Geology Reference
In-Depth Information
(t) = {1/2 ( u/ t)
2
+1 / 2 E (
u/ x)
2
+
gu}dx
(4.2.31)
representing the total
distributed
kinetic, elastic strain and gravitational potential
energy of the drillstring per unit cross-sectional area.
For brevity, “ ... dx” denotes integration over 0 < x < L, where the
drillstring length L is assumed to be constant. During drilling, L will vary with
time, as sections of pipe are added or removed. However, this occurs when
drilling terminates and vibrations are absent. Thus, while dL/dt vanishes in our
modeling, we need to remember that L does increase during the drilling; in
mathematical terms, it is “piecewise constant” with time. Next, time
differentiation yields yields the rate of change
'(t) =
d/dt {
{1/2 ( u/ t)
2
+1 / 2 E (
u/ x)
2
+
gu}dx }
(4.2.32)
=
u
t
u
tt
dx + E ( u
t
/ x ) u
x
dx +
g
u
t
dx
(4.2.33)
=
u
t
(
u
tt
- Eu
xx
+
g)dx + E u
x
u
t
(L,t) - E u
x
u
t
(0,t)
(4.2.34)
where we have integrated by parts, collected distributed properties within the
integrand, and separated end effects from the integral.
At this point, we invoke the partial differential equation governing the
drillstring, that is, Equation 4.2.25, so that
'(t) =
u
t
(-
u
t
+ F
e
) dx + E u
x
u
t
(L,t) - E u
x
u
t
(0,t)
(4.2.35)
= -
u
t
2
dx +
u
t
F
e
dx + E u
x
u
t
(L,t) - E u
x
u
t
(0,t)
(4.2.36)
The right side of Equation 4.2.36 contains all the mechanisms by which energy
can exit or enter the system. We will describe these physically, but first, we
derive an important surface result.
The surface.
We digress and reconsider the surface mass-spring-damper
boundary condition
M
2
u/ t
2
+ u/ t + AE u/ x +ku + Mg = 0 (4.2.26)
which is valid at x = L for all times t. We can rewrite this local momentum
balance in a different but equally enlightening form, by multiplying Equation
4.2.26 throughout using u/ t,
M u
tt
u
t
+
u
t
2
+ AE u
x
u
t
+ kuu
t
+ Mg u
t
= 0
(4.2.37)
so that
d/dt {1/2 Mu
t
2
+ 1/2 ku
2
+ Mg u} + u
t
2
+ AEu
x
u
t
= 0 (4.2.38)
This energy description for the traveling block at x = L is analogous to
Equations 4.2.35 and 4.2.36 describing the drillstring.
Combined drillstring/surface system.
Now, we combine Equations
4.2.32, 4.2.35 and 4.2.36, and multiply each term by A to give
Search WWH ::
Custom Search