Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 174

our interpretation. Second, we assumed semi-infinite drillstrings in order to
circumvent issues related to multiple reflections. One way of extending this
approach to realistic conditions requires a two-pronged attack. First, we need to
supplement our limited rigid and fixed end models with more general classes of
reflection solutions. Second, we need to provide for (simulated) semi-infinite
conditions so that comparisons between incident and reflected waves can be
made. We discuss these two issues separately.

4.2.11.5 More rock-bit interaction models.
So far, we have noted simple phase changes, but changes other than 0 o and
180 o are possible, corresponding to rocks of intermediate hardness. Our
amplitudes have remained unchanged, but in general, amplitude changes and
shape distortions are both possible. We will give formulas that allow us to
determine , and from measured displacement, impact speed, and tip stress
data. Two separate impact models are considered. From these values, rock type
is inferred from (to be) catalogued properties for the particular drillbit used.
An inelastic impact model. To develop the necessary formulas, we
follow a differential equation framework similar to one in Morse and Feshbach
(1953). We first substitute the general solution u(x,t) = f(ct-x) + g(ct+x) into the
proposed rock/bit interaction model
u x (0,t) + u t (0,t) + u(0,t) = 0

(4.2.120)

to obtain
(c - )df(z)/dz + (c + )dg(z)/dz + f(z) + g(z) = 0 (4.2.121)
If the incident wave g(ct+x) is assumed to be known, the above can be viewed as
an equation for the function f, which has the solution
z
f(z) = exp(- z) { dg(

)/d - g( )} exp(+ ) d (4.2.122)

a
where

= /(c - ) (4.2.123)
= ( +c )/( -c ) (4.2.124)
and “a” is an arbitrary constant. This exact solution allows us to extract the
values of the rock-bit interaction constants , , and when discrete sampled

values of F and G are available.
For clarity, we first consider the inelastic limit where = 0. This case
describes problems where forces resulting from elastic displacements are
insignificant. Here, the downhole boundary condition reduces to

u t (0,t) = - ( / ) u x (0,t) (4.2.125)