Mud Acoustics in Modern Drilling - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 259

5.1.2 Differential equation.
As before, let u(x,t) represent the instantaneous axial displacement from
equilibrium conditions. The acoustic pressure in a fluid is given by the formula
p(x,t) = - Bu x (this relationship is analogous to Hooke' s law = E = Eu x from
elementary strength of materials ). If the pressure -Bu x acts on the left side of
our one-dimensional control mass, with -Bu x + (-Bu x ) x dx acting on the right,
then following Newton's law F = m a , the net force differential Bu xx dx must
equal u tt dx per on a per unit area basis. Thus, we obtain the classical wave
equation for the displacement function u tt - c 2 u xx = 0, where c is the sound
speed satisfying c 2 = B / . As we have shown in Chapter 1, the general solution
to this equation is a superposition of right and left-going waves. It is tempting to
use our earlier results for axial vibrations without modification, and in many
applications for uniform waveguides with unchanging properties in the axial
direction, this may be possible and legitimate.
5.1.3 Area and material discontinuities.
The analogies, however, break down whenever cross-sectional changes in
area or material are found along the one-dimensional system; in drillstrings, this
discontinuity always exists, e.g., at the pipe and collar interface. That the
correspondence between axial drillstring vibrations and mud column acoustics
does not apply to problems with area changes will not be apparent from
differential equation derivations alone. To understand why, we must appreciate
the fact that one-dimensional wave models really represent approximations to
three-dimensional physical phenomena. If we recognize that information is
always lost in any averaging process, and that the reduction in the number of
spatial dimensions is tantamount to averaging, it is clear that one-dimensional
models lead to physical compromises. Lost information, at best, must be re-
introduced using auxiliary matching conditions.
Let us consider an example from strength of materials. In stress analysis, a
bar with an area discontinuity is associated with a jump in the stress . This is
so because the force through the transition point does not change; hence,
(p) A (p) = (c) A (c) , assuming that external forces do not act at the transition
point, where the “p” and “c” superscripts denote pipe and collar. In short, we
choose force continuity in the overall picture, so that stress differs on either side
of the jump. Thus we assumed, following this line of reasoning, that A (p) E (p)
u (p) / x = A (c) E (c) u (c) / x in Equation 4.2.101. In addition, the displacement
function itself was assumed to remain unchanged through the transition point,
since tearing is disallowed, and we are led to u (p) = u (c) in Equation 4.2.102.
These two conditions, together, completely defined the pipe-to-collar matching
conditions for axial drillstring vibrations, to the extent allowed in one-