Geology Reference
In-Depth Information
different cross-sectional properties. For such problems, separate PDEs apply to
each section, say, with different subscripted terms
1
,
1
E
1
, A
1
, and u
1
, and
2
,
2
E
2
, A
2
, and u
2
.
4.2.9.1 Matching conditions.
The solutions at either side are connected by special matching conditions
u
1
= u
2
and A
1
E
1
u
1
/ x = A
2
E
2
u
2
/ x (we will show later that these do
not
apply to fluid columns). The first, continuity of displacement, applies because
the system does not “tear” at the interface. The second, continuity of force,
holds because there is no additional external loading applied at the point; stress,
incidentally, is double-valued or discontinuous. Double-valued stresses are well
known in strength of materials; normal stresses abruptly change at changes in
cross-sectional area.
Let us consider, for example, a two-section drillstring consisting of the
pipe (p) and collar (c). Each section of drillpipe, as discussed, is governed by its
own unique partial differential equation, that is,
(p) 2
u
(p)
/ t
2
+
(p)
u
(p)
/ t - E
(p) 2
u
(p)
/ x
2
+
(p)
g = 0 (4.2.99)
(c) 2
u
(c)
/ t
2
+
(c)
u
(c)
/ t - E
(c) 2
u
(c)
/ x
2
+
(c)
g = 0 (4.2.100)
Again, at the pipe-to-collar interface, the PDEs break down and do not apply,
since sudden changes in the medium lead to rapid variations for which the
spatial derivatives do not exist. There they are replaced by matching conditions,
that allow each wave solution to “analytically continue” into the other, while
conserving physically correct quantities. Now we derive the difference model.
4.2.9.2 Finite difference model.
The boundary or matching condition describing
force continuity
can be
expressed as
A
(p)
E
(p)
u
(p)
/ x = A
(c)
E
(c)
u
(c)
/ x (4.2.101)
We have used Hooke' s law = E , which relates the stress to the strain (in
our notation, = E u/ x, and force varies like A). In addition, since the
drillpipe and drill collar are rigidly held together, the
displacement must remain
continuous
, so that
u
(p)
= u
(c)
(4.2.102)
Now, we importantly
require
that Equations 4.2.101 and 4.2.102 adapt to
the finite difference formulation derived earlier for uniform drillstrings in a
stable
manner, so that divergent numerical results do not appear during the
iterations. Ideally, the matching condition will be similar in appearance to the
difference equation for the PDE and retain diagonal dominance.
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