Overview and Fundamental Ideas - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 2

This is accomplished by offering a synopsis of the conventional
mathematics taught in partial differential equations courses, covering standard
solution methods, plus their strengths and weaknesses. We introduce pertinent
analysis areas to give students resources to pursue additional study. We will be
correct but not rigorous in a formal sense. Our objective is not the teaching of
classical mathematics per se . Thorough understanding can only be achieved
through intensive diligent exercise. We only introduce these methods, but in a
manner that demonstrates their powerful capabilities or their severe limitations.
This objective allows us to communicate “the big picture,” so that the
reader can at once grasp an appreciation of what has been made possible by
generations of eminent scholars. But since it turns out that many of the wave
propagation issues encountered in modern petroleum engineering do not draw
directly upon these techniques, this overview suffices insofar as providing the
basic foundation needed for more specialized analysis. The physical problems
particular to our rapidly changing industry are reviewed for each of the titled
applications, and mathematical and numerical models are developed to handle
the specialized circumstances these specific applications demand.
By taking this approach, the requirements for new analysis methods and
the mathematical issues associated with their implementation appear naturally,
and without intimidating theorems, proofs and corollaries. Modern subjects such
as finite difference modeling, multiple-scaling, stationary phase, monopole and
dipole source properties, Fermat' s principle of least time, kinematic wave theory
and group velocity are introduced and developed naturally ... and in a simple and
intuitive way. This approach to introducing specialized subjects in the simplest
manner has posed the greatest challenge. But this philosophy reflects the trials
and tribulations of the author' s own learning process, quickly taking advantage
of the opportunities the Oil Patch offered after a false start in aerospace
engineering. For the reader willing to endure the formalities, the rewards are
enticing: innovative ways to study borehole electromagnetics, new approaches
to MWD telemetry, out-of-box ways to extract permeability from Stoneley wave
motions in borehole acoustics. Their extensions will become apparent. The
author believes that new lines of research will be defined that will lead to
improved efficiencies needed in modern exploration and production.
1.1 The Classical Wave Equation
1.1.1 Fundamental properties.
Many students learn wave propagation by way of the undamped classical
“wave equation” (e.g., see Hildebrand, 1948, or Tychonov and Samarski, 1964)
2 u/ t 2 - c 2 2 u/ x 2 = 0 (1.1)
for u(x,t), where x is the propagation direction and t denotes time. We may
think of Equation 1.1 as describing the transverse vibrations of a string.