Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 122

simpler problems are given, which highlight the character of the equations. The
extent to which these solutions are consistent with the proposed physical model
determines the utility of the mathematical model.
4.1.6 Objectives and discussion plan.
Researchers traditionally force-fit axial, torsional and lateral vibration
problems into simple categories treated in elasticity topics. Early papers applied
rod and beam models to downhole vibrations, examining classical resonances
and mode shapes, but recent MWD data, suggests that downhole phenomena are
more complicated dynamically than simple models suggest. As we have
indicated, researchers have attempted to drill at the calculated classical
resonances with little success; they have tried to relate violent vibrations to the
same resonances, but little evidence supports the idea that natural frequencies
seen in field bear any relation to existing theory.
Work in wave propagation and the stability of dynamical systems has
proceeded rapidly in other disciplines, especially in hydrodynamic stability (e.g.,
see Chapter 3), and workers now concede that instabilities, being many-faceted,
must not be restricted to limiting classical resonances. We will re-evaluate the
meaning of stability, develop a physical model that explains the mysteries of
violent lateral vibrations, and integrate inter-disciplinary ideas into drillstring
modeling. We combine axial, torsional and lateral vibrations with insight
acquired through practical experience.
In addition to the work cited earlier, we will survey the literature,
consolidating only those relevant papers that address important unifying themes.
We will cite, though, only a portion of the literature if only to give a flavor of
drilling problems: our review is not a detailed literature survey. We mainly
stress physical ideas and fundamental modeling concepts, some introduced long
ago, having stood the test of time and the benefits of hindsight. Thus, this
chapter is not about directional drilling, nor is it a “how-to” manual on damage
avoidance; we will not catalogue the voluminous experimental data published in
recent years. We will, in our presentations, emphasize time rather than
frequency domain analysis, since drillstring vibrations appear to be highly
transient. The final algorithms cover, of course, dynamically steady oscillations;
if such exist under sinusoidal excitation, our computational procedures will
model them. But we will not attempt to generalize, understanding well that
exact details depend on the bottomhole assembly, bit wear and design, and
formation properties. To keep our discussions fundamental and introductory, we
will not translate our findings into stabilizers, reamers, drilling operations
procedures, build and walk rates, or inclinations, directions and depths. Nor will
we design shock subs and improved drill bits.
On the modeling side, our mathematics is rigorous and readable, covering
only those points needed to explain the application. A background in calculus,