Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 144

d/dt {1/2 ( u/ t) 2 +1 / 2 E ( u/ x) 2 + gu} A dx - AE u x u t (L,t) (4.2.39)

= - u t 2 A dx - AE u x u t (0,t) + u t F e A dx (4.2.40)

Simplifying with Equation 4.2.38 leads to
d/dt {1/2 ( u/ t) 2 +1 / 2 E ( u/ x) 2 + gu} A dx
+ d/dt {1/2 Mu t 2 + 1 / 2 ku 2 + Mg u} x=L
= - u t 2 A dx - u t 2 | x=L - AE u x u t (0,t) + u t F e A dx (4.2.41)
The left side represents the timewise rate of change of total energy for the
combined surface system and drillstring. For the drillstring, the contributions in
the integrand are the distributed kinetic, elastic strain and gravitational potential
energies. For the surface system, we have the kinetic energy of the traveling
block, the spring potential energy and the block gravitational potential energy, in
that order. The right side , shows how work interaction changes the total energy
balance as the complete system evolves in time. Since and are both positive,
as are the “positive-definite” integrals multiplying them, the first two (negative)
terms represent dissipation due to internal viscous effects.
The AEu x u t (0,t) term represents the work introduced at the formation as a
result of rock-bit interaction. Here, AEu x is the dynamic weight-on-bit (equal to
the reactive force on the formation), and u t (0,t) is the instantaneous bit velocity;
their product represents power input due to rate of penetration. This power is
not strictly positive or negative. Its sign depends on relative cone-to-rock
displacement pattern phasing, whose details are affected by cone geometry,
formation hardness, drill rate and surface rpm. This local argument is only part
of the explanation; the effect of drillstring dynamics, whose qualitative features
are less obvious, are also important (of course, when the bit bounces off bottom,
vanishing stress implies zero power transfer). Finally, u t F e A dx represents the
work done by external forces other than those applied at the bit. If distributed
wall friction or point contacts (such as doglegs) are present, these will contribute
to the integral. But the displacement source describing drillbit motions does not
directly contribute to net energy, since the sum of equal and opposite forces
completely vanishes; of course, it does dictate the form of the wave patterns
created along the drillstring.

4.2.5.3 Detailed bit motions.
Of interest is the form for the displacement function f(t) in Equation
4.2.30. In field operations, rotation rates are specified in rpm , where 1 rpm
represents one revolution per 60 seconds, or 1/60 th Hz . If N rpm is the
drillstring rpm, and a factor of 3 is introduced to simulate tricone effects, it