Transient Variable Area Downhole Inverse Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 143

5.1.2 Theory.
In Figure 5.1a, four Lagrangian displacement functions “f,” “g,” “h” and
“q” are introduced. The first three describe reverberant fields in the locations
shown, while the last represents a propagating wave in the drillpipe. In Section
“1,” 0 < x < L m , the superposition of up and downgoing waves takes the form
u 1 (x,t) = h(t - x/c) + h(t + x/c). The corresponding pressure p 1 = - B wu 1 /wx =
B/c [h'(t - x/c) - h'(t + x/c)] vanishes at the assumed open end x = 0. In Section
“3,” with x > L c , we assume a pure upgoing wave u 3 (x,t) = q(t - x/c) which
satisfies standard radiation conditions. This ignores small reflections at pipe
joints whose effects, from field experience, are known to be minimal; also, if
reflections (and other noise) at the surface travel downhole, this model assumes
that attenuation renders them insignificant by the time they reach the MWD drill
collar (thus, modifications may be needed for very shallow wells). The
corresponding acoustic pressure is p 3 = - B wu 3 /wx = B/c q'(t - x/c). In Section
“2,” L m < x < L c , we more generally assume a linear superposition of the form
u 2 (x,t) = f(t - x/c) + g(t + x/c), which supports up and downgoing waves, with
acoustic pressure p 2 = - B wu 2 /wx = B/c [f '(t - x/c) - g'(t + x/c)]. So far, we
have used the far-left and far-right boundary conditions. Note Sections “1” and
“2” support left and right-going waves, while Section “3” only supports a right-
going propagating wave.
At the “2-3” collar-pipe junction x = L c , continuity of volume velocity and
of pressure require A c wu 2 /wt = A p wu 3 /wt and wu 2 /wx = wu 3 /wx, respectively. The
A's denote collar and pipe cross-sectional areas. The dummy time independent
variables in the resulting algebraic equations can be adjusted so that explicit
solutions for “f” and “g” can be obtained in terms of “q.” These matching
conditions imply that f '(t - L m /c) = ½ (A p /A c +1) q'(t - L m /c) and g'(t - L m /c) =
½ (A p /A c - 1) q'(t - L m /c - 2L c /c). The first equation states that when A p = A c ,
“f” and “q” are identical, since there is no impedance mismatch due to area
change; the second result confirms this, stating that the left-going “g” wave
vanishes identically.
At the “1-2” junction x = L m , the MWD pulser supports a general transient
acoustic pressure difference 'p(t) satisfying p 2 (t) - p 1 (t) = 'p(t). In addition,
continuity of volume velocity requires wu 1 /wt = wu 2 /wt since cross-sectional area
does not change. These conditions together lead to g'(t - L m /c) - h'(t - L m /c) =
- ½ (c/B) 'p(t - 2L m /c) and also f '(t - L m /c) - h'(t - L m /c) = + ½ (c/B) 'p(t) (in
the foregoing matching conditions, appropriate changes in dummy variables
have been used to transform f, g and h arguments to “t - L m /c” form so that
independent displacement solutions can be obtained). Subtraction of one
equation from the other leads to a relationship connecting “f” to “g” that is
independent of the function “h.” Then, use of the equations in the above
paragraph and replacing “q'” by its equivalent in terms of p 3 yields a