Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 247

.
.
C V LATERAL MODE, FIRST BIT BOUNDARY CONDITION
B(1) = 1.
C(1) = 0.
E(1) = 0.
W(1) = PRERAD*SIN(WMGPRE*T)
.
.
C W LATERAL MODE, FIRST BIT BOUNDARY CONDITION
B(1) = 1.
C(1) = 0.
E(1) = 0.
W(1) = PRERAD*COS(WMGPRE*T)
.
.

Figure 4.5.9. Precession boundary condition logic at the bit.

4.5.4.11 Drillstring vibrations in horizontal wells.
Having discussed and motivated our model for axial, torsional and lateral
vibrations in detail, we assume at this point that the ideas rest comfortably with
the reader and that he is ready to continue the work outlined here. In this
concluding section, we suggest various modifications needed to extend the
algorithm to deviated and horizontal wells. Because we are now familiar with
the derivations and their limitations, the suggestions incorporated here are given
with a minimum of elaboration. The Fortran listing given earlier may serve as a
starting point for a more complete computer program for arbitrary drillstring
inclinations and areal discontinuities.
In highly deviated and horizontal wells, the effect of gravity must be
properly included. Thus, only the appropriate component of gravity along the
drillstring axis should be incorporated in the axial vibrations model. In the
vertical drillstring model considered here, gravity played no significant role in
lateral vibrations; however, in deviated drillstrings, gravity now acts as a
transverse load that must be accounted for in bending modeling. Deviated
drillstrings very often rest on the formation. In addition to gravity, a second
transverse loading is provided by formation support, which manifests itself
through variable elastic supports and damping factors. Borehole friction will
affect torsional oscillations, but only at the lower side of the hole; thus, the

e

excitation term in 2 / t 2 - / t - c s 2

2 / x 2 =

e will in general be a

function of , rendering the governing equation nonlinear.
Borehole wall contacts can complicated. Earlier, again for vertical holes,
we suggested that when computed lateral displacements are large, taking on
displacements with (v 2 +w 2 ) > R hole , Fortran if-then logic can be invoked to
apply a formation restoring force. Under some conditions, an additional logical
test based on the value of may be needed, to distinguish upper from lower
sides of the hole. When a drillstring lies on its side, it is possible that, in