Geology Reference
In-Depth Information
The left hand side of Equation 4.2.10 is a force; in particular, note AE du s (x)/dx
= -AE{Mg/AE + gL/E} or -Mg -A gL < 0 at the bit x = 0, while du s (x)/dx
= -Mg < 0 at the surface x = L (the -A gL arises from drillstring weight). Our
convention assumes that AEu x < 0 for compression, with AEu x > 0 for tension;
here, the negative signs indicate that the drillstring is entirely in compression.
4.2.3.2 Static weight on bit.
In reality, the weight “-Mg-A gL” is not completely applied at the drillbit,
since the overwhelming force would precipitate drillbit bearing damage and
excessive torques required for rotation. Instead, a tensile force P > 0 is applied
at the surface that reduces compressive loads everywhere. In other words, the
term “-Mg -A gL” is reduced, or rendered less negative, by the addition of a
positive P, so that Equation 4.2.10 is replaced by
AE du s (x)/dx = A gx + {P - (M +A L)g} (4.2.11)
The force AE du s (x)/dx at the surface x = L is now P-Mg, while the
corresponding force AE du s (0)/dx at the drillbit x = 0 is now the “weight-on-bit”
P - (M +A L)g. For our purposes, the transition point where the sign of the
axial stress reverses is the “neutral point” defined by setting the right side of
Equation 4.2.11 to zero, that is,
x neutral point = - { P- ( M +A L)g}/A g (4.2.12)
Other definitions are found in the drilling literature, but Equation 4.2.12
complements our objectives later on. Also, it will be convenient to define
W 0 = P- (M +A L)g
(4.2.13a)
so that
AE du s (x)/dx = A gx + W 0
(4.2.13b)
where W 0 is the negative of the static weight-on-bit WOB.
Again, the weight-on-bit in Equation 4.2.13a is not necessarily the value
observed because Equation 4.2.5 was selected to simplify Equation 4.2.4. In
doing so, we ignored the possibility that external forces exist which contain
static components of friction (such forces will need to be included in Equation
4.2.6). These would alter the force balance as implied by Equation 4.2.13a,
which does not contain distributed borehole wall friction or concentrated forces.
Equation 4.2.5 was chosen to simplify the mathematics. This static WOB
is not the dynamic value measured at the bit, which may vary from highly
compressive to zero when the drillbit bounces off the formation. Finally,
Equation 4.2.13b, under these restrictive conditions, can be integrated to give
u s (x) = 1/2 gx 2 /E + W 0 x/AE + C 2
=1 / 2
gx 2 /E + W 0 x/AE
(4.2.14)
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