Kinematic Wave Theory - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 50

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Kinematic Wave Theory
The most important innovation in wave description in recent times is
undoubtedly the kinematic wave theory (or simply, “KWT”) formalism proposed
by Gerald B. Whitham at the California Institute of Technology in the 1970s and
described in his classic topic Linear and Nonlinear Waves (Whitham, 1974). In
the decades since, KWT has been applied to water waves, optics, acoustics,
traffic flow and many areas of engineering with success (incidentally, Whitham
also created industry-standard nonlinear aerodynamic models used to study
sonic booms at the earth' s surface). The present author, a student of Whitham' s
at Caltech, subsequently extended KWT to high-order at the Massachusetts
Institute of Technology in his doctoral work under aerospace pioneer Marten T.
Landahl. In this topic, three important applications are given, namely, highly
damaging lateral drillstring vibrations that occur at the neutral point; quantitative
characterization of the Stoneley waves used in acoustic logging for permeability
prediction; and ocean waves on variable currents for offshore load analysis.
Early sonic boom theories recognized that small local changes can lead to
large cumulative farfield alterations following long ray paths: characteristics that
are parallel on a linear basis converged in the farfield. Since properties remain
constant along characteristics, intersections would imply multi-valued, shock-
like behavior - in the acoustic fields beneath airplanes flying near the speed of
sound, they did, resulting in “sonic booms” that damaged surface structures and
windows - and the eventual ban on near-transonic speeds over inhabited
stretches of land. This early sonic boom work would motivate rapid
developments in perturbation methods to appear in the 1970s at Caltech, M.I.T.
and Stanford, in particular, multiple scaling methods, singular perturbation
analysis, advanced WKB schemes, and other asymptotic techniques (e.g., see
van Dyke, 1964; Cole, 1968; Nayfeh, 1973). Importantly, while the new KWT
formalism provided powerful modeling tools, detailed recourse to differential
equations was unnecessary. This requires a different outlook on mathematical
physics, and here, we develop the ideas for used later in our petroleum
applications.
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