Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 139

We have emphasized that the physical relationship between u, u t and u x obtained
at impact is the missing link needed for rock-bit interaction and correct
boundary condition definition. The exact forms for Equation 4.2.21 and 4.2.22
will require experimental study; several models, together with their analytical
and numerical consequences, are considered later.

4.2.4.5 Modeling drillbit kinematics using “displacement sources.”
We still have not discussed the (extension of) “u 0 sin t” needed to
describe rotating drillbit kinematics, stating only that Equation 4.2.19 should not
be used at x = 0. In general, a periodic vertical excitation characterizes the
geometry of the bit, its cones, offsets, and inserts, and the way it is rotated by the
rotary table from the surface or downhole by a mud motor. The rotation rate and
bit geometry affect the displacement excitation imparted to the formation and
the power inputted into the system. Ma and Azar (1985, 1986) provide excellent
examples of studies focusing upon the kinematics and dynamics of the overall
bit motion.
These and similar results are can provide useful data for “displacement
source” modeling. Let us take a conceptual view, following ideas developed in
Chapter 1, and consider the “accordion” in Figure 4.2.3. This accordion can
expand and contract vertically with prescribed length variations, simulating the
up-and-down motion of a rotating tricone bit. This displacement creation is
exactly the displacement source model, used whenever the bit is drilling ahead.
The desired physical effect is a periodic expansion and contraction cycle which
creates a length [ u] = u 0 sin t at the location x bit of the bit centroid (this
notation is discussed in Chapter 1). Displacement source modeling allows us to
prescribe sinusoidal excitations without prescribing actual bit positions, which
should be found as part of the solution.
Now, Equation 4.2.1 (that is, 2 u/ t 2 + u/ t - E 2 u/ x 2 + g = 0)
describes longitudinal displacements . From Chapter 1, the application of a
single force associated with a single delta function (x-x bit ) will create a sudden
change in the value of the spatial derivative u/ x. This implies that an external
force exists, but it doesn't. However, the use of a couple , with equal and
opposite forces situated infinitesimally close will cause a change in the function
u , representing length. Thus, the kinematics of the drillbit can be easily modeled
by an external couple with a jump in displacement or strength

[u(x bit ,t)] sin t (4.2.24)

provided the drillbit is in contact with the formation. This model is flexible
since the exact level of u(0,t) is not itself prescribed at x = 0; the final
determination of bit position is made jointly with Equations 4.2.21 to 4.2.23.