in acoustic textbooks
obvious. Certainly a case can be made for continuity of mass flow velocity, and
in compressible aerodynamic applications, this does lead to improved agreement
with measured results (Chin, 1977). At any rate, we will abide by the
conventional acoustic matching conditions in this topic. Unlike our formulation
for axial vibrations, where two wave equations are generally needed to
characterize different material properties in the drillpipe and drill collar, we