Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 161

i*- i*+

x

i*-1 i* i*+1

Figure 4.2.7 . Pipe-to-collar indexing convention.
We designate by i* the x-index of the interface, as shown in Figure 4.2.7 and
assume uniform grids x. Equation 4.2.101 can be differenced as follows,

A (p) E (p) (u i*+1,n - u i*+,n ) x = A (c) E (c) (u i*-,n - u i*-1,n ) x (4.2.103)
where i * + and i * - represent infinitesimally close positions on either side of i *
(we again assume that the index increases from i = 1 at the bottom x = 0, to i =
i max at the top of the drillstring x = L). Equation 4.2.102 indicates that the
displacement function is continuous through the interface, so that both u i*+,n and
u i*-,n can be written as as u i*,n . If we multiply by x throughout, we can rewrite

Equation 4.2.103 in the form

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 (4.2.104)

4.2.9.3 Generalized formulation.
We modify our earlier finite difference recipe as follows. First , we replace
Equation 4.2.88 for uniform drillstrings with the differenced forms of Equations
4.2.99 and 4.2.100 as required for different values of the index i, noting that the
coefficients in these equations are different. Second , at the index i = i *
corresponding to the pipe-to-collar interface, we replace the PDE-based
difference equation we (may) have written, with Equation 4.2.104 above. Th i r d ,
above the pipe-collar interface at the surface, Equation 4.2.90 applies to u (p) .
Fourth , below the pipe-collar interface, our rock-bit boundary conditions and
displacement source model given by Equations 4.2.92, 4.2.93, 4.2.96, and 4.2.97
apply to u (c) . Because Equation 4.2.104, by construction, is diagonally
dominant, we are assured that the resulting time integrations are stable. This
does not guarantee convergence, but again, stability is at least essential to having
a robust algorithm. Near the interface, the quantities u i*-1,n and u i*+1,n refer to
collar and pipe values, respectively, while u i*,n applies equally to both sections.

4.2.9.4 Alternative boundary conditions.
The downhole boundary condition formulations we have given are quite
general, but for completeness, we will list without discussion other models that
have been considered in the literature.

Fixed end:
u(0,t) = 0 (4.2.105)