Geology Reference
In-Depth Information
4.2.5 Global energy balance.
We now summarize our
momentum-based
differential equations, but
before considering solutions, it is instructive to consider
energy arguments
,
since they provide insight into the physical mechanisms at work.
4.2.5.1 Formulation summary.
The complete AC/DC axial displacement u(x,t) satisfies Equation 4.2.25,
together with Equation 4.2.26 at the surface, and the rock-bit interaction in
Equation 4.2.27.
2
u/ t
2
+
u/ t - E
2
u/ x
2
+ g = F
e
(4.2.25)
M
2
u/ t
2
+
u/ t + AE u/ x + ku + Mg = 0 at x = L
(4.2.26)
G( u, u/ x , u/ t) = 0 at x = 0
(4.2.27)
Equations 4.2.26 and 4.2.27 represent
boundary
conditions. Additional
initial
conditions are needed to “start” the time integrations. If the axial vibrations are
initiated from rest, Equation 4.2.14 might be used at t = 0; that is,
u(x,0) = u
s
(x) =1/2 gx
2
/E + W
0
x/AE (4.2.28)
u(x,0)/ t = 0 (4.2.29)
The boundary condition in Equation 4.2.27 describes the mechanical impedance
condition at the rock-bit interface; it does not model the kinematics of the
drillbit. As discussed, the “accordion-like” displacement variations due to the
bit are naturally modeled by displacement sources, i.e.,
[u]
x=xbit
= f(t) ... {e.g., u
0
sin t} (4.2.30)
where x
bit
represents the centroid of the bit, and f(t) describes the vertical
displacement history of the bit.
4.2.5.2 Energy considerations.
Differential equations describe dynamics from a
momentum
viewpoint, a
by-product of their “
F
= m
a
” origins. We now reinterpret our drillstring
formulation from an
energy
perspective; this sheds insight into the type of
constraints needed to reproduce observed phenomena. It is clear that numerical
solutions to formulations that appear reasonably correct can lead to disaster if
lurking sign errors go undetected. Energy checks provide additional safety; if
both momentum and energy descriptions are consistent, greater confidence can
be placed in the results.
The drillstring.
By learning how energy enters and exits the system, we
understand the ingredients required for modeling rock-bit interaction and other
quantities of physical interest. For this purpose, consider the integral
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