Geology Reference
In-Depth Information
Again, the excitation in Equation 4.2.94 is formed by the action of two
forces of magnitude F ds , acting in opposite directions, and separated by a
distance x. We will model these forces directly, with each force numerically
modeled as follows. Since a single force is responsible for a discontinuity in the
first spatial derivative (refer to Chapter 1 for a detailed discussions), we have in
the notation of the present chapter
F ds /E = u x + - u x - = ( U ibit+1,n - U ibit+,n )/ x - (U ibit-,n - U ibit-1,n )/
x (4.2.95)
where i = i bit represents the spatial index of the bit centroid x = x bit . Now, the
displacement values U ibit+,n and U ibit-,n to the immediate right and left of i = i bit
are identical, since the displacement function is continuous for isolated forces.
Thus, we denote U ibit+,n and U ibit-,n by the single-valued quantity U ibit,n . Hence,
the action of a single force leads to
U ibit-1,n - 2U ibit,n + U ibit+1,n = F ds
x/E = - [u] x=xbit
(4.2.96)
where we have used Equation 4.2.94. The second force required to complete the
construction of the desired couple can be taken a single grid block away, with
the required opposite sign, in the form
U ibit,n - 2U ibit+1,n + U ibit+2,n = - F ds x/E = + [u] x=xbit (4.2.97)
The “strength” of the displacement source [u] x=xbit can, for simplicity, be taken
in the form
[u] x=xbit = u 0 sin t, = 2 (3N rpm )/60 (4.2.98)
where u 0 represents (the absolute value of) the maximum axial displacement
generation due to the drillbit. Here, N rpm is the drillstring rpm , and the factor 3
models the tricone nature of the bit. PDC bits, on the other hand, may require
factors anywhere from ranging 9 to 20, with smaller values of u 0 . Finally,
initial conditions are required to start the transient calculations, as discussed
earlier. Because the right side of Equation 4.2.89 indicates that information
from two earlier time steps are required before the time-iterations may begin, we
initialize both of these initial solutions using the static displacement function
given by Equation 4.2.28 (this automatically satisfies the static requirement in
Equation 4.2.29).
4.2.9 Modeling pipe-to-collar area changes.
So far, we have assumed a uniform drillstring to illustrate the basic ideas.
If area discontinuities and changes in material properties exist, e.g., at the pipe-
to-collar interface, the partial differential equations (PDEs) to the left and right
break down at the interface because spatial derivatives taken through it do not
exist. In reality, the drillstring is composed of numerous sections, such as
drillpipe, shock sub, jarring devices and drill collar, each being characterized by
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