Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 126

characterized the dissipative nature of the drillstring, would converge with the
work of Dareing and Livesay (1968). This paper solved a more realistic
vibrations formulation satisfying the weakly-damped wave equation 2 u/ t 2 +
u/ t - E 2 u/ x 2 = 0, which replaced the simpler 2 u/ t 2 - E 2 u/ x 2 = 0
model. Mass-spring surface boundary conditions were again assumed to be
undamped; also, matching conditions again assumed continuity of displacement
and force at the pipe-to-collar interface. The reader is encouraged to study the
above works, since they emphasize the basic physics of downhole vibrations.
This background is important to understanding modern papers, which stress
real-time data acquisition methods, and more complicated phenomena such as
stress-reversals, stick-slip torsional oscillations and bit whirl.

4.2.2 Governing differential equation.
Axial vibrations are significant because drillstrings “make hole” parallel to
the drillstring axis. Many drilling related results appear in the literature on
resonant vibrators, pneumatic jackhammers and area-contoured wave
amplification devices; however, none are relevant to modeling rotary drilling.
Here, the fundamental governing equations are given; recent literature is
reviewed in the context of specific issues. Key ideas introduced in other
technical disciplines will be adapted for use in our work; these include ideas in
earthquake seismology and physical acoustics. Then, applying methods from
finite difference analysis, the general formulation is solved numerically.

4.2.2.1 Damped wave equation.
To introduce the basic concepts, let us consider the classical equation for
longitudinal or axial vibrations

2 u/ t 2 +

u/ t - E 2 u/ x 2 + g = F e (x,t,u, u/ x, u/ t) (4.2.1)

where is the mass density per unit volume, is the damping factor, E is
Young' s modulus, and g is the acceleration due to gravity. In this equation,
u(x,t) is the longitudinal displacement of a material element from its equilibrium
position at x for a particular time t. Note that Dareing and Livesay (1968)
employ the form 2 u/ t 2 + u/ t - AE 2 u/ x 2 + g = 0 where is mass per
unit length ; thus, the two damping factors are related by A = , with ours being
= /A, where A is the cross-sectional area. The vertical drillstring is
schematically shown in Figure 4.2.1, where x = 0 describes the formation
contact point, with x increasing upwards. Equation 4.2.1 follows from Newton' s
“ F =m a ,” which relates the resultant acceleration to the applied external force.
The “m a ” is just “ 2 u/ t 2 ” on a per unit volume basis, while the “E 2 u/ x 2 ”
arises from the difference between the normal stress E u/ x acting on one face
of the control mass, and E u/ x + (E u/ x)/ x acting on the opposite. Their
difference leads to (E u/ x)/ x or E 2 u/ x 2 . The “ u/ t” refers to viscous