Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 121

models can be found in the classic work of Love (1944). Applications appear in
the petroleum industry and in offshore engineering problems, work involving
pipelines, risers and mooring lines. For example, Nordgren (1974) formulated a
nonlinear, three-dimensional, large-amplitude transient problem, and solved it
by finite differences. Garrett (1982) gave a finite element model that modeled
large amplitude deflections, finite rotations and tension variation along the
length. Walker and Friedman (1977) proposed, for directional drilling
applications, a static mathematical model of the bottomhole assembly that
handles arbitrary hole inclination, geometrical and material properties. We will
develop the dynamic extension to their work. We emphasize that modern
vibrations, to include its sophisticated mathematics, is well developed; however,
the applications are acoustical and aerospace-oriented. Classic rod and beam
papers, for example, include Clark and Reissner (1951), Abramson, Plass, and
Ripperger (1958), Eisner (1964) and Clough and Penzien (1975).
4.1.5.4 Our focus.
The above cited works cannot be extended to deal with certain oilfield
boundary conditions, e.g., rock-bit interactions, drill bit modeling, and
penetration rate analysis. In subsequent sections, ideas from acoustic and
mechanical impedance modeling and earthquake seismology are introduced and
combined with rod and beam theory to model real drillstring motions. In our
discussions of elastic line modeling, we focus on integral elements of the
drilling process. For example, in axial vibrations, we separately study the
surface system, the drillbit kinematics, the rock-bit interaction and reflections at
area discontinuities, in this one-dimensional limit.
The same approach applies to torsional oscillations. While torsional and
axial wave equations remain independent of each other, dynamic coupling is
achieved at the bit by means of boundary conditions. When the bit bounces, a
torsional free end exists; when it drills ahead, the dynamic torque-on-bit is
proportional to the dynamic weight-on-bit as calculated above plus other
empirical factors. Axial and torsional calculations are taken beyond
conventional analysis by allowing energy transfer between mean and dynamic
states. Thus, increases in torsional strain energy due to initial wind-up, its
subsequent release to drive dynamic rotations, plus bit friction effects, will
manifest themselves through calculated results that display axial bit bounce and
torsional stick-slip. Once distributed axial and torsional loads are available at
any instant, they are used to couple both lateral vibration modes. We give a
stable numerical algorithm which solves the complete problem; in an illustrative
calculation, we demonstrate how the simultaneous effects of drillstring
precession, stick-slip, bit bounce and nonzero rate-of-penetration can be
realistically modeled. Numerical models, being computational, do not shed
insight into physical processes. Whenever possible, analytical solutions to