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6.5 Applications to Negative Pulser Design in Fluid Flows
and to Elastic Wave Telemetry Analysis in Drillpipe Systems
We have studied MWD signal analysis and reflection cancellation
assuming dipole sources, that is, signal generators such as positive pulsers and
mud sirens which create disturbance pressures whose polarities are
antisymmetric with respect to source position, and additionally, assuming fluid
flow systems. We emphasize that the methods are equally applicable, with some
re-interpretation of variable names, to negative pulser design, and then, to both
fluid and elastic systems. These notions and applications are developed next.
The ideas behind “conjugate harmonic functions” are well documented in
the theory of elliptic partial differential equations, the best known and simplest
application being that for Laplace's equation. In short, whenever the equation
U
xx
+ U
yy
= 0 holds, there exists a complementary model V
xx
+ V
yy
= 0
describing a related physical problem; these are connected by the so-called
Cauchy-Riemann conditions U
x
= V
y
and U
y
= -V
x
. These relationships lie at
the heart of the theory of complex variables.
For example, the “velocity potential” describing ideal, inviscid, irrotational
flow past a two-dimensional airfoil satisfies Laplace's equation; the
complementary model for the “streamfunction” describes the streamline pattern
about the same airfoil. These ideas have been used by this author to study
steady-state pressure distributions, torque characteristics and erosion tendencies
associated with MWD mud sirens (Chin, 2004), while detailed applications to
Darcy flows in petroleum reservoirs are developed in Chin (1993, 2002). The
siren application is developed in detail in Chapter 7 of this topic.
The conjugate function approach, though, has never been applied to
hyperbolic equations. However, simple extensions for the classical wave
equation used in this topic to model the acoustic displacement function u(x,t)
lead to powerful practical implications. A rigorous derivation is easily given.
Recall that for long waves, u(x,t) satisfies w
2
u/wt
2
- c
2
w
2
u/wx
2
= 0, which can be
rewritten in the form w(wu/wt)/wt - c
2
w (wu/wx)/wx = 0 (again, x is the direction of
propagation, t is time and c is the speed of sound). We introduce, without loss
of generality, a function I(x,t) defined by wI/wx = wu/wt and wI/wt = c
2
wu/wx.
This merely restates the identity w
2
I/wtwx = w
2
I/wxwt. However, the definition
importantly implies that I(x,t) satisfies w
2
I/wt
2
- c
2
w
2
I/wx
2
= 0. In other words,
the function I likewise satisfies the wave equation. But what do mathematical
boundary value problems similar to those for u(x,t), for which we have already
developed numerical solvers, model in engineering practice?
Consider first the formulation w
2
u/wt
2
- c
2
w
2
u/wx
2
= 0, with the jump or
discontinuity [wu/wx] specified through the source position (that is, a “delta-p”
function of time is prescribed at the pulser) and then wu/wx = 0 at the drillbit (a
uniform pipe is assumed and outgoing wave conditions are taken at x = f). This
is the dipole formulation previously addressed, assuming an opened acoustic
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