Geology Reference
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any friction will need to be modeled when considering u
d
(x,t). This topic-
keeping is natural for vertical wells, where borehole wall contact is minimal.
Thus, combination of Equation 4.2.5 and Equation 4.2.4 yields
2
u
d
(x,t)/ t
2
+
u
d
(x,t)/ t - E
2
u
d
(x,t)/ x
2
= F
e
(4.2.6)
The static equation for our assumed u
s
(x) integrates to
du
s
(x)/dx = ( g/E) x + C
1
(4.2.7)
u
s
(x) = ( g/2E) x
2
+ C
1
x + C
2
(4.2.8)
where C
1
and C
2
are constants of integration.
4.2.3.1 Sign conventions.
Suppose that a concentrated mass M rests on top of a vertical drillstring of
length L, and that the system in Figure 4.2.2, in static equilibrium, is entirely
supported by the formation.
Figure 4.2.2.
Drilling surface system.
We require that the axial force du
s
(x)/dx then equal -Mg at x = L.
Thus, Equation 4.2.7 can be written in the form
du
s
(L)/dx = A gL + AE C
1
= - Mg
(4.2.9)
leading to C
1
= -Mg/AE - gL/E or
du
s
(x)/dx = A gx - AE{Mg/AE + gL/E}
(4.2.10)
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