Geology Reference
In-Depth Information
5.2 Method 5-2. Problems with collar-drillpipe area
discontinuity, with drillbit assumed as closed end, solid
drillbit reflector
(software reference, collar-pipe-closed-*.for)
.
5.2.1 Theory.
Figure 5.1a from Method 5-1 applies to this problem, however, the drillbit
is assumed as a solid reflector in the present model. If the drillbit can be
modeled as a solid reflector, e.g., if nozzle areas are truly small or if very hard
rock is being drilled, then the acoustic displacement satisfies u
1
(0,t) = 0 at the
bit. This requires that u
1
(x,t) = h(t - x/c) - h(t + x/c) which vanishes at x = 0.
Minor changes to the analysis of Method 5-1 lead to the three applications
formulas below -
'p(t) - 'p(t - 2L
m
/c) =
= (A
p
/A
c
+1) p
3
(t - L
m
/c) + (A
p
/A
c
- 1) p
3
(t - L
m
/c - 2L
c
/c)
(5.2.1)
p
pipe
(t) + {(A
p
-A
c
)/(A
p
+A
c
)} P
pipe
(t - 2L
c
/c) =
= {- 'p(t - L
m
/c - L
c
/c) + 'p(t + L
m
/c - L
c
/c)}/(A
p
/A
c
+1)
(5.2.2)
p
surface
(t) + {(A
p
-A
c
)/(A
p
+A
c
)} p
surface
(t - 2L
c
/c) =
= {- 'p(t - L
m
/c - L/c) + 'p(t + L
m
/c - L/c)}/(A
p
/A
c
+1) (5.2.3)
Identical cases to those performed in Method 5-1 were run. Refer to Method 5-1
for a description of the 'p(t) functions assumed. Excellent recovery is achieved
in all instances, that is, black and green curves identical.
5.2.2
Run 1. Phase-shift-keying, 12 Hz carrier wave.
Figure 5.2a
. Phase-shift-keying, 12 Hz carrier wave.
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