Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 132

where C 2 vanishes because the drillbit is rigidly fixed to the bottom x = 0 at time
t = 0 (that is, we assume that u s (0) = 0). Since u(0,t) = u s (0) + u d (0,t) = u d (0,t),
the u d (0,t) computed later is the exact static plus dynamic position of the bit.
Recapitulation. Let us summarize our discussion thus far. If the effects of
borehole wall friction and contact are disregarded in the static model, then the
static axial force distribution is given by

AE du s (x)/dx = A gx + {P-(M +A L)g} = A gx - WOB (4.2.15)

Dynamic oscillations superimposed on this static field satisfy

2 u d (x,t)/ t 2 +

u d (x,t)/ t - E 2 u d (x,t)/ x 2 = F e (4.2.16)

Having explained the conventional separation of static and dynamic forces, we
now consider the solutions to some earlier dynamical models.
Equation 4.2.16 governs u d (x,t). It is the classical damped wave equation
considered in Chapter 1, and it must be solved subject to initial and boundary
conditions. Different types of auxiliary conditions are possible. For example,
the initial conditions u d (x,0) = 0 and u d /(x,0)/ t = 0 describe a drillstring at rest
at t = 0. The boundary condition u(0,t) = A sin t at the bit x = 0, not used in
this topic, describes prescribed sinusoidal displacements with amplitude A and
frequency , while u d (0,t)/ x = 0 models a free-end during bit bounce. We
later show why the u d (x,t)/ t in Equation 4.2.16, and a similar term in the
surface condition, represent damping (dissipation in the draw works and power
swivel is small). This crude frequency-independent model (Chapters 1 and 2)
accounts for energy losses due to drillstring movement, formation and mud
attenuation, internal viscous losses in the metal, and sound radiation, if
significant. Damping and power transfer at the bit are accounted for directly
through rock-bit interaction, which we model through an impedance condition.

4.2.4 Boundary conditions - old and new ideas.
We will discuss three topics: surface boundary conditions, rock-bit
interactions, and drillbit kinematic modeling. The first is straightforward; the
latter two, however, require some development. In each case, we describe
existing methods, critique them, and extend them to remove any limitations.

4.2.4.1 Surface boundary conditions.
At the surface, the derrick and draw works are usually modeled by a mass-
spring-damper system, this assumption resulting in the boundary condition
M

2 u/ t 2 +

u/ t + AE u/ x + ku + Mg = 0 (4.2.17)