Transient Constant Area Surface and Downhole Wave Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 132

The black curve is the pressure trace created ahead of the valve. At the
same time, a negative trace at the valve propagates down toward the drillbit and
reflects upward without a change in sign. This is shown flipped over in the red
curve so that it can be compared with the black more easily to show the time
delay. Green is the actual pressure ahead of the pulser that is transmitted to the
surface and shows the superposition of the two. It is this green pressure trace
that is detected uphole. At 12 Hz, the green curve looks somewhat like the
black curve, i.e., they both have two large positive crests, except the green curve
is stretched out somewhat. A robust receiver algorithm might be able to read this
and interpret it correctly. If the green data is available, the blue curve shows
how the shorter black curve is recovered successfully.

4.5.8 Run 4. 24 Hz, Coarse sampling time.

Here we increase our carrier frequency to 24 Hz and continue with phase-
shift-keying. The time delay between pulser and bit is 0.01 sec, same as the
sampling time.

C CASE 3. PHASE-SHIFTING (F = FREQUENCY IN HERTZ)
PI = 3.14159
A = 0.25
F = 24.
C G = A*SIN(2.*PI*F*T)
C Means 2*PI*F cycles in 2*PI secs, or F cycles per sec, F is Hz.
C One cycle requires time PERIOD = 1./F
PERIOD = 1./F
IF(T.GE.0.0. AND.T.LE. PERIOD) SIGNAL = +A*SIN(2.*PI*F*T)
IF(T.GE.PERIOD.AND.T.LE.2.*PERIOD) SIGNAL = -A*SIN(2.*PI*F*T)
IF(T. GE.2.*PERIOD) SIGNAL = 0.

Figure 4.5e . 24 Hz, coarse sampling.
Here, the phase shift is clearly seen in the black signal created ahead of the
valve and which propagates uphole. The reversed-sign red signal travels
downhole (shown as a positive for easy comparison with the black trace) and
reflects at the bit with sign unchanged and travels uphole, appending itself to the
wave created ahead of the valve, to produce the green signal. The green signal