Examples from Classical Mechanics - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 94

2 / t 2 - c 2 2 / x 2 = 0 (3.59)

where the constant speed of sound c satisfies c 2 = 1 / . Thus, the velocity

m

potential also satisfies the wave equation.

3.2.11 Modeling MWD sources.
Wh y should we deal with Equation 3.59 for (x,t), when the one-
dimensional form of Equation 3.27 for p(x,t) is already available? It turns out
that pressure, useful in conventional engineering analyses, has limited value in
modeling positive or negative MWD pulsers ! This observation motivated our
detailed discussion of wave equation sources in Chapter 1. In order to model
pulser sources correctly, describing their velocity and pressure symmetries or
anti-symmetries accurately, different classes of dependent variables are needed.
Many topics discuss acoustic pressure only, which satisfy the wave equation in
Equation 3.27. But other physical properties, e.g., the velocities u, v and w, the
disturbance density ', the fluctuating temperature, and the velocity potential,
are also wave-like. The dependent variable appropriate to a particular problem
depends on specific boundary condition details. For the oscillating end piston
popular in simple examples, the axial velocity may be natural; or, the
Lagrangian fluid displacement itself may be used with its time derivative
prescribed at the end.
In many petroleum engineering applications, especially those in
Measurement-While-Drilling, the excitation source actually resides internally
within the waveguide and not at the very end. Mathematical modeling is more
challenging in several respects. Acoustic energy is created within the medium,
and propagated in both directions; then, all subsequent reflections must be
allowed to pass transparently through the source point. But not all sources
function similarly, even on a qualitative level. For example, as we will see later,
the created pressure on either side of a positive pressure (or siren) pulser is
antisymmetric: when one side is over-pressured relative to ambient conditions,
the other is under-pressured. For negative pressure pulsers, both are either over-
pressured or under-pressured, and the created pressure field is symmetric.
Symmetries and antisymmetries must be correctly accounted for since their
associated signs become very important after reflections: they affect
constructive and destructive wave interferences.
We explore these subtleties by considering specific models. In Chapter 1,
we indicated that sources can be introduced into wave equations by appending
delta functions . Therefore, let us naively add a delta function to the right side of
Equation of 3.27 for pressure,

2 p/ t 2 - c 2 2 p/ x 2 = A (x-x * ) (3.60)

where A is a normalizing constant and 0 < x, x * < L.