Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 140

This formulation for drillbit kinematics, together with earlier rock-bit
interaction ideas, meets our modeling objectives. Importantly, its solutions will
include problems satisfying the “u(0,t) = u 0 sin t” if they evolve as part of a
transient solution (that is, for hard enough rock, if the bit does not bounce). But
in general, computed values of u(0,t) are unlikely to be purely sinusoidal; they
will contain fluctuating AC/DC components, a consequence of the complicated
dynamic displacements at the bit. This usage was first reported in Chin
(1988a,b) where computed solutions showing rate-of-penetration and bit bounce
were given; detailed results for a stabler scheme are given later towards the end
of this chapter. However, the rudimentary ideas underlying displacement
sources are not new; as noted in Chapter 1, they are well known in earthquake
seismology, where different types of source singularities model symmetric and
antisymmetric triggering events.
The first use of the displacement source concept in drilling actually
appeared in Lutz, Raynaud, Gstalder, Quichaud, Raynal and Muckleroy (1972).
Drawing upon extensive field and test bench measurements, and detailed
kinematical interpretation of drillbit motions, the authors concluded that the
tricone bit was more than a passive creator of longitudinal and torsional
vibrations. Unlike earlier studies which modeled bit excitation with prescribed
vertical displacement histories, e.g., Equation 4.2.19, Lutz et al reiterated the
role of the tricone as a source of axial displacement or as a displacement
generator characterized by the up-and-down lift involved in passing from one
tooth to the other. This concept is important since sources , following Chapter 1,
adhere to different rules than do boundary conditions , and lend greater modeling
flexibility. The power of the idea was not immediately realized, and its only
application since then were this author's 1988 papers. The notion of
displacement sources was recently revived in Clayer, Vandiver, and Lee (1990),
which clearly described the excitation process at the bit. These authors
specifically described the action of a tricone bit as a relative displacement
source situated between the rock and the end of the drillstring.
As the cones and teeth roll over irregular formations, relative movement
between the body of the bit and a fixed, unaffected, faraway reference point in
the rock is observed. Close to the drill bit, the rock may deform elastically or
inelastically, failing in a small region close to the teeth. The action of a tricone
bit parallels that of a displacement source, e.g., the accordion shown in Figure
4.2.3, inserted between the bit and a flexible formation, which is modeled with
an equivalent spring and damper. This type of source can model mathematically
both the periodic lifting action of the teeth as well as the displacement caused by
the tri-lobed pattern which develops in the rock when one uses tri-cone bits,
while at the same time allowing some rock-bit interaction.