Geology Reference
In-Depth Information
We will add to this classical equation the right-hand-side forcing function
“'p” G(x-x
s
) where we refrain from conjecturing the meaning of “'p” for now.
We do, however, note that G(x-x
s
) represents the Dirac delta function - that is, a
concentrated force - situated at the source point x = x
s
; in physical terms, the
assumption that the imposed excitation lies at a single point in space requires
that the dimensions of the MWD pulser be small compared with a typical
wavelength, a requirement presently satisfied by all commercial tools. In doing
so, we obtain the inhomogeneous partial differential equation
U
mud
u
c
tt
- B
mud
u
c
xx
= “'p” G(x-x
s
) (2.1)
The superscript “c” refers to the MWD drill collar, of course, which houses the
siren source (we discuss drill collar acoustics first in order to show why the
Lagrangian displacement is the proper dependent variable for use in the
remainder of the analysis). Note that Equation 2.1 does not include damping.
Again, this is not to say that attenuation is unimportant; it is, but attenuation is
only significant over large spatial scales, as would be the case when signals
travel through the drillpipe. In the present section, which deals with interference
dynamics near the source point, interactions act only over short distances on the
order of the bottomhole assembly length and the mathematically complicating
effects of damping can be justifiably neglected.
Let us now multiply Equation 2.1 throughout by the differential element
dx, and integrate the resulting equation from “x
s
- H” to “x
s
+ H,” where H is a
small positive length on the order of the source dimensions. Since u(x,t) and its
time derivatives are continuous through the source point, that is, there is no
“tearing” in the fluid, we obtain the “jump condition”
- B
mud
u
c
x
(x
s
+H,t) + B
mud
u
c
x
(x
s
-H,t) = “'p” (2.2)
since the integral of the delta function is unity. If we note that the acoustic
pressure is in general defined by
p = - B u
x
, (2.3)
it follows that “'p” = p
c
x
(x
s
+H,t) - p
c
x
(x
s
-H,t). In other words, “'p” is the
jump
in
acoustic pressure
through the MWD source and it appears naturally
only
in a
fluid displacement formulation using u(x,t). This jump is created not only by
our aforementioned speakers, but by poppet valves and mud sirens which stop
the oncoming flow in one direction, thus creating high pressure, while
permitting the flow at the opposite position to pull away, thereby producing low
relative pressure. It is also physically clear that the created pressures on either
sides of the dipole source must be equal and opposite in strength, measured
relative to hydrostatic background levels. We emphasize that this 'p is not to be
confused with a statically unchanging pressure differential, e.g., the viscous
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