Transient Constant Area Surface and Downhole Wave Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 99

4.1 Method 4-1. Upgoing wave reflection at solid boundary,
single transducer deconvolution using delay equation, no
mud pump noise (software reference, XDUCER*.FOR).

4.1.1 Physical problem.
Consider an upgoing pressure wave originating from downhole, containing
encoded mud pulse information. It travels up the standpipe and is assumed to
reflect at a solid reflector, the mudpump piston (this model does not apply to
centrifugal pumps), and then propagates downward. Both incident and reflected
waves are found in the surface standpipe where pressure transducers are
installed. We wish to extract the upgoing wave from the total signal. We will
not consider mudpump noise, random noise, or other noise sources here, as these
subjects are deferred to Chapter 6. Multiple transducer methods are available to
remove downward reflection and other down-going signals as given in Methods
4-3 and 4-4. Here, we describe a single transducer method that assumes a solid
reflector - convenient for standpipe use when two transducers cannot be
installed. The solid reflector in Figure 4.1a represents the pump piston.

Schematic

Upgoing
wave

Solid
reflector
x = L

x

Transducer
x = 0

Figure 4.1a. Wave reflection at solid reflector.

The surface reflector is assumed to be solid for the single transducer
approaches of Methods 4-1 and 4-2. This applies to positive displacement
pumps only - an excellent assumption for almost all mud pumps that is
validated by numerous experiments. For centrifugal pumps, an open end, zero
acoustic pressure boundary condition would apply; centrifugal pumps and free-
end conditions are not treated in this topic because they are not very common,
although an exact analysis would only require minor theory and software
changes. However, our multiple-transducer deconvolution models in Methods
4-3 and 4-4 apply to both solid (u = 0) and open-end (u x = 0) reflectors.