Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 239

4.5.4.7 Example calculations: bit-bounce, stick-slip, rate-of-penetration,
and drillstring precession.
Our integrated axial, torsional and bending vibrations formulation has been
implemented in a fully debugged Fortran algorithm, at least programming-wise.
Various ideas and coding details are still in research phase and are under
continual testing; thus, Tests A and B represent preliminary results subject to
refinement. For this reason, the details of the drillstring assumed in the
calculations are not discussed, since any computed results are qualitative at best.
Our vibrations solutions are only intended to provide a flavor of eventual
capabilities, and for this reason, the reader should not place undue emphasis on
the exact numbers themselves. But we do emphasize that real drilling vibrations
are complicated motions that encompass varied types of observations reported
by the myriad of authors cited - and that these observations can be simulated
rather directly. Modeled phenomena include bit bounce, stick-slip oscillations,
forward rate-of-penetration, torque reversal and drillstring precessional motions.
We will demonstrate this with two contrasting calculated examples.
Test A. Smooth drilling and making hole. In this first calculation, the
rock-bit interaction model (a.k.a., the bottomhole impedance boundary condition
or rock-bit model) u x + u t + u = 0 assumed in our model for axial vibrations
is executed with the numerical values shown in Figure 4.5.4, as extracted from
the Fortran listing of Figure 4.5.3 for coupled motions.
C DEFINE UNITS OF ROCK-BIT INTERACTION
RBALPH = -1.
RBBETA = +1.
C RBLAMB = +20 GIVES BIT BOUNCE,-20 GIVES FORWARD DRILL ONLY.
C RBLAMB = +20.
RBLAMB = -20.
Figure 4.5.4. Rock-bit impedance model, smooth drilling.
The RBALPH, RBBETA, and RBLAMB parameters shown in the Fortran
above represent , and respectively. Our computed results for the axial
displacement u (at i = 1, the bit position) demonstrate that the drillstring is
“making hole” in a well-behaved manner: u descends vertically, with the descent
rate or rate-of-penetration du/dt always negative. The torque-at-bit ,
interestingly, reverses sign at regular intervals, and is therefore indicative of
stick-slip and torsional unwinding; in Figure 4.5.5, we separately identify four
blocks of computed time data, taken whenever torque reverses in sign. The
strain u/ x at the bit, for the parameters used in u x + u t + u = 0, is
compressive, but sometimes tensile, depending upon the formation interaction
(recall that u x < 0 for compression and u x > 0 for tension). For our fictitious
rock, both push and pull exist, and descent is due to a good extent to gravity.
Bending displacements v and w at the bit appear to be stable computationally
over numerous time steps despite rapid changes in axial and torsional solutions.