Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 211

simple tridiagonal form. By removing the two outermost left and right bands,
the resulting matrix can be inverted using the tridiagonal Fortran algorithm
given for axial vibrations, to yield solutions for the lateral displacements v i .
Although the computational procedure is similar to that axial disturbances, the
physics requires a somewhat different perspective. For axial vibrations,
disturbances propagate at a constant speed “c,” theoretically anyway, and time
step sizes are chosen consistently with the Courant-Friedrichs-Lewy (or, CFL)
condition t < ( x)/c. In the case of lateral vibrations, we have shown that
propagation speeds depend upon the wavenumber, the disturbance waves being
dispersive; thus, the choice of t will automatically eliminate a range of lateral

waves from consideration.

4.3.7.3 Additional modeling considerations.
We will briefly address several subjects that are important in modeling
simple bending vibrations, which apply also to the dual torque-coupled lateral
modes considered later.
Borehole wall contacts. Closed form transverse solutions for static
drillstrings in boreholes are difficult to obtain because wall constraints due to
finite hole radii lead, possibly, to multiple contact points, or none at all. This is
compounded by “static indeterminacy,” thus forcing the use of more
complicated solution methods. Nor are static computational approaches any less
troublesome. At contact points, if any, our difference model does not apply
directly, and modifications to include local shear force effects (at positions
unknown a priori ) in the overall force balance are needed. But treating static
problems as large-time limits of transient processes completely circumvents the
problem. Finite hole radii with formation contact is simply modeled with
Fortran if-then logic that invokes a spring-like restoring force if the borehole
radius is exceeded at any time step (the distributed force model can be used with
elastic, inelastic or plastic spring responses to model different formations).
This transient approach is useful in three dimensions, when torque couples
both lateral bending modes, and spiral etch patterns have been computed.
Moreover, this device is well grounded physically, because it is exactly the same
process or path that nature takes. Suitable initial conditions are needed, e.g., a
starting deflected or undeflected state, with or without initial velocity. The end
displacement pattern, and number and position of contact points, incidentally,
may vary with initial condition. Aldred and Sheppard (1992), interestingly,
propose a mechanism by which transverse drillstring motions couple to rotary
ones by way of borehole wall contact. They suggest that harmonic resonant
behavior is not primarily responsible for observed high shock levels, and that
these shock levels can lead to an increase in surface torque and a decrease in the
rate-of-penetration. Their ideas can be tested using the algorithm given here.