Geology Reference
In-Depth Information
What, exactly, does this blind application of delta functions model? From
Chapter 1, we understand that A (x-x * ) produces an internal discontinuity in
the spatial derivative p/ x; the right side of Equation 3.60 is responsible for a
sudden jump in the pressure gradient p/ x. Now, positive pressure and siren
pulsers will produce internal jumps in the pressure level itself. Thus, it is not
clear what Equation 3.60 models, and how it can be used, unless a mechanical
device capable of generating pressure gradient discontinuities can be designed.
Next, we repeat this exercise with the velocity potential of Equation 3.59,
and add a delta function to its right side. Thus, we consider
2 / t 2 - c 2 2 / x 2 = A (x-x * ) (3.61)
Equation 3.61 now models a physical system where the spatial first derivative
/ x is discontinuous; that is, the axial velocity u is discontinuous, on noting
Equation 3.58a. How might this be accomplished in practice? For instance, a
“pulsating balloon” introduced into our drillstring at x = x * would produce such
acoustic disturbances. The balloon surfaces at the sides of x * always head away
from (or toward) each other symmetrically, thus providing the required axial
velocity discontinuity. For example, if u(x * -) = -100 ft/sec and u(x * +) = +1 0 0
ft/sec, then the difference u(x * +) -u(x * -) = +200 ft/sec 0 is nonzero, providing
the discontinuity in axial velocity. Thus, Equation 3.61 represents one possible
way to model negative pressure pulsers. The reader might consider how pulsers
that create pressure discontinuities should be modeled; certainly, (x,t) and
p(x,t) are inappropriate. This problem will be considered in detail later.
The role of source point modeling is also important in drillstring
vibrations. Conventional boundary condition models prescribe periodic
displacements at the drillbit x = 0 in order to simulate bit motions. But periodic
displacements have zero time averages; by invoking such models, non-zero rate-
of-penetration and bit bounce cannot be modeled ! In Chapter 4, we will show
how “displacement sources” can be used to model tricone bit kinematics; the
additional degree of freedom that this flexibility offers then permits us to apply
rock-bit interactions as boundary conditions at x = 0, thus opening up new
classes of possible dynamical motions at the bit. Since we do not force
sinusoidal displacements at the bit (which have zero time average and thus no
penetration rate), this formalism allows us to construct bit models that, in the
language of electrical engineers, allow DC in addition to AC changes. Since the
first publication of the foregoing ideas in the 1990s, applications to mud pulse
telemetry have been developed to a high degree of sophistication. These are
described in detail in the author' s new topic Measurement While Drilling Signal
Analysis, Optimization and Design (Chin et al, 2014).
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