Borehole Acoustics –New Approaches to Old Problems - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 377

x 2 (t)

d (t)/dt = E(x,t)/ t dx + E(x 2 ,t) dx 2 /dt - E(x 1 ,t) dx 1 /dt

x 1 (t)
x 2 (t)

= { E(x,t)/ t + {dx/dt E(x,t)}/ x dx (10.21)

x 1 (t)
We next invoke Equation 10.11, which states that E/ t + (c g E)/ x = 2 i E. If
we identify the velocity dx/dt in Equation 10.21 with the group velocity c g here,
we can replace the integrand of Equation 10.21 with 2 i (k) E. Note that,
because k is a constant, then so is i (k). The imaginary frequency can be
moved across the integrand, leaving leaving an anticipated

i

d (t)/dt = 2 (t) (10.22)

or i = ( d

(t)/ ) /(2 dt) (10.23)

Now , “d (t)/ ” represents the fractional change in total wave energy following a
wave packet, and can be determined (from amplitude spectral data taken over
the entire length of the waveform from measurements) at successive receivers,
where dt is the corresponding transit time. Thus i can be calculated.
Approach 1. Since i is now available, Equation 10.7 can be used to
solve for the permeability when the remaining parameters are known.
Actually, any one of the parameters on the right-side of Equation 10.7 can be
calculated when the remainder are known.
Approach 2. The formula

r (k) = Vk - k ½ from Equation 10.3 can be

approximated by 2 f 0 Vk where f 0 is the center frequency in Hertz. Hence,

we obtain
d (t)/dt =

i

2 (t)

k 1/2

= - 2 (t)

(2 f 0 /V) 1/2

= - 2 (t) (10.24)

so that, on solving for , we find
= -1/2 (V/2 f 0 ) 1/2 (t) -1 d (t)/dt
= -1/2 (V/2 f 0 ) 1/2 t -1 d (t)/ (t) (10.25)
This value for can be used in Equation 10.2b for the required solution for the
permeability k.