Transient Variable Area Downhole Inverse Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 145

P pipe (t) - {(A p -A c )/(A p +A c )} P pipe (t - 2L c /c) =
= {'p(t - L m /c - L c /c) + 'p(t + L m /c - L c /c)}/(A p /A c +1) (5.1.4)

In the above form, the indexes on the right-side of the equation contain the
correct space-time dependencies, with the solution P pipe (t) applying to x = L c .
The resulting P pipe , of course, will decrease in amplitude as it travels to the
surface, on account of attenuation (at the surface, the above P pipe signal is
reduced by the attenuation factor e -Dx where D depends on sound speed,
kinematic viscosity, drillpipe radius and frequency - more on this in Chapter 6).
The amount of attenuation depends on frequency, and fluid rheology, density
and kinematic viscosity.
Finally, for our third application, note that we can shift arguments in
Equation 5.1.2 by a different amount to obtain, instead of Equation 5.1.3, the
following
p 3 (t - L/c) - {(A p -A c )/(A p +A c )} p 3 (t - L/c - 2L c /c) =
= {'p(t - L m /c - L/c) + 'p(t + L m /c - L/c)}/(A p /A c +1) (5.1.5)

where L is a length such that L > L c (L is the distance from the origin x = 0).
The term p 3 (t- L/c) is the drillpipe pressure at any x = L location uphole, e.g.,
very far away at the surface standpipe or any other intermediate position along
the drillpipe (this only accounts for the wave originating from downhole and not
any downgoing reflections). If we denote p 3 (t - L/c) = p surface (t), then Equation
5.1.5 becomes
p surface (t) - {(A p -A c )/(A p +A c )} p surface (t - 2L c /c) =
= {'p(t - L m /c - L/c) + 'p(t + L m /c - L/c)}/(A p /A c +1) (5.1.6)
Attenuation effects along x >> 0, that is, the long path in Section “3,” are
easily modeled. We had assumed a pure upgoing wave u 3 (x,t) = q(t - x/c)
without attenuation only for the purposes of studying local wave interactions in
the relatively short bottomhole assembly. Once the signal leaves the drill collar
and enters the drillpipe, it is subject to a decay factor e - Dx where D > 0 depends
on frequency, fluid kinematic viscosity and pipe radius. Over large distances,
pressure takes the form p 3 (x,t) = - e - Dx B wu 3 (x,t)/wx = + (B/c) e - Dx q'(t - x/c).
The “x” in this exponential actually refers to the distance from the source
located at x = L m , however, since L m is very small compared to the surface
location x = L, we can use e - DL with very little error. In summary, if in
Equation 5.1.6, L is very far away from the source, then the corresponding
pressure is obtained by multiplying the undamped solution of Equation 5.1.6 by
e - DL to give e - DL p surface (t).
Analogously, the reverse applies at the surface for 'p determination. We
can use our surface reflection cancellation filters to separate waves originating
from downhole from surface reflections traveling downhole. The waves
originating from downhole would consist of everything from downhole (the