Kinematic Wave Theory - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 51

Numerous examples show that well-characterized waves can arise from
complicated localized phenomena. Consider the irregular motions due to a
pebble tossed into a lake; at some distance from impact, an organized circular
wave pattern that propagates radially outward always emerges. Or the myriad of
local flow uncertainties about an exploding firecracker - faraway, what is
observed is a clean acoustic field defined by a simple spherical source. The
three-dimensional details related to cutting, shearing and rocking motions at a
rock-drillbit interface, for example, always vanish at some distance from the
tricone or PDC bit; at the surface, axial and torsional disturbances having
distinct wave-like character are always observed. The point explosions set off
by geophysical sources enter the earth as strong, highly three-dimensional
disturbances, but they ultimately undergo reflections and transmissions as well-
defined plane waves. These observations suggest that the end result of transient
localized disturbances are waves that vary slowly in space. Is it necessary to
model engineering problems using complicated and labor-intensive finite
element analysis? If some situations, yes. However, these may not provide the
insight needed to develop engineering insight and physical understanding.
2.1 Whitham's Theory in Nondissipative Media *
Early KWT methods applied to nondissipative waves only, although the
formalism did include uniform and nonuniform media, as well as linear, weakly
nonlinear and fully nonlinear motions. The general KWT equations were low-
order in that high-order spatial derivative terms did not appear. It is known that
singularities (infinite amplitudes) obtained from low-order theory may not exist
in reality, since neglected high-order terms, which become important as
amplitudes increase, may alter the instability structure (e.g., see Chin (1993)).
For instance, in a sonic boom, pressure never jumps from one level to another: it
is always smeared by locally important diffusion. Low and high-order effects of
dissipation can introduce changes not anticipated from the simpler model.
Chin (1976, 1980) extended KWT to include high-order wave dispersion
and dissipation, showing how several contrasting models really appeared as
special limits of a broader unifying theory. That work covered fully nonlinear
waves, however, the complete subject falls beyond the scope of this topic. In
this section, we discuss KWT for linear, nondissipative systems only, first for
uniform, and then for nonuniform media, using Whitham' s original
nomenclature. In the next section, more complete equations are given; there, we
develop new mathematical techniques that generalize early results.
* In introducing KWT, an intuitive style conveying mathematical principles is used. The physical
consequences of several theorems, initially stated without proof, are developed first, to set the stage
for broader discussion. Their inadequacies are explored, and only when the need for improved
models is demonstrated, are more formal derivations given that rigorously extend the initial claims.