Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 128

derivatives. This force may be concentrated, e.g., at mild dogleg; or, it may be
distributed, e.g., when continuous wall friction exists. For brevity, our “F e ”
designation will represent both of these effects, plus others to be discussed,
collectively. It turns out that F e will play more than its conventional role in
describing force excitation. We introduce its use as a couple internal to the
drillstring, following the ideas developed in Chapter 1, to create “displacement
source” effects that model the up-and-down kinematics of rotating tricone bits.
This displacement source completely describes the 3 rpm motions imposed by
the rotary table, freeing up the boundary condition (applied at x = 0 by early
authors) so that “rock-bit interactions” (a.k.a., “mechanical impedance boundary
conditions”) can be prescribed there instead.
4.2.2.3 Dynamic and static solutions.
Equation 4.2.1, we emphasize, models the complete dynamic and static
longitudinal fields for the total displacement u(x,t). We will also refer, from
time to time, to dynamic and static as AC and DC when analogies to electric
circuits are suggested. In mechanical engineering analyses, these AC and DC
component fields are (correctly) modeled separately, as we will demonstrate,
and subsequently superimposed. But in drilling vibrations where rock-bit
interactions are allowed, the two dynamically interact through non-standard
boundary conditions, and the advantages of any AC/DC separation are lost. For
example, as will be shown, AC motions will produce changes in DC level; thus,
the need to combine both displacements into a single unambiguous variable is
seen. This is the case when we model transient shocks, bit-bounce and rate-of-
penetration, and especially so, when nonlinear back-interaction effects due to
high amplitude lateral vibrations are considered.
4.2.2.4 Free-fall as a special solution.
Our sign convention for the gravity term is chosen consistently with the
free-fall limit: any particular “dx slice” satisfies “ 2 u/ t 2 = - g,” which leads
to “u = -1/2 gt 2 ” if the remaining formulation terms were absent. This situation
represents a drillstring dropping unimpeded, in the limit of vanishing formation
support. The free-fall limit is used as a check on the numerical scheme, to
ensure that the drillstring drills ahead (or, falls) under obvious conditions. In
modeling rate-of-penetration, the sign convention u/ t < 0 means “making
hole,” in which case empirical rock-bit boundary conditions are used, while
u/ t > 0 means “bit bounce,” in which case a stress-free end is assumed. The
presence of the full “ g” body force term in Equation 4.2.1, usually neglected in
dynamic analyses, is again consistent with the need to consider the complete
dynamic and static field, that is, with the requirement to have the drillstring fall
freely in the limit when the formation disappears.