Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 111

transition). The physical quantities that are “conserved” through the
discontinuity depend on the physics of the particular problem - these properties
may include pressure, total mass flow rate, force and so on. The drillstring is a
“waveguide” supporting many types of vibrations. General waveguide motions
represent sums of distinct mode shapes or eigenfunctions, which are either
“propagating” or “evanescent” (see Chapter 3). Different modes contain
different amplitude distributions in the cross-space; the lowest (propagating)
mode is the constant amplitude “plane wave” mode.
We showed earlier that waveguide effects and dissipation may preclude
certain waves from propagating; thus, full three-dimensional analysis is not
always required, if the end motions of interest are planar one-dimensional waves
- for instance, flow details near a MWD pulser are unimportant to large-scale
wave transmissions and reflections, although they will affect the strength of the
created signal. Similarly, we need not model the inserts on a drillbit; a
macroscopic boundary condition accounting for rock-bit interaction and its
effect on long waves suffices if we are not interested in three-dimensional events
near the bit. Wave equation models may be combined with “lumped parameter
descriptions” of small-scale physical phenomena in order to understand the
overall problem. This is the approach taken in this topic for all wave motions.
4.1.2 Drillstring vibration modes, axial, torsional and lateral.
The three primary classes of drillstring motions are “axial,” “torsional” and
“lateral vibrations.” Secondary classifications are often made; for example,
“stick-slip” oscillations form one subset of torsional vibrations, while “whirling”
and “precessional instabilities” are subsets of lateral vibrations. “Bit bounce” is
a severe instance of axial vibration. Drillstring vibrations can be “fully
transient,” or they may be “dynamically steady,” that is, constant in frequency.
Furthermore, energy transfer between the mean and harmonic components of the
strain field of a single axial, torsional, or lateral vibration mode, and energy
transfer between these latter modes, are both possible. Thus, numerous degrees
of freedom exist which may superpose linearly or nonlinearly.
4.1.2.1 Axial vibrations.
The vibrations of the bar described above are axial or longitudinal ones
because they occur along the longitudinal axis of the rod. It is not necessary for
the rod to be straight. A rod bent in the shape of a pretzel, so long as the
“intersections” are restricted from contact, will produce axial vibrations if struck
at the ends or excited from within. The mathematics is one-dimensional,
utilizing a single coordinate “x” measured along the axis. The drillpipe system
generally contains discontinuities in properties along its length, e.g., cross-
sectional area, material density, Young' s modulus, sound speed, rectangular and
polar moments of inertia, and so on. At these sudden changes, incident waves