Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 120

to PDC bits; inputs included stabilizer, bit cutter and formation models.
Although the effort undertaken in such analyses is credible, there are
disadvantages to this line of attack. Computer times required for stand-alone
runs were approximately one hour on a Cray supercomputer. Significant labor
and facilities resources are needed, rendering any product beyond the reach of
many drillers. In this topic, only reliable methods requiring seconds and
minutes on personal computers are developed.
4.1.5.2 Elastic line simplifications.
Despite the apparent generality of the above work, all of the three-
dimensional details in drillstring simulation need not be treated equally. In
Chapter 3, we showed that fine scale phenomena in three-dimensional
waveguides ( here , the drillstring and BHA) are likely to decay without ever
seeing propagation as a plane wave. Thus, they can be ignored. While
drillstrings undergo extremely complicated dynamically coupled vibrations,
their geometries are fortunately simple because they are much longer than
typical diameters. As most wavelengths are likewise large, we are additionally
justified in modeling the drillstring as a lineal system.
Thus, modeling the inserts on a tricone bit is unnecessary if the important
large-scale effects along the drillstring are alone sought. The vertical response
of a bit on a given formation, determined from vibration-isolated lab tests, such
as those pursued early on by Amoco Production Company, suffice for a
macroscopic description of the rock-bit interaction. In fact, we follow this
approach. We will model axial, torsional and lateral components as vibrations
of an idealized “elastic line,” where the only spatial independent variable is the
length coordinate x. This is not a single degree-of-freedom system: we allow
static and dynamic superpositions of axial, torsional and two modes of lateral
vibrations, everywhere along our elastic line.
Cross-sectional details appear by way of properties such as area,
rectangular and polar moments of inertia, while lengthwise variations are easily
incorporated by allowing explicit dependencies on x. Of course, the correct
conservation form of the differential equations must be used for gradual
changes; sudden variations are modeled using global impedance matching
conditions that correctly fix reflection and transmission coefficients. This
elastic line approach will be generalized to include nonlinear coupling between
axial and transverse vibrations, while retaining the simplification that planar
cross-sections remain so upon bending. We also introduce nonlinear boundary
conditions for wall contact and rock-bit interaction simulation.
4.1.5.3 Historical precedents.
This approach is not new; for example, this philosophy is used in the
vibration modeling of machine shaft oscillations. The origin behind elastic line