Mud Acoustics in Modern Drilling - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 273

which applies to each of the internal nodes i = 2 to i = i max -1. Boundary
conditions relating U 1,n to U 2,n at x = 0, and U imax-1,n to U imax,n at x = L
complete the system of tridiagonal equations. For example, if we are modeling
acoustic wave propagation in a borehole, an infinitely rigid bottom can be
specified by setting U 1,n = 0. If far uphole, the annular fluid discharges in a
continuous stream into a large tank and the acoustic pressure approximation p =
- B u/ x 0 implies that U imax-1,n - U imax,n = 0 .
More complicated models are easily created. If the pressure uphole near
the surface is known with certainty (say, p s (t)), then p = - B u/ x implies that
p s (t) = - B(U imax,n - U imax-1,n )/ x or U imax-1,n - U imax,n = ( p s (t) x)/B. On the
other hand, if the uphole conduit is extremely long, and only an outgoing wave
“radiation condition” is desired, the fact that u(x,t) must be a function of “x-ct”
only faraway means that u/ t + c u/ x = 0 at some i = imax, so that we have
(U imax,n - U imax,n-1 )/ t + c (U imax,n - U imax-1,n )/ x = 0. The finite difference
recipes given in Chapter 4, therefore, apply with little modification and are just
as easily implemented. The greatest subtleties, as indicated early on in this
chapter, are posed by the existence of area discontinuities in the annular wave
propagation problems dealing with swab and surge, and in drillpipe acoustics
models for MWD mud pulse telemetry. We will discuss the required matching
conditions, which we emphasize, are different from those of Chapter 4.

5.3.2 Modeling area discontinuities.
Equation 5.52 applies to muds within uniform fluid columns in uniform
pipes; also, it applies within each section of a drillpipe containing multiple area
changes, with the same constants , B and because the same mud is present
throughout. This contrasts with axial vibrations, where we required different
partial differential equations for the pipe and drill collar, since the elastic
properties in each system can be different. At an area discontinuity, differential
equations do break down and fail to apply, since sudden changes imply rapid
wave variations for which the requisite derivatives may not exist. There, they
are replaced by matching conditions that analytically continue one solution into
the second, while globally conserving the physically correct quantities.

5.3.2.1 Axial vibrations.
For such problems, force must be continuous from one side to the next
since external loads do not exist. Since the areas change through the interface,
stress must be discontinuous . Force continuity can be expressed via Hooke' s
law “ = E ” as A (p) E (p) u (p) / x = A (c) E (c) u (c) / x, as shown in Equation
4.2.101. If we designate the index of the interface location by i*, we can write
A (p) E (p) (u i*+1,n - u i*+,n )

x = A (c) E (c) (u i*-,n - u i*-1,n ) x, where i*+ and i*-

refer to positions infinitesimally close to i*.