Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 227

4.5 Coupled Axial, Torsional and Lateral Bending Vibrations
We have discussed the fundamentals behind axial, torsional and lateral
vibrations, and the physics of modal interaction through drillbit boundary
conditions. Here, we integrate all of our previous models into a single one that
allows axial, torsional and two lateral vibration modes to dynamically interact.
A transient coupled formulation is given which builds upon all of the ideas
developed thus far - this model is solved numerically and computed results
corresponding to “smooth” and “rough drilling” are discussed. We warn against
the dangers of accepting the Fortran listing at face value. Like all research
products, its underlying implicit assumptions are numerous. For example, the
back-interaction of large lateral vibrations on the axial stress field as discussed
previously is not included in the printed listing, although simple modifications
will extend the program to handle such nonlinear effects.
Also, the code may not be directly useful to certain applications. For
instance, there are essential differences between rotary drilling where the entire
drillstring rotates, and turbodrilling, where the drillstring does not rotate,
although the drillbit does. On the other hand, vertical, deviated and horizontal
drillstrings see different kinds of gravity effects and formation reactions. The
emphasis of this topic is therefore tutorial, and not, software oriented. In the
drilling literature, the meaning of linearity is sometimes confused with the
presence of dynamic mode coupling. We emphasize that mode coupling does
not imply nonlinearity, when simple classical boundary conditions are used.
This is not to say that the resulting problem is simple! But our use of if-then
rock-bit boundary conditions, which clearly relate to changes triggered by
transient threshold amplitude levels, renders the complete boundary value
problem nonlinear, despite any apparent linearity in the governing partial
differential equations. This observation is crucial: computed results may appear
similar, but results from one run will generally not scale into those for another.
4.5.1 Importance to PDC bit dynamics.
At the present time, a significant commercial driver for understanding
coupled vibrations is costly PDC bit failure. The failure of such bits in hard
formations is often blamed on uneven thermal expansion rates between the
binder and the diamonds during rapid frictional heating (e.g., refer to an early
paper by Cohen, Maurer, and Westcott (1993)). However, part of the blame
rests with bit-induced coupled axial, torsional and lateral vibrations in hard
formations: conventional PDC bits tend to whirl backward, resulting in reduced
penetration rates and drillbit life. When a bit whirls, it “walks” around the well
bore backwards, increasing hole size and reducing penetration rate, and
decreasing directional control. Bit whirl shows how a “simple boundary
condition” can control the entire drillstring dynamics.