Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 146

4.2.6.2 Simple analytical solution.
Proper boundary conditions are important in modeling correct energy
partitioning at the bit: part of the wave energy destroys the rock, another part
reflects into the drillstring toward the surface, while the remainder dissipates in
the form of heat. How much of each is the problem. We first attempt to
understand the consequences of end boundary conditions used in mechanical
engineering, and demonstrate why they are unsuitable to drillstring vibrations.
For simplicity, the kinematics of the bit are ignored: we simply examine the
reflections of an incident pulse and study its implications insofar as penetration
is concerned. Our results are qualitative, since surface interactions and finite
drillstring length are neglected for now. The results are intended to demonstrate
only the range of possibilities available with different math models.
In Chapter 1, we showed how f(ct+x) and g(ct-x) represent axial
displacement waves propagating in opposite directions. Here, we will suppose
that f(ct+x) represents a known wave heading towards x = 0. We will construct
the complete solution comprising of both incoming and outgoing waves, and
from it, determine and explain different elements of the bottom response.
This exploratory work ascertains the form of boundary conditions needed
to model desired physical effects. In particular, we seek classes of boundary
conditions for which sinusoidal inputs will create DC levels of displacement
change. For brevity, we refer to u x as the stress (that is, the normal stress, or
Eu x ). To motivate the final result, we consider some inappropriate models first
in order to explain their limitations.
4.2.6.3 Classic fixed end.
The rigid end boundary condition u(0,t) = 0 is implemented by the
construction u(x,t) = f(ct+x) - f(ct-x), which solves u(0,t) = f(ct) - f(ct) = 0 at x =
0. Consequently, the speed u t (0,t) vanishes for all time. The stress u x (x,t) = f
'(ct+x) + f '(ct-x) takes on the value 2f '(ct) at x = 0, which is twice the f '(ct)
value due to the incoming wave alone. In this zero rate-of-penetration rigid wall
model, the power input term u x u t vanishes for all time.
4.2.6.4 Classic free end.
Here, we satisfy u x (0,t) = 0. The construction u(x,t) = f(ct+x) + f(ct-x)
leads to u x (x,t) = f '(ct+x) - f '(ct-x), which satisfies u x = 0 at x = 0. The speed u t
= cf '(ct+x) + cf '(ct-x) = 2cf '(ct) at x = 0, which is twice the cf '(ct) value due to
the incident wave. The displacement at the bit is u(0,t) = 2f(ct), which is
generally nonzero; however, for a given sinusoidal Fourier wave component, its
time-averaged value over a period identically vanishes.