Geology Reference
In-Depth Information
4.5.4.3 Computational recipe.
As in our earlier discussion for simple lateral vibrations, the required finite
difference equations are obtained by writing Equations 4.5.7 and 4.5.8 for the
internal nodes i = 3, 4, ... , imax-3, imax-2. This process yields, for each of v
and w,
imax-4
equations in
imax
unknowns at the indexes i = 1 to imax. For
each dependent variable, the additional four equations are obtained from
boundary conditions. The two corresponding to the indexes i = 1 and 2
represent downhole bit conditions, whereas those at i = imax-1 and imax are
surface constraints.
The right sides of Equations 4.5.7 and 4.5.8 represent known functions
from previous time steps that have been stored in memory. The variable
coefficients related to N(x,t) and T(x,t) on the left sides of these equations are
computed at the same time step using the algorithms given in for uncoupled
axial and torsional vibrations. Once the coefficients of pentadiagonal matrixes
have been defined, the two outermost diagonals in the five-banded matrix are
reduced (using standard operations from linear algebra) to tridiagonal form.
Then, the tridiagonal matrix solver given earlier can be used to solve for v
i
and
w
i
for i = 1 to imax.
4.5.4.4 Modes of coupling.
Dynamic coupling appears at several distinct points in the analysis. We
have indicated earlier that axial and torsional coupling can occur through
boundary conditions at the bit. The foregoing equations emphasize that v(x,t)
and w(x,t) coupling also occurs through the explicit presence of T(x,t) and
N(x,t) in both bending partial differential equations; these two variable fields
represent known quantities. They are first determined by solving
2
u/ t
2
+
u/ t - E
2
u/ x
2
= 0
(4.5.9)
for axial displacement vibrations, and
2
/ t
2
-
/ t - c
s
2
2
/ x
2
=
e
(4.5.10)
for torsional vibrations, using numerical methods already described. As
discussed earlier, the complete boundary value problems associated with
Equations 4.5.9 and 4.5.10 provide the capability of modeling bit-bounce, rate-
of-penetration, and rock-bit interaction in the case of axial vibrations, and stick-
slip oscillations and torque reversals for torsional vibrations. Should lateral
mode back-interaction on the axial strain field become important, an equation of
the form A
2
u/ t
2
- EA
2
u/ x
2
= {EI (
3
v/ x
3
) ( v/ x)}/ x derived
previously, extended to handle dual bending modes with torsion, would replace
Equation 4.5.9 above.
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