Transient Constant Area Surface and Downhole Wave Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 101

P upgoing (t). This simple “delay equation” can be solved in closed analytical form
and its solution is easily implemented digitally. Before proceeding with
example solutions, we emphasize the physical assumptions.
Equation 4.1d applies to solid reflectors only and assumes that the
functional form and amplitude of the reflection is unchanged. This type of
reflection does not necessarily apply to desurgers, for which there may be shape
distortion and damping, particularly at low frequencies and high amplitudes
(detailed discussions are offered in Chapter 6). Thus, it is only applicable to
positive displacement mudpumps with piston reflectors, and only when the
roundtrip signal attenuation between the transducer and the pump pistons is
negligible, a condition satisfied in practice. It is a trivial matter to handle
attenuation by inclusion of a fractional scale factor in Equation 4.1d if required,
for instance, if significant attenuation due to the rotary hose is present.
In summary, if the pump is a centrifugal pump and attenuation is
negligible, the above equation is replaced by
P upgoing (t) - P upgoing (t-h) = P measured (t)

(4.1e)

because an acoustically-opened (zero pressure, “wu/wx = 0” ) boundary condition
applies instead of the solid reflector “u = 0.” If attenuation is not negligible, the
above equations are replaced by P upgoing (t) + GP upgoing (t-h) = P measured (t) and
P upgoing (t) - GP upgoing (t-h) = P measured (t), respectively, where 0 < G < 1 is a positive
loss factor, with GP upgoing (t-h) describing the reduced pressure at the transducer
after the roundtrip travel time t = h (the constant G is measured separately).
Note that the distance L from the standpipe to the mudpump is typically 50
ft or less, since space on offshore rigs is limited (the total length of the standpipe
is about 30 ft). If water or brine is used as the drilling fluid, then approximately,
c = 5,000 ft/sec and h = 2L/c = 2(50)/5,000 ft or 0.02 sec. For typical drilling
muds, c = 3,000 ft/sec and the delay time h may increase to 0.03 or 0.04 sec.
Results for typical field numbers are presented next. Again, no other noise
excepting reflection at the solid piston is assumed in these examples.

4.1.3 Run 1. Wide signal - low data rate.
The upgoing signal assumed is a rectangular pulse with rounded edges,
with higher R values increasing the sharpness of the corners. This is
constructed as shown below using two hyperbolic tangent functions. In this
topic, the Fortran code used is shown in Courier font, as seen below.

C CASE 1. BROAD PULSE WIDTH
C Clearly see upgoing and reflected pulses enhance the signal
A = 1.0
R = 20.0
G = A*(TANH(R*(T-0.100))-TANH(R*(T-0.600)))/2.