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Note that we consistently over-predict the permeability by about 20%.
There are several reasons for this difference. The analytical model assumes a
monochromatic frequency, whereas the Kelley source used to generate the
synthetics is a transient excitation about a given center frequency, which is taken
as the monochromatic frequency - presumably, better filtering will provide an
improved working frequency. Secondly, the present analytical model assumes
that the borehole does not contain a tool. In arriving at the above predictions,
we replaced the wellbore radius with an “effective radius” that subtracts out the
area occupied by the tool; this is an adhoc procedure. Obviously, we a different
definition of effective radius, the agreement would improve. In using the
permeability formulas, it is important that the receivers be close in proximity,
because the underlying math model assumes a homogeneous formation;
waveform differences obtained for wide separations will contain the effects of
heterogeneities which were discussed but not modeled here. Actual results from
the published patent are given in Figures 10.5a-i.
Figure 10.5a. Input 0.0001 darcy, predicted 0.0002 darcy, = 54%
(Note - top, time traces, i.e., pressure versus time; and bottom, amplitude
spectra, i.e., pressure versus frequency).
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