Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 172

penetration and weight-on-bit. The latter two studies gave detailed time
oscillograph traces relating displacement, force and impact velocity. These
results indicate that predictive models characterizing drillstring response to
elastic or brittle rock contact exist and can be defined; for this reason, it is not
surprising that the SNAP Lo g proved successful in its lithology correlations.
Taken together, these studies suggest a drillstring response at the bit taking
the general linear form u x (0,t) + u t (0,t) + u(0,t) = 0, at least over a limited
range of displacements, as suggested in Equation 4.2.22. Such relationships are
often used in jack-hammer and resonant drilling rod design; they are mechanical
impedance models which relate force, displacement and velocity through
constants characteristic of the rock and drilling rod.
Figure 4.2.9. Linear WOB versus u t (0,t).
However, the use of impedance models is new to inverse formation
evaluation problems, and Equation 4.2.22 is significant in two respects: the
model does describe reality and it is linear. From a practical viewpoint, it is
also important that such relationships can be obtained in the laboratory under
closely controlled conditions absent of drillstring reverberation.