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emphasize that a single partial differential equation suffices because the same
mud is present everywhere. Also, all of our mud acoustic observations and
derivations apply to swab-surge problems in the annulus and to MWD mud
pulse applications in the drillpipe.
5.1.4 Mud acoustic formulation.
In summary, we state the following long wave results for acoustic wave
motions in one-dimensional swab-surge and MWD applications,
u
tt
- c
2
u
xx
= 0 (5.1)
c
2
= B / (5.2)
p = -Bu
x
(5.3)
u
x
(p)
= u
x
(c)
(5.4)
A
(p)
u
t
(p)
= A
(c)
u
t
(c)
(5.5)
where damping (for now) is ignored. If, in swab-surge problems, the drillbit is
off-bottom, a third matching condition A
(c)
u
t
(c)
= A
(h)
u
t
(h)
would apply at the
collar and borehole interface, where “h” refers to the hole radius. A complete
boundary value problem formulation would consist of Equations 5.1 to 5.5,
boundary conditions at the lower drillbit or mud motor end x = 0, boundary
conditions at the surface x = L, initial conditions at time t = 0,
plus
a suitable
model for the excitation source. Before dealing with modern applications, we
turn to some well known classical results.
5.1.5 Example 5-1. Idealized reflections and transmissions.
The complete wave field in the drilling channel solving the general
formulation can be complicated analytically, and voluminous, if tackled
numerically. For these reasons, elementary topics often study the problem in the
simpler limit where multiple reflections do not appear. We examine these
simpler problems because they help us formulate matching conditions at
impedance junctions for use in more complicated problems. Consider two semi-
infinite pipe Sections “1” and “2” joined at x = 0, having areas A
1
and A
2
,
containing a fluid with bulk modulus B, density and sound speed c. No
restrictions on the relative sizes of A
1
and A
2
are required below and the
physical problem is sketched in Figure 5.1. There, a plane wave is incident from
the left; it approaches the junction x = 0, where the incoming wave will partially
transmit and partially reflect. Note that x = 0 does
not
represent the drill bit in
the present analysis (this zero value is taken in order to simplify required
algebraic manipulations). Thus, in Section 1, the pressure and displacement
functions take the form
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