Transient Constant Area Surface and Downhole Wave Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 109

4.2.2 Software note.

In Method 4-1, we did not allow pump noise; the purpose was to evaluate
the echo cancellation or deconvolution method under ideal circumstances.
Method 1 utilizes the XDUCER*.FOR software series. Method 4-2 here with
mudpump noise utilizes our HYBRID*.FOR software series, a modification of
the single transducer XDUCER*.FOR model to include downward traveling
pump noise. By pump noise, we mean traveling pressure waves excited by the
pump pistons and not turbulence or other noise associated with surface facilities.

Schematic

Upgoing
wave

Pump
noise

Solid
reflector
x = L

x

Transducer
x = 0

Figure 4.2a. Wave reflection at solid reflector, with pump noise.
4.2.3 Theory.
In this section, consider the reflection of an upgoing wave at a solid pump
piston, to include the effect of pump noise propagated in the downgoing
direction from the piston location. Again a single transducer method is
assumed. Let p measured (t) now denote all pressures recorded by the standpipe
transducer, upgoing and downgoing, which would include mud pump noise. We
also assume that another pressure transducer independently records the mud
pump noise function N(t) at the pump , where “t” is the same time instant at
which the transducer records its signals. This is, in a sense, a two transducer
method. However, from an operational viewpoint, it only requires a single
transducer located on the standpipe, which is more convenient (and less
dangerous) for rigsite operations. The second transducer can be placed near the
mudpump and will not inconvenience any drillers on the rig floor. Then, if p(t)
is the upgoing signal, and p(t-h) is the downgoing pressure reflected at the solid
piston, where “h” is the roundtrip delay time, it follows that

p(t) + p(t-h) = p measured (t) - N(t - h/2) (4.2)