Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 123

plus exposure to differential equations is helpful. Numerical methods are
similarly developed so that their limitations, strengths and conceptual foundation
are understood. We will not dwell on integration rules, or improved spline-fit
algorithms. Rather, we demonstrate how the appropriate equations are
physically correct, and how they are differenced, programmed and solved. Once
the formalism for vertical wells is successfully tested against empirical
observation, qualitatively at least, the required modifications extending the
framework to deviated and horizontal wells are presented.
No claim is made to have “solved” the general problem. In fact, our
limited objective aims at providing only the analytical and computational tools
needed to model new phenomena and evolving concepts. This approach is
necessarily so with a subject that, despite its long-standing interest, is
nonetheless seeing rapid development. Drillstring vibration, as presently
envisioned, draws from numerous disciplines: mechanical vibrations, partial
differential equations, earthquake seismology, numerical analysis, rock
mechanics, acoustics and evolving empirical knowledge available from MWD
measurements taken near the bit. Drillstring vibration modeling, in conclusion,
is an integrated science, and hence, from it derives the personal challenge that I
have undertaken: a readable, usable, and down-to-earth volume that brings state-
of-the-art physical, mathematical and numerical ideas to practical engineers,
without emphasizing prerequisites, formality or academic credentials.
4.2 Axial Vibrations
“Axial” or “longitudinal vibrations,” terms used interchangeably, result in
displacements parallel to the drillstring axis. In this section, we formulate the
general boundary value problem for longitudinal vibrations, to include drillbit
kinematics, time-varying rate-of-penetration and rock-bit interaction. But first,
we review existing models, and explain their strengths and limitations. Axial
vibrations and stresses are considered first for several reasons. For one, their
solutions draw upon the classical wave equation analyses given in Chapter 1, but
some inadequacies of these techniques motivate the development of new
mathematical methods in a natural setting.
Second, an understanding of axial vibrations is essential to properly
modeling lateral vibrations, and in particular, why the latter can be devastating
downhole, yet undetectable at the surface. Often, nonlinearities are blamed for
these violent instabilities, while their mysterious disappearance is attributed to
borehole wall contact. Later, the mystery behind lateral vibrations is solved, and
a single linear model resolving both of the foregoing field observations is given,
which yields to analytical solution via kinematic wave theory (see Chapter 2).
Then, a nonlinear model which couples both axial and lateral vibrations is
formulated. Numerical finite difference algorithms for axial and lateral
vibrations are given in both chapters which simulate fully transient motions.