Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 141

Analogies from earthquake seismology. Displacement sources were
first used to model seismic wave generation due to explosion-induced volume
changes in the earth; underground rumblings and wave interactions admit
multiple reflections, not unlike those in drillstrings, that ultimately pass
transparently through the source on re-reflection. The seismology text by Aki
and Richards (1980), for example, introduces the subject of displacement
discontinuities and volume sources in a readable fashion, especially with respect
to symmetries and antisymmetries in the displacement and stress variables about
the source point. For instance, over large times, the normal modes of the earth' s
oscillations depend on whether they are excited by the slippage of a fault plane
or by the volume creation due to a hidden nuclear blast. The type of source (or,
generating mechanism) is inferred from initial measured data; once known, the
strength of the source and the subsequent vibration history can be determined.
Of course, seismologists deal with three-dimensional problems, utilizing vector
forces, couples and dipoles to model different kinds of excitation. Hudson
(1980) provides a discussion, showing how various point loadings combine to
model physical phenomena having different symmetries or antisymmetries.
Similarly, the axial vibrations of a rod excited internally will depend upon
the nature of the source; e.g., if it is a diaphragm that moves to-and-fro, or if it is
an “accordion” that creates volume. In the former, stress is antisymmetric about
the diaphragm; a net force acts on the system. In the latter, no net force acts:
equal and opposite forces are present, which create nonzero displacement -
hence, our “displacement source.” Consider, as an example, air-driven
jackhammers where pneumatic effects create internal displacement sources
without creating net applied force: these sources are not associated with nonzero
external applied forces (it not necessary to have a nonzero force in order to
drill). Of course, one can assist the drilling process by yanking on the drill with
one' s foot, in which case the net excitation is the superposition of a
displacement source and an external applied force ! Again, the displacement
source nature of rotating tricone bits is modeled, applying two equal and
opposite periodic axial forces at the bit centroid, assuming displacement as the
dependent variable (on the other hand, a single delta function can be used if the
-model in Chapter 1 is assumed). In summary, we have invoked a rock-bit
interaction model u x + u t + u = 0 as a boundary condition at x = 0. The
displacement source, unlike Equation 4.2.19, is not a boundary condition, but an
excitation creating axial length without specifying bit position, while allowing
the transparent passage of waves - this allows us to prescribe a rock-bit
interaction model at the end of the drillstring and calculate quantities like bit-
bounce and rate-of-penetration. Also, we have emphasized the need to describe
AD/DC solutions by a single unified transient dependent variable. These
discussions were detailed because the approaches are new.