Geology Reference
In-Depth Information
solutions even on modern personal computers are not always possible, since
stack overflow is typically encountered in the execution of the required complex
transcendental arithmetic. A computer program was therefore written and
structured to accommodate memory limitations expected of desktop machines;
the program, written in standard Fortran for portability, delivers almost
instantaneous acoustic solutions on all desktop computers, and, as noted earlier,
can be used to optimize MWD source placement and telemetering frequency.
Obvious checks were used to validate the production code against programming
error. For example, in the uniform waveguide limit where all cross-sectional
areas are identical and materials do not change, calculated pressures
immediately to the left and right of the source point are found, as required, to be
equal and opposite; their difference, of course is found to be the inputted 'p.
Other checks included agreement with transmission and reflection coefficients
obtained for simple waveguide geometries for which simple analytical solutions
were available.
2.4.2.7 Final data analysis.
Once the numerical values for the elements of [
C
] are available from
matrix inversion, the primary quantities of physical interest can be obtained by
straightforward post-processing. The Lagrangian displacement and acoustic
pressure of fluid particles in the drillpipe are obtained by taking real parts as
indicated in Equations 2.17a - 2.17c,
u
p
(x,t) =
Re
U
p
(x) e
iZt
(2.17a)
p
p
(x,t) = - B
mud
Re
dU
p
(x)/dx e
iZt
(2.17b)
=
Re
{(iZB
mud
/c
mud
) C
1
exp iZ(t-x/c
mud
)} (2.17c)
noting that the argument “t - x/c
mud
” signifies an upward propagating wave. For
the borehole annulus, the corresponding “t + x/c
mud
” solutions traveling away
from the drillbit nozzles are
u
a1
(x,t) =
Re
U
a1
(x) e
iZt
(2.18a)
p
a1
(x,t) = - B
mud
Re
dU
a1
(x)/dx e
iZt
(2.18b)
=
Re
{(-iZB
mud
/c
mud
) C
10
exp iZ(t+x/c
mud
)} (2.18c)
While pressure levels themselves are important to signal-to-noise
determination in signal processing, as well as to surface transducer selection and
placement, very often some measure of signal generation efficiency is required,
for different reasons. For example, one might reasonably ask, “How optimal is
the created signal that is propagating up the drillpipe for a given 'p?” This
question is important in applying downhole constructive interference to
maximizing carrier wave strength, a technique that is all the more significant
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