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Test B. Rough drilling with bit bounce. In our second calculation, the
rock-bit interaction model
u x + u t + u = 0 takes on the values shown in
Figure 4.5.6, that is,
C DEFINE UNITS OF ROCK-BIT INTERACTION
RBALPH = -1.
RBBETA = +1.
C RBLAMB = +20 GIVES BIT BOUNCE, -20 GIVES FORWARD DRILL ONLY
RBLAMB = +20.
C RBLAMB = -20.
Figure 4.5.6. Rock-bit impedance model, rough drilling.
The sign change in the term introduces some computationally interesting
results, namely, the existence of bit bounce and rough drilling. From the
following results, we find torque reversal as before, plus well-behaved bending
displacements that confirm numerical stability. However, we find that the bit
speed u/ t may also reverse sign, demonstrating that bit-bounce is possible for
the u x + u t + u = 0 assumed. When this is so, the Fortran logic enforces zero
bit strain, and continues to the next time integration; gravity ultimately pulls the
drillstring down, at which point the rock-bit interaction model again applies.
The computational results in Figure 4.5.7 are again conveniently broken out
accordingly at significant transition points in the physics. These are displayed
graphically in Figure 4.5.8, reproduced from the author' s original publication
“Fatal Tubular Bending Motions Difficult to Detect Uphole” in Offshore
Magazine , April 1988. The lateral vibrations in “Figures 6 and 7” are coupled
through transient drillstring torques that are in turn excited by axial vibrations
with “rate of penetration” and “bit bounce” shown in “Figures 4 and 5.”
Model limitations and extensions. There are various areas for
improvement and research, insofar as the modeling presented thus far is
concerned. For one, significant research remains in properly defining the
parameters in u x + u t + u = 0 to ensure that they are realistic and simulate
actual bit behavior. The empirical data cited in our axial vibrations discussion
for single tooth impacts represent a first step in this direction. As coded, our
coupled “one way” model (at any given time step) computes axial solutions first,
torsional ones next, followed by those in bending. This “cascading effect” does
not allow “nonlinear back-interaction” of lateral bending vibrations on the axial
and subsequent torsional solutions. The fix needed is easily (but not yet)
implemented: simply replace our axial wave equation with the corrected one
containing nonlinear right-side bending terms, as given earlier (e.g., for a single
bending mode, A 2 u/ t 2 - EA 2 u/ x 2 = EI {( 3 v(x,t)/ x 3 ) ( v/ x)}/ x).
This approach to nonlinear back-interaction was first proposed by Chin and
Rizzetta (1979) and Chin (1981) for unsteady transonic flows in high speed
transonic wing aerodynamics, where significant interactions between mean and
harmonic flows exist.
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