Geology Reference
In-Depth Information
To accomplish this, we now model the desurger more precisely. We
envision the desurger as a mass-spring-damper system with a mass0 (e.g., the
bladder mass plus a time-averaged fluid mass of the partially filled volume), a
spring constant k (due to the elasticity of the bladder and the charge pressure of
the compressed gas, as compared to the standpipe pressure), and an attenuation
factor J (due to orifice losses and internal friction). The rubber membrane is
excited by the acoustic pressure p
a
= - Bu
x
(x,t), so that the ordinary differential
equation satisfied by the end value u(0,t) at the boundary x = 0 now takes the
form Mu
tt
+ J u
t
+ ku = (BSD
2
/4)u
x
(in this right-side term, having units of
force, D is an effective bladder diameter). Away from the desurger, both
incoming and outgoing waves exist, and radiation conditions do not apply.
The general solution to this boundary value problem is difficult if we
attack the partial differential equation directly. Instead, we will solve the
desurger ordinary differential equation at x = 0 exactly, and use the general
solution u(x,t) = f(ct-x) + g(ct+x), plus the Convolution Theorem, to analytically
continue the solution into the domain x > 0. This equally rigorous approach
allows us to construct the exact general solution as
u(x,t) = g(ct+x) - g(ct-x)
(6.1b)
ct-x
+ {2(BSD
2
/4)/[0c
2
(a-b)]} ³ g(V) [ae
a(ct-x-V)
- be
b(ct-x-V)
] dV
0
It is important to recognize that M, J, k, B, D, c and U
mud
do not appear
individually in the complete solution. Rather, they appear implicitly through the
lumped parameters
a = {- (Jc+(BSD
2
/4)) - [(Jc+(BSD
2
/4))
2
- 4k0c
2
] }/20c
2
(6.1c)
b = {- (Jc+(BSD
2
/4)) + [(Jc+(BSD
2
/4))
2
- 4k0c
2
] }/20c
2
(6.1d)
In the general solution for the displacement “u,” the “g(ct+x)” term represents
the known incident waveform, whereas the term second term -g(ct-x) represents
the reflection at a rigid interface, e.g., the piston faces of a positive displacement
mud pump if the desurger were not functioning (again, u = 0 at piston faces).
The last term in Equation 6.1b represents the distortion of signal due to
reflection at the desurger; this distortion consists of a phase shift and a shape
change that is again exponential in nature. In order to determine its effects
quantitatively, we give exact solutions for a class of important incident signals.
Let us consider the incident upcoming displacement wave taking the form
“A sin Z(ct+x)/c.” This function is particularly important to MWD, since a
general transient signal can always be written in terms of its harmonic Fourier
components. The exact solution corresponding to this assumption is
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