Geology Reference
In-Depth Information
Modeling steady state oscillations.
Dynamically steady oscillations
were discussed for axial vibrations and the same separation of variables
techniques apply to Equation 4.3.42. Naturally, because the beam equation is
significantly more complicated than the classical wave equation, the resulting
complex variables based results will be more unwieldy. The early paper of
Huang and Dareing (1968) considers critical buckling loads and natural
frequencies of lateral vibration modes for long vertical pipes, suspended in fluid,
simply supported at the top and vertically guided at the bottom. As discussed,
wall contacts cannot be modeled, thus limiting the usefulness of the results. The
authors also restricted the drillstring analysis to uniform cross-sectional areas.
Simulating area changes.
In our discussion thus far, we have ignored the
fact that real drillstrings contain changes in cross-sectional properties,
particularly at the interface separating the drillpipe and the drill collar. At these
discontinuities, the differential equation breaks down because not all derivatives
exist, and recourse to separate matching conditions is necessary. For transverse
vibrations, these include continuity of displacement, moment and shear; slope,
of course, may or may not be conserved through the transition. These matching
conditions can be developed into a diagonally dominant finite difference model,
as in our treatment of axial disturbances, to ensure stable time calculations.
4.3.8 Example Fortran implementation.
We list the Fortran needed to describe transverse vibrations, but limit
ourselves to uniform drillstrings for simplicity. Therefore, we will concentrate
on the finite difference representation in Equation 4.3.64, and describe its matrix
implementation as defined by the pentadiagonal system in Equation 4.3.65.
Again, we will refer to statements in the general coupled program for axial,
torsional and (two) lateral vibrations developed much later in this chapter. Since
portions of this general code exist in the sections on axial, lateral and torsional
modes, only those excerpts directly applicable to the subject of this section will
be presented. The relevant Fortran statements are listed immediately below.
Figure 4.3.3 deserves elaboration. If we compare Equations 4.3.64 and 4.3.65,
the matrix coefficients
D
,
A
,
B
,
C
,
E
and
W
(for the right-hand-side of Equation
4.3.65) are easily identified; these appear in the “VBEND” logic shown in the
600 “do-loop.” Also shown in this loop are right-side effects that deal with
coupled torque and W-mode bending that the reader should ignore for the
purposes of this section. Once Loop 600 is executed, matrix coefficients related
to i indexes 1, 2, imax-1, and imax must be defined, subject to the boundary
conditions assumed at the bit and the surface. This process is similar to that
discussed in detail for axial vibrations. The implementation described here
appears to be unconditionally stable on a von Neumann basis, and transient
calculations at this writing have consistently remained stable. Again, examples
for smooth and rough drilling will be given towards the end of this chapter.
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