Geology Reference
In-Depth Information
u(x,t) = A sin {Z(ct+x)/c} - A sin {Z(ct-x)/c} (6.1e)
+2A(BSD
2
/4)b{b sin (Z(ct-x)/c) + (Z/c) cos (Z(ct-x)/c)}/{0c
2
(a-b)(b
2
+Z
2
/c
2
)}
- 2A(BSD
2
/4)a{a sin (Z(ct-x)/c) + (Z/c) cos (Z(ct-x)/c)}/{0c
2
(a-b)(a
2
+Z
2
/c
2
)}
+{2A(BSD
2
/4)Z/[0c
3
(a-b)]}{ae
a(ct-x)
/(a
2
+Z
2
/c
2
) - be
b(ct-x)
/(b
2
+Z
2
/c
2
)}
The first line, again, represents the incoming wave and its solid wall
reflection, where A is the signal amplitude corresponding to the fluid
displacement “A sin {Z(ct+x)/c}” leaving the pulser. The second and third lines
represent the phase distortion (of the sinusoidal signal) introduced by the
desurger, while the fourth line shows that the desurger produces a non-
sinusoidal distortion that will contain exponential smearing.
A more detailed examination of Equation 6.1e indicates that these
distortion effects vanish in the limit of high frequency Z, because “the waves do
not have sufficient time” to act on the system, especially in the limit of a “heavy
desurger” with large 0or a “small amplitude A.” How high a frequency is
needed so that desurger distortion is not important? Apparently, the carrier
frequency of 12 Hz used in present mud siren operations is high enough - thus
higher frequency operations should be safer insofar as signal distortions are
concerned. Of course, the mathematical structure of the terms in Equation 6.1e
reveals the complicated nature of the dimensionless parameters controlling the
physical problem.
In electric engineering, the resistance, capacitance, and inductance values
of the circuit elements determine the non-dimensional time scales associated
with the particular circuit. Analogies exist here. The physical time scales “ac”
and “bc” inferred from the exponential terms above define two of the time scales
important to the problem; the third is the period 1/Z. The mud density U
mud
does not explicitly appear in our solutions, but the dependence can be easily
recovered if we note that c
2
= B/U
mud
. This leads to B/c
2
= U
mud
. If the B/c
2
coefficient multiplying the integral is replaced instead by U
mud
, the distortion is
seen to be proportional to mud density, or more precisely, a ratio that depends
on U
mud
./M, among other quantities. The dependence is complicated by the
appearance of the same variables in “a” and “b.” As noted above, the distortion
appears to increase with increasing mud weight. All quantities being equal, the
distortion is largest for low frequencies Z.
Interestingly, in recent mud flow loop experiments, measured pressure
signal levels and shapes in the 1-2 Hz range were particularly affected by the
desurger, whereas for 5 Hz and above, the effect of the desurger was
insignificant. In wind tunnel experiments designed to understand near-static
pressure response, manometer-based results were noticeably affected by fluid
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