Harmonic Analysis: Six-Segment Downhole Acoustic Waveguide - Measurement While Drilling Signal Analysis, Optimization and Design - page 64

2.4.1 Nomenclature.

p

... Internal cross-sectional area in drillpipe

c

... Internal cross-sectional area in MWD drill collar

... Internal cross-sectional area in mud motor (exclude metal rotor)

m

b

... Internal cross-sectional area in “bit passage”

a2

... Cross-sectional area, annulus surrounding drill collar

a1

... Cross-sectional area, annulus surrounding drillpipe

mud ...

Bulk modulus in mud

mm ...

Bulk modulus in mud motor (weighted average, rubber and mud)

c mud ... Speed of sound in mud

c mm ... Sound speed in mud motor (weighted average, rubber and mud)

f ... Frequency of dipole source (in Hertz)

p ... Acoustic pressure

'p ... Dipole signal strength (for positive pulser or siren, not explicitly

used, since it is “p/'p” that is actually considered here)

U mud ... Mass density of mud

U mm ... Mass density in mud motor (weighted average, rubber and mud)

t ... Time

u ... Lagrangian displacement of fluid element from equilibrium

Z ... Circular frequency of dipole source

x ... Axial coordinate, “x = 0” at mud-motor/drill-collar interface

x s ... Distance to MWD acoustic dipole source, located in drill collar

x c ... Drill collar length

x m ... Length of mud motor

x b ... Length of “bit passage” beneath mud motor

x a ... Length of annulus, from floor of hole to pipe-collar interface

The foregoing long-wave property is significant to MWD telemetry
modeling. Since the passage of a long acoustic wave down the drillpipe and
drill collar and then up the annulus can be regarded as the kinematic equivalent
of two consecutive 90 o bends separated by a short “drill bit passage,” we can
topologically “unwrap” the collar-bit-annulus telemetry channel to approximate
Figure 2.4a by the equivalent one-dimensional waveguide in Figure 2.5. In this
representation, the origin “x = 0” is taken as the interface between mud motor
and MWD drill collars; at the far left, a single left-going wave travels up the
annulus, while at the far right, a single right-traveling wave travels up the
drillpipe. The lengths and cross-sectional dimensions in Figure 2.5 do not
represent actual geometries, but are intended for display purposes only; they are
not drawn according to scale. The drillbit, mud motor and MWD collar lengths
are, respectively, x b , x m and x c .
High-data-rate mud pulse telemetry is governed by the equations of
classical acoustics, e.g., Morse and Ingard (1968). The role of the wave