Transient Constant Area Surface and Downhole Wave Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 128

Schematic

"1" "2"

Drillbit
x = 0

MWD
pulser
x = L

Upgoing
signal to
surface

x

Figure 4.5a. Solid boundary reflection, waves from dipole source.
4.5.4 Theory.

The formulation for the difference delay equation model used is derived
here referring to Figure 4.5a. In Section “1,” the Lagrangian displacement
function is assumed as u(x,t) = h(t - x/c) - h(t + x/c) where “h” is unknown,
with u(0,t) = h(t - 0) - h(t + 0) = 0 assuming that the bit at x = 0 is a solid
reflector. Then p(x,t) = -B wu/wx = + (B/c) [h'(t - x/c) + h'(t + x/c)] represents
the corresponding acoustic pressure function. In Section “2,” we assume
radiation conditions, that is, u(x,t) = f(t - x/c), a wave traveling to the right
without reflection. The function p(x,t) = -B wu/wx = + (B/c) f '(t - x/c)
represents the acoustic pressure in Section “2.”
Boundary matching conditions apply at the MWD source point. At the
pulser x = L, we have p 2 - p 1 = 'p(t). The exact time dependencies in the given
transient function are determined by the telemetry encoding method used, while
the peak-to-peak strength is determined by valve geometry, rotation rate, flow
rate, fluid density, and so on. Substitution of the foregoing wave assumptions
leads to
f ' (t - L/c) - h'(t - L/c) - h'(t + L/c) = (c/B) 'p(t) (4.5a)
The requirement u 1 = u 2 at x = L implies continuity of displacement, for which
we obtain h(t - L/c) - h(t + L/c) = f(t - L/c). Thus, taking partial time
derivatives, we obtain
h'(t - L/c) - h'(t + L/c) = f '(t - L/c) (4.5b)
If we eliminate f ' between the Equations 4.5a and 4.5b, 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, we have