Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 175

an interaction model depending on the single ratio / only. The resulting
formula f(z) = g(z), using Equation 4.2.122, leads to the complete solution
u(x,t) = g(ct+x) + g(ct-x) (4.2.126)
showing that, as a consequence of the boundary condition in Equation 4.2.125,
the reflected waveform propagates with its shape unchanged, although its
amplitude and sign (that is, phase) may be different. Equation 4.2.126 contains
a reflected wave that is identical with the incident one (that is, both terms
contain the same g ), except modified by a (positive or negative constant) factor
: its amplitude is generally not unity. How are inverse solutions obtained?
Assuming that the net contributions g(ct+x) and g(ct-x) propagating in opposite
directions are individually available (we will demonstrate how in the next
section), is completely determined as follows.
Since the incident wave g(ct+x) is known, the function g(ct-x) can be
constructed. And because g(ct-x) is known, and g(ct-x) is available, is
obtained by division. Once is known, the definition = ( +c )/( -c )

obtained from Equation 4.2.124 yields
/ = -c(1+ )/(1- ) (4.2.127)
which completely determines rock/bit mechanical impedance condition in
Equation 4.2.125. Reference to the (hypothetical) rock-bit interaction database
will then identify the formation corresponding to the known rock bit. The
foregoing discussion assumes displacement measurements. The same argument
applies to stress data. Since u x (x,t) = g x (ct+x) - g x (ct-x), the (known) ratio
between g x (ct-x) and g x (ct+x) again yields . The foregoing derivation
contains the limiting results used in our earlier illustrations. In particular, = +1
leads to a free end having infinite / , while = -1 corresponds to a rigid fixed
end rock having a vanishing / . Intermediate values of / and model
formations of intermediate hardness, whenever elastic effects are unimportant.
Elastic impacts, with stress effects. Downhole or surface signal
processing might first determine, in field implementations, if shape distortions
between incident and reflected waves exist; if not, one would conclude that the
simple rock-bit interaction model of Equation 4.2.125 holds. If distortions exist,
Equation 4.2.129 below, which includes elastic responses, or its extension
allowing for impact velocity, may apply. In the inelastic rock-bit interaction
model of Equation 4.2.125, space and time derivatives of axial displacement
appear, but the exact level of u(x,t) is unimportant. In this limit, the reflected
wave shape is undistorted relative to the incident one; general amplitude
changes are allowed, but phase changes are restricted to 0 o and 180 o . Here, we
assume that an elastic response at the bit is balanced purely by the normal stress
(impact velocity is ignored). In the foregoing example, we permitted our
incident waves to have general shape; now, though, we will need to interrogate