Geology Reference
In-Depth Information
1.8.3 Example 1-17. Splitting of an applied initial disturbance.
Here, we consider an
infinite
pipe containing an MWD pulser that creates a
“delta-p” signal at the valve x = 0. Obviously, half of the signal will propagate
in one direction, with the remaining half in the opposite direction. This splitting
also applies in
finite
conduits initially, since the effects of boundaries are not
immediately felt. When terminations such as drill bits and mudpumps are
present, reflections will complicate the information contained in the “delta-p”
signal. Later in this topic, we show how a wave equation
2
u/ t
2
- c
2 2
u/ x
2
= 0 (1.169)
for the fluid displacement u(x,t) applies, where c is the sound speed. Here, the
acoustic pressure satisfies p = - B u/ x, where B is the bulk modulus.
Let us consider disturbances periodic in time with a frequency
. For such
excitations, we introduce the separation of variables
u
1
(x,t) = X
1
(x) e
i t
, x < 0 (1.170a)
u
2
(x,t) = X
2
(x) e
i t
, x > 0 (1.170b)
Substitution in Equation 1.169 leads to the unidirectional
Helmholtz equations
d
2
X
1
(x)/dx
2
+ (
2
/c
2
) X
1
(x) = 0
(1.171a)
d
2
X
2
(x)/dx
2
+ (
2
/c
2
) X
2
(x) = 0
(1.171a)
Their solutions may be taken in the complex exponential forms
X
1
(x) = C
1
e
i
x/c
+ C
2
e
-i
x/c
(1.172a)
X
2
(x) = C
3
e
i
x/c
+ C
4
e
-i
x/c
(1.172b)
Thus,
u
1
(x,t) = C
1
e
i( x/c + t)
+ C
2
e
i(- x/c + t)
(1.173a)
u
2
(x,t) = C
3
e
i( x/c + t)
+ C
4
e
i(- x/c + t)
(1.173b)
The exponentials were chosen to facilitate the use of directional filtering or
radiation conditions. Since the x < 0 and x > 0 solutions for u(x,t) must be left
and right-going, respectively, it is clear that C
2
= C
3
= 0. In determining the
remaining C
1
and C
4
, we require that the fluid remain continuous at x = 0,
u
1
(0,t) = u
2
(0,t)
(1.174)
Now, the mud pulser applies its “delta-p” signal strength P
s
, with
B u
1
(0,t)/ x - B u
2
(0,t)/ x = P
s
e
i t
(1.175)
Substitution of Equations 1.173a,b into Equations 1.174 and 1.175 leads to
u
1
(x,t) = -(icP
s
/2
B) e
i
(t +x/c)
(1.176)
u
2
(x,t) = -(icP
s
/2
B) e
i
(t -x/c)
(1.177)
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