Transient Constant Area Surface and Downhole Wave Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 134

4.6.3 Theory.

In Section “1,” we now take the Lagrangian displacement u(x,t) in the form
u(x,t) = h(t - x/c) + h(t + x/c), which no longer vanishes at bit, since it is not
solid reflector, but which allows the acoustic pressure to vanish at the bit, that is,
p(x,t) = -B wu/wx = + (B/c) [h'(t - x/c) - h'(t + x/c)] = 0 at x = 0. In Section “2,”
we retain our upgoing wave assumption u(x,t) = f(t - x/c), which implies that
p(x,t) = -B wu/wx = + (B/c) f '(t - x/c) or p(L,t) = + (B/c) f '(t - L/c). At the
pulser x = L, the enforced pressure drop is p 2 - p 1 = 'p(t), a given function
dictated by the pulser position-encoding scheme. These assumptions lead to
f '(t - L/c) - h'(t - L/c) + h'(t + L/c) = (c/B) 'p(t) (4.6a)
Continuity of displacement through the MWD source, that is, u 1 = u 2 at x = L,
requires that h(t - L/c) + h(t + L/c) = f(t - L/c). Thus, taking partial time
derivatives, we have
h'(t - L/c) + h'(t + L/c) = f '(t - L/c) (4.6b)
If we eliminate f ' between the two Equations 4.6a and 4.6b, the “t - L/c” terms
cancel, and we obtain h'(t + L/c) = + {c/(2B)} 'p(t). In terms of the dummy
variable W = t + L/c,
h'(W) = + {c/(2B)} 'p(W- L/c) (4.6c)
Next we subtract Equation 4.6b from 4.6a, but this time introduce the dummy
variable W = t - L/c, to obtain
f '(W) = h'(W) + {c/(2B)} 'p(W + L/c) (4.6d)
If we eliminate h'(W) between Equations 4.6c and 4.6d, we obtain 'p(W + L/c) +
'p(W - L/c) = (2B/c) f '(W) which we can express in the more meaningful
physical form
½ ['p(W + L/c) + 'p(W - L/c)] = (B/c) f '(W) (4.6e)
We now recast the Equation 4.6e in a more useful form, taking the dummy
variable W = t - L/c. Since it is clear that (B/c) f '(t - L/c) is just the upgoing
pressure p 2 (L,t), we can write, noting p 2 (L,t) = + (B/c) f '(t - L/c),
½>'p(t) + 'p(t - 2L/c)] = (B/c) f '(t - L/c) = p 2 (L,t) (4.6f)

or

p 2 (L,t) = ½>'p(t) + 'p(t - 2L/c)] (4.6g)
This can be easily interpreted physically. Suppose that a pulser creates a 'p
signal. A signal 'p/2 travels uphole. It simultaneously sends a signal - 'p/2
downhole, so that the net pressure differential is 'p. This -'p/2 travels
downward to the bit, which is assumed as an open ended reflector, and the signal
reflects with an changed pressure sign, that is, + ½ 'p(t - 2L/c), now including
the roundtrip time delay 2L/c. This derivation applies only to dipole pulsers