Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 222

oscillations are completely absent, static torque must be accounted for in the
overall dynamical picture, if bending effects such as “precessional instabilities”
are to be modeled. Again, drillstrings unwind when they are off-bottom; the
amount of unwinding is in part set by the initial torque-up or static torque.
Thus, modeling the complete torque variable is of paramount importance.

4.4.2.3 Finite difference modeling.
Equation 4.4.6 is identical to the axial wave equation discussed earlier.
Thus, the same tridiagonal finite difference equations, implementation recipes
and diagonal dominance considerations apply, with obvious changes in
nomenclature, and identical matrix inversion processes can be used.
Simplifications, of course, are found in the auxiliary conditions. For instance,
the mass-spring-damper surface model used to simulate traveling block
oscillations is now replaced by the simpler time-dependent displacement
condition = t, where is the applied rotation rate. For example, we might
have an of 60 rpm , written in terms of radians; note that we do not use 3 60
rpm , which applies to tricone-induced axial vibrations (also, the displacement
source model used to simulate vertical tricone excitation does not apply here).
In our discussion so far, we have considered a purely uniform drillstring,
ignoring cross-sectional discontinuities along drillstring lengths, particularly at
the interfaces separating drillpipe from drill collar. In our earlier study of
longitudinal vibrations, we showed that continuity of force and of longitudinal
displacement apply to axial motions , and derived stable finite difference
formulas for solution matching at impedance change junctions. The required
matching conditions for torsional vibrations are analogous, that is, we assume
continuity of torque and of angular displacement . Like axial strain, the torsional
strain is double-valued at the impedance junction: it suddenly jumps in value.
The analytical and numerical results for axial vibrations apply analogously, with
only obvious changes in nomenclature; for example, observe the similarity
between force AE u(x,t)/ x for axial vibrations, whereas torque GJ

(x,t)/ x for torsional vibrations.

4.4.2.4 WOB/TOB.
The ratio of average weight-on-bit to torque-on-bit provides useful
information for bit bearing failure detection and formation identification. This
parameter, when available in real-time, is therefore useful in anticipating drilling
problems, and in providing mechanically-based “second opinions” in support of
lithology predictions founded on gamma ray, resistivity and porosity
measurements.