Geology Reference
In-Depth Information
But since continuity of displacement requires u
(p)
= u
(c)
, it follows that
A
(c)
E
(c)
u
i*-1,n
- (A
(p)
E
(p)
+ A
(c)
E
(c)
) u
i*,n
+ A
(p)
E
(p)
u
i*+1,n
= 0, as we had
obtained in Equation 4.2.104. In the final recipe, the tridiagonal equations
corresponding to all internal nodes are written, but that associated with the pipe-
collar location is replaced by the matching condition just given.
5.3.2.2 Mud acoustics.
Here, the matching condition is different, and alternate finite difference
results are obtained. Assumed
pressure and volume velocity continuity
imply
that u
(p)
/ x = u
(c)
/ x and A
(p)
u
(p)
/ t = A
(c)
u
(c)
/ t (our arguments apply
within drillstrings for MWD applications, as well as in borehole annuli for
swab-surge modeling). In difference equation form,
(u
i*+1,n
- u
i*+,n
)
x = (u
i*-,n
- u
i*-1,n
)
x
(5.53)
A
(p)
(u
i*+,n+1
- u
i*+,n
)
t = A
(c)
(u
i*-,n+1
- u
i*-,n
)
t
(5.54)
The second equation indicates that
A
(p)
u
i*+,n
= A
(c)
u
i*-,n
(5.55)
in general. Therefore, since u
i*+,n
and u
i*-,n
are
not
equal as a result of Equation
5.55, the dependent variable u
i*,n
must be
double-valued
at the interface. Since
we prefer to deal with a single unknown at i = i* in our matrix of algebraic
equations, we
define
(without any loss of generality, as we will show) that
unknown to be the arithmetic mean value
u
m
n
= ( u
i*-,n
+ u
i*+,n
)/2 (5.56)
so that the pressure matching condition given in Equation 5.53 can be rewritten
in the form
u
i*-1,n
- 2u
m
n
+ u
i*+1,n
= 0
(5.57)
Equation 5.57 is
completely different
from the matching condition
A
(c)
E
(c)
u
i*-1,n
- (A
(p)
E
(p)
+ A
(c)
E
(c)
) u
i*,n
+ A
(p)
E
(p)
u
i*+1,n
= 0 (5.58)
obtained for axial vibrations. In either case, the matching condition, like the
difference approximation to the wave equation, was designed to be “diagonally
dominant” and hence numerically stable.
A consequence of our definition and use of a mean displacement is this:
the two difference equations at the neighboring sides of the interface,
corresponding to the indexes i = i* + 1 and i = i*-1, must be rewritten in terms
of u
m
. For example, the u difference equation at i = i*+1 involves the value of u
at i = i*+. Since u
i*+,n
= { A
(c)
/A
(p)
}u
i*-,n
, we find that u
m
n
= ( u
i*-,n
+ u
i*+,n
)/2
= 1/2 {1 + A
(c)
/A
(p)
}u
i*-,n
, or
u
i*-,n
= 2 u
m
n
/{1 + A
(c)
/A
(p)
}
(5.59)
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