Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 179

4.2.11.6 Separating incident from reflected waves.
In the above rock-bit interaction examples, we assumed that we could
distinguish between a single incoming pulse and its reflected (altered) wave. If
both incident and reflected signals are known, the “transfer function” (or
boundary condition, e.g., Equation 4.2.129) describing rock-bit interaction can
be determined. Let us consider how a hypothetical downhole imaging device
might work while drilling. Suppose that the distance-to-bit for a downhole
sensor array mounted along the drillstring is known, and that the sound speed of
the metal is given; thus, the wave transit time t between sensor position and bit
can be calculated. We also assume that a means of separating up and
downgoing waves (to be derived) is available for real-time use.
Now, at any time t while drilling, the sensor-microprocessor array is turned
on momentarily; it determines the up and downgoing waves at that location for
that time t, and stores only the downward incident wave. This wave will reflect
at the bit, and should return to the sensor in a time 2 t. At that instant, the
sensor unit self-activates; again, both oppositely traveling waves are extracted,
but this time, the upward traveling reflected wave is instead stored. Since the
incident wave pulse and its reflection have been isolated, the rock-bit interaction
constants are in principle known. Of course, in any real system, the
complications introduced by torsional waves and dispersive lateral waves must
be considered. But at least for now, the inverse problem has been reduced to
mathematical one which calls for the extraction of up and downgoing waves
from a general transient field.
Many techniques are available which separate or filter superposed waves
that travel in opposite directions; these abound in the seismics, communications
and electrical engineering literature. They are “directional filters,” which must
be contrasted with “frequency filters.” In one-dimensional drillstrings, the
filtering problem is actually less demanding than those in three-dimensional
earth imaging, despite the strong reverberant fields; Example 1-12 summarizes a
well known gas-dynamic model. This is so because, locally anyway, the
undamped wave equation u tt - Eu xx = 0 applies. We now discuss only the
simplest, that is,
u tt - c 2 u xx = 0 (4.2.145)
Although our discussion will focus on the displacement u(x,t), we emphasize
that it applies equally to velocity and stress, since these latter quantities also
satisfy the wave equation. This is easily demonstrated. If we differentiate
Equation 4.2.145 with respect to x, we find that (u x ) tt - c 2 (u x ) xx = 0 . Li ke w i s e ,
if we differentiate respect to t, we have (u t ) tt - c 2 (u t ) xx = 0. Both of these results
represent wave equations, that is, for stress and velocity.