Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 216

4.4 Torsional and Whirling Vibrations
There are dangers associated with generalization, but it is often true that
axial vibrations are associated with tricone bits , which are vertically excited by
displacement sources, while torsional vibrations are associated with PDC bits
having flatter faces that are characterized by high levels of friction. Torsional
oscillations represent twisting actions about the axis of the drillstring, while
whirling oscillations consist of drillstring motions about the central borehole
axis. Whirling motion, a subset of lateral bending vibrations, is not torsional;
however, whirl is included in this chapter because it does not exist without some
rotational driving mechanism.
Why study torsional vibrations? These dynamic oscillations are
associated with “stick-slip” phenomena, that is, a winding and unwinding of the
drillstring that proves detrimental to bit life. Stick-slip conditions are discussed
in detail, and simple mathematical models are proposed. Then, there are more
obvious problems, e.g., over-tightening of drillpipe connections over a period
which can lead to equipment fatigue and premature wear.
Why study whirling? The “backward whirling” of PDC bits is detrimental
to cutter life. When a bit whirls, its instantaneous center moves about the bit
face rather than around its true center, as in smooth rotation. The path taken by
any single cutter is not circular: cutters impact the formation at off-design
angles, leading to damage in form of spalling, chipping, and premature bit
failure (Bobrosky and Osmak, 1993). Whirling will be discussed in detail and
modeled mathematically here and in the final presentation. In this section, we
present the differential equation describing torsional waves, and show how
solutions for “drillstring wind-up” and “solid body rotation” appear as special
limits. Then, the dynamic process consisting of static “torque-up,” “torsional
wave generation,” and “winding and unwinding” due to momentary “bit bounce
or bottom slippage” is outlined. This is followed by a broad discussion on
“stick-slip” oscillations and its modeling.
4.4.1 Torsional wave equation.
The partial differential equation governing torsional displacements is
similar to that describing longitudinal displacements along a drillstring.
Because of these similarities, we will not rehash the fundamentals in any
significant detail; analogous global energy balances, static-to-dynamic energy
mode transfers, finite difference schemes, and so on, apply with obvious
changes in nomenclature. If (x,t) represents the “angle of twist” along our
“elastic line” at any point x at a given time t, the transient torque T(x,t) satisfies
T = C / x (4.4.1)
where C is known as the “torsional rigidity” or the “torsional stiffness.” For
circular cross-sections,