Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 127

dissipation (we will demonstrate this later), while “ g” refers to static weight.
Our use of “x” does not require a straight hole, but drillstrings in deviated wells
should be modeled using only that component of weight along the borehole axis.
We will elaborate on this later.

x

Surface

Drillstring

+ Neutral point

Bit

x = 0 Roc k

Figure 4.2.1. Drillstring coordinate system.
For now, the drillstring is assumed to be uniform, without any changes in
properties in the x direction, although these restrictions will be relaxed later.
Thus, the properties , , E and A are constants. Later, we will discuss
impedance matching conditions that must be enforced at discontinuities, and
develop numerical models that simulate partial reflections and transmissions.
Note that A represents the cross-sectional area of the drillstring, that is, the area
occupied by actual metal if the pipe is hollow, or the entire area if the drillstring
is an idealized solid rod. To keep the discussion simple, we will not consider
the effects of the mud inside or outside of the drillpipe, nor will we consider the
effects of buoyancy. To obtain “typical” strain estimates, consider a drillstring
with cross-sectional area A = /4 (4 2 - 3.34 2 ) 4 in 2 and E=30 10 6 psi, so that
AE = 120 10 6 lbf. If the F in “F = AE u/ x” were 30,000 lbf, then u/ x =
0.0003 approximately. We might speculate that u/ x O(10 -4 ), give or take
an order of magnitude, depending on static and dynamic conditions.

4.2.2.2 External forces and displacement sources.
We have also introduced an axial force excitation F e (x,t,u, u/ x, u/ t)
which may depend linearly or nonlinearly on the displacement function and its