Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 115

consequences might be. For example, the failure of a real-time MWD logging
tool due to bounce on a flatbed truck may not presage downhole catastrophe,
since the failure modes may be completely different; one might ask, rightly so,
what the roles of borehole constraints are and how drillbit motions excite the
system. In designing MWD tools, for instance, vibration table testing is of little
value unless the actual conditions and shock loadings encountered downhole
while drilling are simulated. Similar comments apply to computer models and
numerical simulators.
Before we discuss the mathematics in detail, it is useful to define the most
basic time and space scales. Consider a drillstring rotating at 1 rpm; its rotary
speed is therefore 1 revolution per 60 seconds, or 1/60 th Hz. If it rotates at
N rpm and drills with a tri cone bit, the axial excitation frequency takes on the
value f = 3N rpm /60 Hz or N rpm /20 Hz. Thus, for N rpm = 60, we have f = 3 Hz;
higher frequencies are possible depending on the construction of the bit. The
wavelength associated with a frequency f and a sound speed c is = c/f. If c =
15,000 ft/sec in the metal pipe, and f = 3 Hz, then = 5,000 ft. Since real
drillstring lengths range from a significant fraction to many multiples of , wave
equation (as opposed to lumped mass) modeling is appropriate. It is difficult,
and even dangerous, to assign “ballpark numbers” to different modes of
downhole vibration, since much depends on the BHA used, the formation being
drilled, the drillbit design, and the degree of cone wear (e.g., the average human
weight is 150 lbf, but the range covers 1-1,500 lbf!). We therefore avoid the
temptation to quote “real numbers.”
4.1.3 Long-standing vibrations issues.
We have described the vibration modes of drilling interest, but before we
pursue the subject of modeling, it is helpful to consider “the big picture” and
discuss the long-standing issues in vibrations analysis. Philosophically, one
always learns and executes more effectively if real world problems are posed.
4.1.3.1 Example 4-1. Case of the missing waves.
It is known that far downhole, severe lateral vibrations are the predominant
reason for drillstring failure; yet it is generally acknowledged that bending
oscillations cannot be detected at the surface. Lateral vibrations with bending
moments routinely exceeding 20,000 ft-lbfs, for example, have been measured
in drill collars. However, simultaneous surface measurements for axial and
torsional waves, which propagate effortlessly, provide few clues as to their
occurrence. These observations, true even in vertical holes where borehole wall
damping is minimal, seem contradictory and mutually exclusive. Where is the
fallacy?