Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 234

4.5.4.5 Numerical considerations.
Our numerical model was implemented using time-dependent, implicit,
finite difference schemes, initialized by static start-up conditions. Transient
terms were approximated using three-point time difference formulas to
minimize storage requirements; thus, only three time levels of axial, torsional
and lateral wave data are required in computer memory at any given time. Bit
displacement, stress and velocity histories are written to output graphics files
and tables immediately upon solution. All spatial derivatives were modeled
using central difference approximations. These lead to computationally efficient
tridiagonal matrixes in the case of axial and torsional vibrations, and only
slightly more complicated pentadiagonal banded matrixes for both coupled
lateral bending modes. All of the component modules, that is, axial, torsional
and bending, are believed to be stable. Thus, practical PC-based simulations can
be performed in minutes for typical bottomhole assemblies without difficulty.
Again, as explained in Chapter 1, numerical models are often the final
resort in modeling reality. Morse and Feshbach (1953) and Graff (1975) clearly
state that classical normal mode analyses are often inappropriate, when certain
complicated boundary conditions are used, because completeness requirements
are not fulfilled by the corresponding mathematical eigenvalue problem.
However, solution can be obtained by solving (numerical) transient
formulations; these also yield dynamically steady solutions, if they exist.
Numerical approaches offer other advantages. For example, the problem
of borehole wall contact, in the static limit, is ripe with computational
complexity because static indeterminacies arise at contact points that are
unknown a priori. One particularly simple device that avoids these conceptual
difficulties requires us to imagine that a continuous transverse spring loading
exists, with a spring “constant” k acting on the beam. In the Fortran logic, this
spring constant may be assumed to be vanishingly weak. Only if the net local
transverse displacement {v 2 (x,t) + w 2 (x,t)} equals or exceeds the wellbore
radius, at any instant during the course of the computations, at the required
angle, is k(x) set to an arbitrarily large or possibly finite number.
Computed results are only as accurate as the assumed boundary conditions,
and these, we emphasize, are continually under development. Cook, Nicholson,
Sheppard, and Westlake (1989), following exhaustive study of data obtained in
the Gulf of Mexico, conclude that considerable uncertainty in actual downhole
boundary conditions make accurate BHA simulation difficult, indicating the
value of real-time downhole measurements. This, of course, reinforces the role
of real-time MWD measurements in refining our understanding of downhole
vibrations. The integrated model proposed in this chapter, therefore, should be
viewed as a vehicle by which hypotheses can be tested, validated, or refuted. It
is also clear that boundary condition types must change as the drillbit enters