Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 218

dynamic angle of twist. Earlier we emphasized the significance of the combined
static and dynamic axial displacement field, since energy transfer between the
two in general exists through coupling at the bit. This, as we will find, is
similarly the case with torsional oscillations. In a purely static problem, the
simplification / t = 0 leads to the ordinary differential equation 2 / x 2 = 0 ,
which integrates to s (x) = Ax + B, with A and B being constants. This static
solution shows that successive cross-sections along the drillstring displace
azimuthally with a linear variation.
In a purely transient limit, the assumption / x = 0 leads to 2 / t 2 = 0 , s o
that we obtain the solid body rotation (t) = t (plus an arbitrary integration
constant) where is a constant drillstring rotation speed. This type of motion,
ideally anyway, exists when all other kinds of compressible transients have
decayed. In early drilling publications, this unrealistic solid body assumption
was actually quite popular. Wave-like torsional disturbances ride on these solid
body rotations, the same way, as in axial vibrations, where dynamic
displacements ride on the static stress field due to weight. In the simplest
dynamical problems, the static plus dynamic book-keeping (x,t) = s (x) +
d (x,t), by substitution in Equation 4.4.6, shows that the two component strain
fields can be treated independently via linear superposition , at least when partial
differential equations alone are considered.
This approach, natural in mechanical engineering, is assumed in the early
drilling literature. However, it is not representative of actual drilling operations.
In drillstring vibrations, the static mean is never clearly defined. At the start of
drilling, an unstressed drillstring will not rotate until some threshold torque-up
level that overcomes static frictional effects at the face of the formation is
achieved. The exact threshold depends on the bit type, the formation, and the
static weight-on-bit. Only when it is sufficiently wound is the drillstring ready
to rotate; once rotating, dynamic rock-bit interaction will excite the drillstring
with torsional waves that propagate to the surface or to areal changes, and
reflect. The complete picture does not end here. As we have seen in axial
vibrations, the dynamic interaction between the rock and the bit can lead to
transient bit bouncing, which implies a free torsional end. Low dynamic weight-
on-bit will also imply free torsional ends, since greater azimuthal slippage is
allowed. Both effects will momentarily remove the source of rock-bit excitation
and allow the drillstring to unwind. Then, the wind-up process repeats itself.
Thus, the transfer between mean potential energy and wave potential and kinetic
energy, due to axial induced boundary condition coupling at the bit, may be
quite active. Consequently, as for axial vibrations, it is more natural to study the
complete transient angular displacement (x,t) rather than separate

s (x) and

d (x,t) components. This is the approach undertaken in this topic.