Geology Reference
In-Depth Information
fundamental time difference delay equation that is customized in different ways
depending on the application. These applications are discussed next.
If reflection signal processing is used at the surface standpipe to remove
reflections at the desurger, the rotary hose and the pump, and also to remove
pump noise, what remains is a signal proportional to p 3 (t), which again, contains
the MWD signal and all the downhole reflections at the drillbit and collar-pipe
junction. (Note that we have not addressed thermodynamic attenuation along
the drillpipe - it is not significant computationally since it is assumed to affect
all parts of the signal uniformly, e.g., any electronic gain may be used but it is
not numerically important.) From this p 3 (t), we wish to extract a fully transient
'p(t) which contains the true downhole well logging information. This is our
first application. To do this, the equation referred to in the above paragraph is
written as follows -
'p(t) + 'p(t - 2L m /c) =
= (A p /A c +1) p 3 (t - L m /c) - (A p /A c - 1) p 3 (t - L m /c - 2L c /c)
(5.1.1)
Equation 5.1.1 is solved using the exact solution and algorithm for difference
delay equations discussed in Chapter 4. Once 'p(t) is obtained, we have the “0”
and “1” information formed by the position-encoding of the pulser.
In the second application, we assume that 'p(t) across the pulser is given,
and that the upgoing p 3 (t) signal is required, say, to estimate the form of the
signal that might be obtained at the surface (a model for attenuation must be
used to account for non-Newtonian losses along the drillpipe, which may extend
miles in directional drilling applications, and this is developed in Chapter 6).
Then, Equation 5.1.1 can be formally written in the form
p 3 (t- L m /c) - {(A p -A c )/(A p +A c )} p 3 (t - L m /c - 2L c /c) =
= {'p(t - 2L m /c) + 'p(t)}/(A p /A c +1)
(5.1.2)
We emphasize that, in this form, Equation 5.1.2 is not completely
meaningful physically. To see why, we return to the mathematical formulation
and note that the pressure function p 3 (x,t) is defined for x t L c only. The
argument in the first term on the left side above refers to x = L m which is less
than L c . To render the equation useful, we need to change dummy variables and
shift all arguments to obtain
p 3 (t - L c /c) - {(A p -A c )/(A p +A c )} p 3 (t - 3L c /c) =
={'p(t - L m /c - L c /c) + 'p(t + L m /c - L c /c)}/(A p /A c +1) (5.1.3)
Now, we recognize that p 3 (t - L c /c) is the acoustic pressure at the very bottom of
the drillpipe (x = L c ) just above the MWD drill collar. It is this “P pipe ” at x = L c
that travels up the drillpipe to the surface. It is more instructive to introduce the
new symbol P pipe (t) = p 3 (t - L c /c) so that the above equation can be rewritten as
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