Geology Reference
In-Depth Information
equation is well known, governing fluctuations in quantities such as mass
density, pressure, temperature and entropy; not only does the equation apply to
the quantities cited, but it applies to more abstract (but often used) mathematical
entities such as the “displacement potential” and the “velocity potential.” The
particular choice of dependent variable to be used is crucial to obtaining useful
solutions. Elementary textbooks emphasize, for example, that the selected
variable must accommodate the type of boundary conditions used; cases in point
are the reflection conditions corresponding to the open or closed ends of an
organ pipe. In MWD telemetry, the modeling problem is more complicated: the
dependent variable used in our case must also accommodate the “dipole nature”
of the source, that is, it must describe in a natural manner the antisymmetric
disturbance pressures about the acoustic source generating the downhole signal.
The simplest example of an MWD dipole source is created by placing a
low-frequency woofer in a pipe filled with stagnant fluid (e.g., air), with the face
of the speaker lying in the cross-sectional plane of the pipe. When the speaker is
excited by electric current, its cone oscillates, thereby creating sound waves that
propagate in both directions. The nearfield mechanics of the speaker are
interesting. When the created pressure on one side is high relative to ambient
levels, the pressure on the opposite side is low by the same amount, and vice-
versa. Several diameters away from the speaker, the three-dimensional details
associated with cone geometry vanish - the resulting so-called “plane wave”
does not vary with cross-section.
In other words, the created pressure (or, disturbance pressure relative to
hydrostatic levels) is antisymmetric with respect to the source point. Because
this pressure field is always antisymmetric, the difference in pressure levels
across the source point, at any instant in time, will typically be nonzero. Hence,
we say that a
jump
or
discontinuity in pressure
(that is, “delta-p”) in pressure
exists. We emphasize that this is not to be confused with a statically unchanging
pressure differential, e.g., the viscous wake behind a bluff body, which does not
propagate as sound; of course, since such a static drop is associated with a
change in surface hydrostatic pressure, it can be used to encode information,
although at extremely low data rates.
The strength of this jump is determined by events that fall outside the
mechanics of acoustic wave propagation. In the foregoing example, the strength
of the speaker signal is fixed by the amplitude of the electrical signal fed to the
wave generator and the contours of the cone. In MWD, the strength of the
created dipole signal will depend the geometry of the poppet valve or mud siren,
the frequency of oscillation, the hydraulics of the problem by way of the flow
rate and density of the drilling fluid, and so on. The (long) wave, upon
reflection and re-reflection from boundaries, will pass freely through the source
because rotor-stator gaps are never fully closed; only the jump in pressure can
be specified with certainty in any formulation, since exact pressure levels at the
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