Overview and Fundamental Ideas - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 12

This is important because many variables in petroleum applications contain
embedded discontinuities, for which spatial derivatives do not exist - for
instance, certain MWD signals are associated with a “delta p” pressure
discontinuity at the pulser. The relaxed piecewise continuity requirement allows
us to obtain analytical solutions for practical problems. Series containing
discontinuities cannot be differentiated, of course, but they can be integrated.
Numerous wave applications are associated with discontinuities. Elaborating
further, pressure discontinuities are associated with “positive pressure” pulsers
and “mud sirens,” while velocity discontinuities model “negative pressure”
pulsers. Discontinuities in axial displacement, i.e., “displacement sources,”
model the kinematics of rotating drillbits in drillstring vibrations.
In order for series representations to be possible, the eigenfunction
sequence must “complete.” This requirement, often construed by engineers and
even mathematicians as mere formality, is actually quite important to problems
such as those cited, where internal discontinuities originate at the source point.
It also turns out that functions with a finite number of discontinuities (e.g.,
patented multiple MWD “sirens-in-series” systems) can be represented by
eigenfunction series in a least-squares sense. But in order that this
representation be easy to accomplish, the sequence must be mutually orthogonal.
For our purposes, it suffices to say that orthogonality is defined by special
integral requirements that must be satisfied by eigenfunction pairs.
It is necessary to introduce these abstract ideas because not all classes of
boundary conditions yield to simple analyses such as those presented in the
earlier examples. For example, in axial drillstring vibrations, both surface and
downhole boundary conditions may take the mixed form u x + u t + u = 0
(similar considerations arise in desurger interactions with MWD signals). For
such problems, the boundary conditions turn out to depend on the eigenvalue
itself and the eigenfunctions will not be mutually orthogonal. Although for the
boundary condition just given, the eigenfunctions are complete, it is still
difficult to compute the coefficients of the eigenfunction series.
Thus, obtaining a solution to any initial value problem for such boundary
conditions using series can prove to be elusive. Usually, the solution is obtained
circuitously, e.g., using Laplace transforms, or numerical simulations, as we will
when we model rate-of-penetration in drillstring vibrations. For a
comprehensive discussion, refer to Morse and Feshbach (1953); there,
alternative but complicated solution methods are given, which apply when
orthogonality breaks down. Leighton (1967) also gives mathematical
discussions on orthogonality and completeness.