Geology Reference
In-Depth Information
hard and soft formations. In general, one might postulate a relationship
connecting the penetration depth u(0,t) to the instantaneous rate-of-penetration
u
t
(0,t) and the local strain u
x
(0,t),
G( u,u
x
,u
t
) = 0 (4.2.21)
which may be linear or nonlinear. The particular form that G(u,u
x
,u
t
) = 0 takes
will depend on many factors, e.g., the type of drillbit used, the kind of rock
being drilled, and intermediate conditions such as bit and bearing wear. Time-
dependence might be included if mud invasion beneath the bit affects formation
hardness. Azimuthal -dependence may be useful (once axial and torsional
analyses are coupled) to model formation dip, bedding plane or anisotropic rock
effects. Models represented by Equation 4.2.21 are impedance boundary
conditions; the impedance relationship G(u,u
x
,u
t
) = 0 is a rock-bit interaction
model. We will discuss possibilities for Equation 4.2.21 later, but for numerical
solutions, few restrictions on functional form need to be imposed (Fortran
if-
then
statements can be used). For analytical solutions, we can take
u
x
+ u
t
+ u = 0 (4.2.22)
which yields to linear analysis, modeling the relative effects of instantaneous
elastic rock displacement, impact velocity, and dynamic strain.
From Chapter 1, we know that Equation 4.2.22 can promote interactions
between AC and DC solutions; thus, there are optimistic reasons to believe that
penetrating values
of
bit elevation
u(0,t) can be found for certain values of , ,
and : solutions for u(0,t) are
not
likely to be purely sinusoidal. But Equation
4.2.22 is not complete, since it does not apply when the drillstring bounces
upward. Thus, we will refine our model, invoking Equation 4.2.22
only
when
the velocity u(0,t)/ t < 0 is downward; however, we insist that
u(0,t)/ x = 0,
if
u(0,t)/ t > 0
(4.2.23)
This represents a stress-free end when bouncing off-bottom; the logic tests,
for example, can be carried out using previous time step values of u(0,t) in a
numerical scheme. The boundary conditions in Equations 4.2.22 and 4.2.23 are
applied at x = 0 as shown in Figure 4.2.3. Henceforth, the statement in Equation
4.2.21 should be interpreted as including Equation 4.2.23.
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