Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 244

4.5.4.8 Precessional instabilities.
The study of drillstring buckling under torsion and drillstring precessional
instabilities was initiated by Dunayevsky and his co-workers in the 1980s, and
reported in Dunayevsky and Judzis (1983), Dunayevsky, Judzis, and Mills
(1984), Dunayevsky, Judzis, Mills and Griffin (1985), Dunayevsky, Judzis and
Abbassian (1989), and Dunayevsky, Abbassian and Judzis (1993). Our “twisted
eraser” thought experiment clearly shows that a long twisted rod under axial
load will bend in two orthogonal planes. Dunayevsky and Judzis (1983), in their
analysis of critical loads for three-dimensional drillstring buckling, importantly
include in the governing bending equations, terms due to torque and axial force.
This early paper dealt with static deflections only, and did not address transient
problems; the torque in their governing elastic line equations was constant,
while the axial force varied linearly as discussed earlier.
Later, Dunayevsky and his colleagues studied drillstring precession in
directional boreholes and drillstring stability, when the rotation is constrained by
hole geometry. Under certain conditions, axial vibration energy will be
transformed into transverse vibration energy; at the same time, drillstring
rotation about the drillstring axis becomes unstable and the transition to rotation
about the borehole axis takes place. This is “precessional” motion. Their
critical conditions are formulated in terms of rotary speeds, amplitude of weight-
on-bit variations, damping and geometry. A finite element procedure reduces
the partial differential equation describing drillstring dynamics to a set of
“Matthew-Hill” ordinary differential equations. The stability boundaries of
“parametric resonances” for these equations determines the onset of precessional
motion of the drillstring. The results can be used to predict stable performance
of drillstrings during directional drilling. We emphasize that the instability
studied by Dunayevsky represents one of many types that can damage the
drillstring; like other instabilities considered in the literature, whether or not it
exists downhole is subject to empirical verification. But this is not easy.
Rewcastle and Burgess (1992) note that precession of the bottomhole assembly,
in reality, is accompanied by high levels of friction and abrasion: the lateral-
tangential collar motion causes the BHA to impact the borehole wall and to
decelerate rapidly whenever maximum deflections are reached. This will damp
out all evidence in measured surface torques.
4.5.4.9 Comments on Dunayevsky model.
The equations governing combined axial, torsional, and lateral bending
vibrations used by Dunayevsky and his colleagues were obtained by specializing
the “Kirchoff vector equations” (Bolotin, 1964) for a general elastic line,
therefore requiring some very unwieldy and error-prone algebra. Several high-
order terms in torque and axial force seem to be inconsistently modeled.