Geology Reference
In-Depth Information
analogous to Equation 4.2.70. The surface boundary condition in Equation
4.2.26 can be differenced as
M( u
imax,n
-2u
imax,n-1
+ u
imax,n-2
)/( t)
2
+ (u
imax,n
-u
imax,n-2
)/(2 t) + AE (u
imax,n
- u
imax-1,n
)/ x
+ k u
imax,n
+ Mg = 0 (4.2.90)
This result can be rearranged so that it occupies the bottom row of the
coefficient matrix shown in Equation 4.2.76, taking the form
(-AE/ x) u
imax-1,n
+{M/( t)
2
+ /(2 t) + AE/ x + k} u
imax,n
= { 2 M/ ( t)
2
}u
imax,n-1
+{ /(2 t) - M/( t)
2
}u
imax,n-2
- Mg (4.2.91)
The boundary condition imposed at x = 0 occupies the top row of the
matrix in Equation 4.2.76. It is somewhat more complicated than the surface
condition above. Because the drillstring may
forge ahead
or
retreat upward
, the
boundary model describes
either
rock-bit interaction
or
a stress-free end. Since
it is impossible to know
a priori
which boundary condition applies at any
instant, the choice itself must be determined as part of the numerical solution.
If, based on results from the preceding time step, the
drillbit
velocity u/ t is
upward
, then the
stress-free
end condition u/ x = 0 will be imposed, that is,
U
1,n
- U
2,n
= 0 (4.2.92)
following Equation 4.2.23. On the other hand, if the
drillbit
velocity u/ t is
downward
into the formation, then the
rock-bit boundary condition
in Equations
4.2.22 or 4.2.45 (or, u
x
+ u
t
+ u = 0) will be discretized in the form
{- / x + / t + } U
1,n
+ { / x} U
2,n
= { / t} U
1,n-1
(4.2.93)
In implementing this switching criterion, we obviously take advantage of the
capabilities offered by Fortran
if-then
logic. Any other programming language
may be used, but in this topic, subroutines are conveniently given in Fortran.
We have referred to , and as constants, for the purposes of analytical
solution, but these quantities can in fact represent
functionals
of u(x,t) and its
space and time derivatives, if empirically justified. This change is trivial
numerically. The extension from a linear model to
any
nonlinear one, for
instance, G(u, u/ x , u/ t) = 0 at x = 0, is simple from a programming point of
view. So far, we have imposed
boundary conditions
at the surface x = L and
downhole at the rock-bit interface x = 0. We have yet to specify the
displacement source (ds) excitation
that models the axial reciprocation due to
the rotating drillbit. Following Chapter 1, with reference to Equation 1.120, we
require that the
jump in displacement
or the
length generation
satisfy
[u]
x=xbit
= - F
ds
x / E
(4.2.94)
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