Downhole nonlinear acoustics measurements in rock formations using dynamic acoustic elasticity and time reversal

1. A method for determining nonlinear hysteretic parameters in a formation through a borehole, comprising:

generating strain in a volume surrounding the borehole by focusing a low frequency, periodic acoustic signal of a first frequency on the volume, the low frequency, periodic acoustic signal having a first amplitude at a first time and a second amplitude at a second time, wherein the first amplitude of the low frequency, periodic acoustic signal generates first strain in the volume and the second amplitude of the low frequency, periodic acoustic signal generates second strain in the volume;

transmitting pulsed, high frequency acoustic signals through the volume simultaneously with the generation of strain in the volume with the low frequency, periodic acoustic signal;

measuring signals generated in the formation in the volume relating to particle velocity or particle acceleration in the formation;

determining the strain in the volume based on the signals generated in the formation in the volume relating to the particle velocity or the particle acceleration in the formation, wherein the strain determined in the volume includes the first strain corresponding to the first amplitude of the low frequency, periodic acoustic signal and the second strain corresponding to the second amplitude of the low frequency, periodic acoustic signal;

measuring time-of-flight of the pulsed, high frequency acoustic signals through the volume as a function of strain within the volume during the generation of strain in the volume with the low frequency, periodic acoustic signal, the time-of-flight of the pulsed, high frequency acoustic signals including a first time-of-flight corresponding to the first strain and a second time-of-flight corresponding to the second strain;

determining change of the time-of-flight of the pulsed, high frequency acoustic signals as the function of the strain in the volume, the change of the time-of-flight of the pulsed, high frequency acoustic signals as the function of the strain in the volume including change from the first time-of-flight corresponding to the first strain to the second time-of-flight corresponding to the second strain; and

determining nonlinear hysteretic parameters α, β, and δ based on the change of the time-of-flight of the pulsed, high frequency acoustic signals as the function of the strain in the volume, wherein the nonlinear hysteretic parameter α relates to strength of hysteresis, the nonlinear hysteretic parameters β, and δ are combinations of third- and fourth-order elastic constants representing acoustoelasticity, and pore pressure in the volume is determined based on the nonlinear hysteretic parameters α, β, and δ.

2. The method of claim 1 , wherein said borehole comprises an open borehole.

3. The method of claim 1 , wherein the low frequency, focused periodic acoustic signal is generated using time reversal.

4. The method of claim 1 , wherein the low frequency, focused periodic acoustic signal is generated using phased arrays of acoustic sources.

5. The method of claim 1 , wherein the high frequency, pulsed acoustic signals are generated in the borehole.

6. The method of claim 1 , wherein the low frequency, focused periodic acoustic signal is generated in the borehole.

7. The method of claim 1 , wherein the particle velocity or the particle acceleration is measured using a non-contact vibrometer or a contact accelerometer.

8. The method of claim 1 , wherein the low frequency periodic acoustic signal is between 1 Hz and 1000 Hz.

9. The method of claim 1 , wherein the high frequency acoustic signal has a frequency between 200 kHz and 1.5 MHz.

10. The method of claim 1 , wherein the hysteresis of the nonlinear hysteretic parameters α, β, and δ as the function of strain in the volume is determined.

11. The method of claim 10 , wherein the pore pressure in the volume is determined further based on the hysteresis of the nonlinear hysteretic parameters α, β, and δ thereof as the function of strain in the volume.

12. The method of claim 11 , further comprising the step of calculating fracture failure of said borehole.

13. The method of claim 11 , further comprising the step of calculating collapse failure of said borehole.

14. The method of claim 11 , further comprising the step of calculating oil recovery performance of said formation.

15. The method of claim 1 , wherein determination of the change of the time-of-flight of the pulsed, high frequency acoustic signals includes determination of a ratio of change in perturbed velocity to linear velocity, and the nonlinear hysteretic parameter α is determined based on the ratio of change in perturbed velocity to linear velocity.

16. An apparatus configured to determine nonlinear hysteretic parameters in a formation through a borehole, comprising:

a transceiver trained to focus time-reversed acoustic signals in a focal volume centered on said borehole and generate strain in the focal volume, wherein a first strain at a first amplitude is generated in the focal volume at a first time and a second strain at a second amplitude is generated in the focal volume at a second time;

a probe source comprising a transmitting transducer configured to transmit high frequency acoustic pulses into the focal volume while the transceiver focuses the time-reversed acoustic signals in the focal volume;

a receiver comprising a receiving transducer configured to receive, from the focal volume, the high frequency acoustic pulses transmitted by the probe source;

a signal processor configured to determine time-of-flight of the received high frequency acoustic pulses; and

a sensor disposed in the borehole configured to generate signals conveying particle velocity or particle acceleration;

wherein:

the strain in the focal volume is determined based on the signals conveying the particle velocity or the particle acceleration, wherein the strain determined in the focal volume includes the first strain corresponding to the first amplitude at the first time and the second strain corresponding to the second amplitude at the second time;

the time-of-flight of the received high frequency acoustic pulses include a first time-of-flight corresponding to the first strain and a second time-of-flight corresponding to the second strain;

change of the time-of-flight of the received high frequency acoustic pulses as the function of the strain in the focal volume is determined, the change of the time-of-flight of the received high frequency acoustic pulses as the function of the strain in the focal volume including change from the first time-of-flight corresponding to the first strain to the second time-of-flight corresponding to the second strain;

nonlinear hysteretic parameters α, β, and δ are determined based on the change of the time-of-flight of the received high frequency acoustic pulses as the function of the strain in the focal volume, wherein the nonlinear hysteretic parameter α relates to strength of hysteresis, the nonlinear hysteretic parameters β, and δ are combinations of third- and fourth-order elastic constants representing acoustoelasticity; and

pore pressure in the focal volume is determined based on the nonlinear hysteretic parameters α, β, and δ.

17. The apparatus of claim 16 , wherein the high frequency acoustic pulses have a frequency between 200 kHz and 1.5 MHz.

18. The apparatus of claim 16 , wherein said borehole comprises an open borehole.

19. The apparatus of claim 16 , wherein said probe source is disposed within the borehole.

20. The apparatus of claim 16 , wherein said receiver is disposed within the borehole.

21. The apparatus of claim 16 , wherein determination of the change of the time-of-flight of the received high frequency acoustic pulses includes determination of a ratio of change in perturbed velocity to linear velocity, and the nonlinear hysteretic parameter α is determined based on the ratio of change in perturbed velocity to linear velocity.