Borehole Acoustics –New Approaches to Old Problems - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 378

10.2.4 Permeability prediction from phase considerations.
Approach 3. The foregoing approach requires direct measurement of
amplitudes and filtering of non-Stoneley waves. Permeabilities can be obtained
from still another approach. A phase velocity based approach, which is
potentially simpler, was also developed for evaluation. Let us now turn to
Equation 10.3, for which

r (k) = Vk - k 1/2 . This leads to c p = r (k)/k = V -

k -1/2 . If we substitute Equation 10.2b, assume that 2

f 0 Vk and solve for

permeability, we have the closed form expression

={4 f 0 (V-c p ) 2 R w 2 /(

mud 2 V 6 )} oil c oil 2

{

oil }

(10.26)

where c oil 2 is the sound speed of the “oil” or (any) pore fluid and c p is assumed
to be known. This formula again predicts permeability well, when noise-free
synthetic seismograms are used to generate Stoneley data for evaluation.
10.2.5 Example permeability predictions.
The theory and results presented in this chapter for Stoneley waves were
developed from Chin (1980) and from United States Patent No. 6,327,538,
entitled “Method and Apparatus for Evaluating Stoneley Waves, and for
Determining Formation Parameters in Response Thereto,” awarded to the author
on December 4, 2001. The predictions given below were reported in the patent
and utilized “exact” Stoneley synthetic data created from an independently
written computer program developed by the Full Waveform Acoustics Logging
Consortium at M.I.T. The procedure is extremely simple and stable. In the
following table, “Actual perm” is the assumed permeability used to generate
synthetic time domain waveforms with the MIT “black box” computer program.
The assumed center frequency of 5,000 Hertz is slightly outside the range of
validity of the analytical model; also, parameters for a “typical mud” and a
“typical rock” were assumed, with only the permeability changing from one
example to the next. This software is, importantly, completely independent of
the formulas developed here. “Predicted kdc” is the predicted permeability
using the present results. As is seen, the agreement is good over six orders of
magnitude and satisfactory also at the extremes.

Actual perm = .0001 darcy, Predicted kdc= .0002 darcy, % error = 54.5424
Actual perm = .0010 darcy, Predicted kdc= .0013 darcy, % error = 30.4819
Actual perm = .0100 darcy, Predicted kdc= .0122 darcy, % error = 22.4089
Actual perm = .1000 darcy, Predicted kdc= .1209 darcy, % error = 20.9475
Actual perm = .5000 darcy, Predicted kdc= .6127 darcy, % error = 22.5442
Actual perm = 1.0000 darcy, Predicted kdc= 1.2414 darcy, % error = 24.1356
Actual perm = 2.0000 darcy, Predicted kdc= 2.4427 darcy, % error = 22.1325
Actual perm = 5.0000 darcy, Predicted kdc= 4.8649 darcy, % error = 2.7024
Actual perm = 10.0000 darcy, Predicted kdc= 5.7577 darcy, % error = 42.4226
Figure 10.4. Permeability predictions.