Sonic well logging methods and apparatus utilizing parametric inversion dispersive wave processing

A method for determining a characteristic of a formation traversed by a borehole, comprising the steps of:

• a) obtaining signals from a plurality of sonic detectors, which signals include dispersive waveform information;
• b) Fourier transforming at least a portion of said signals to obtain Fourier transformed signals;
• c) backpropagating said Fourier transformed signals utilizing a dispersion curve defined by a plurality of parameters to obtain a set of backpropagated signals
• d) modifying at least two of said plurality of parameters according to a search algorithm, and repeating said backpropagating step a plurality of times with modified parameters;
• and
• e) using sets of backpropagated signals, identifying a best-fit dispersion curve for at least one depth of the formation;
• and
• f) conducting steps a) through e) for x-directed signals and y-directed signals in order to obtain two best-fit dispersion curves for said at least one depth of the formation.

A method for determining a characteristic of a formation traversed by a borehole, comprising the steps of:

• a) obtaining signals from a plurality of sonic detectors, which signals include dispersive waveform information;
• b) Fourier transforming at least a portion of said signals to obtain Fourier transformed signals;
• c) backpropagating said Fourier transformed signals utilizing a dispersion curve defined by a plurality of parameters to obtain a set of backpropagated signals
• d) modifying at least two of said plurality of parameters according to a search algorithm, and repeating said backpropagating step a plurality of times with modified parameters;
• and
• e) using sets of backpropagated signals, identifying a best-fit dispersion curve for at least one depth of the formation, wherein
• said dispersion curve defined by a plurality of parameters is Bessel-function based.

A method of analyzing dispersive sonic wave data obtained in a borehole resulting from generated sonic signals, said method comprising the steps of:

• a) determining an initial set of parameter values for a parametric expression having a plurality of parameters and capable of fitting sonic dispersion data, thereby defining an initial dispersion curve, said plurality of parameters including at least one of (i) a slowness at an indication of first frequency and a slowness at an indication of second frequency. (ii) an indication of a kick-in frequency, and (iii) and an indication of a flexion frequency;
• b) first processing said sonic wave data utilizing said dispersion curve and comparing a result of said first processing to obtain an indication of fit;
• c) varying one or more of said parameter values, thereby defining an alternative dispersion curve;
• d) next processing said sonic wave data utilizing said alternate dispersion curve and comparing a next result of said next processing to obtain another indication of fit;
• e) repeating steps c) and d) so that said parameter values for at least two of said plurality of parameters are varied, and until a set of parameters are identified that define a dispersion curve resulting in a best fit.

A method of analyzing dispersive sonic wave data obtained in a borehole resulting from generated sonic signals, said method comprising the steps of:

• a) determining an initial set of parameter values for a parametric expression having a plurality of parameters and capable of fitting sonic dispersion data, thereby defining an initial dispersion curve;
• b) first processing said sonic wave data utilizing said dispersion curve and comparing a result of said first processing to obtain an indication of fit;
• c) varying one or more of said parameter values, thereby defining an alternative dispersion curve;
• d) next processing said sonic wave data utilizing said alternate dispersion curve and comparing a next result of said next processing to obtain another indication of fit;
• e) repeating steps c) and d) until a set of parameters are identified that define a dispersion curve resulting in a best fit;
• and
• f) conducting steps a) through e) for x-directed signals and y-directed signals in order to obtain two best-fit dispersion curves for a depth of the formation.

A method for determining a characteristic of a formation traversed by a borehole, comprising:

• a) obtaining signals from a plurality of sonic detectors, which signals include dispersive waveform information;
• b) Fourier transforming at least a portion of said signals to obtain Fourier transformed signals;
• c) backpropagating said Fourier transformed signals utilizing a plurality of dispersion curves defined by a plurality of parameters including at least two of
• (i) slowness at an indication of first frequency,
• (ii) slowness at an indication of second frequency, (iii) an indication of a kick-in frequency, and
• (iv) an indication of a flexion frequency,
• to obtain a plurality of sets of backpropagated signals;
• d) stacking said backpropagated signals of each set to provide stacked sets;
• e) using said stacked sets to determine an indication of a formation characteristic.

Apparatus for determining characteristics of a formation traversed by a borehole, comprising:

• a) a sonic source which generates substantially non-attenuating dispersive waves;
• b) a plurality of sonic receivers which detect said non-attenuating dispersive waves generated by said sonic source and generate signals in response thereto;
• and
• c) a processor which
• (i) obtains said signals,
• (ii) processes said signals utilizing an initial dispersion curve defined by a parametric expression having a plurality of parameters with an initial set of parameter values and capable of fitting sonic dispersion data to obtain a result, said plurality of parameters including at least one of (I) a slowness at an indication of first frequency and a slowness at an indication of second frequency, (II) an indication of a kick-in frequency, and (III) and an indication of a flexion frequency;
• (iii) analyzing said result to obtain an indication of fit,
• (iv) varies one or more of said parameter values, thereby defining an alternative dispersion curve,
• (v) processes said signals utilizing said alternate dispersion curve to obtain another result,
• (vi) compares said another result to obtain another indication of fit, and
• (vii) repeats (iv), (v) and (vi) so that said parameter values for at least two of said plurality of parameters are varied, and until a set of parameters are identified that define a dispersion curve resulting in a best fit.

Apparatus for determining characteristics of a formation traversed by a borehole, comprising:

• a) a sonic dipole source which generates substantially non-attenuating dispersive waves;
• b) a plurality of sonic receivers which detect said non-attenuating dispersive waves generated by said sonic source and generate signals in response thereto;
• and
• c) a processor which
• (i) obtains said signals.
• (ii) processes said signals utilizing an initial dispersion curve defined by a parametric expression with an initial set of parameter values and capable of fitting sonic dispersion data to obtain a result;
• (iii) analyzing said result to obtain an indication of fit,
• (iv) varies one or more of said parameter values, thereby defining an alternative dispersion curve.
• (v) processes said signals utilizing said alternate dispersion curve to obtain another result,
• (vi) compares said another result to obtain another indication of fit, and
• (vii) repeats (iv), (v) and (vi) until a set of parameters are identified that define a dispersion curve resulting in a best fit, wherein
• said dipole source includes an x-dipole source and a y-dipole source.