Time-reversed nonlinear acoustics for downhole pressure measurements

A method for measuring pore pressure in a formation, the method being performed by a system that includes a first transceiver disposed in a borehole, a second transceiver disposed in the borehole, one or more processors, and electronic storage, the method comprising:

• transmitting, with the first transceiver, a pulsed sinusoidal acoustic signal, the pulsed sinusoidal acoustic signal having a frequency;
• receiving, with the second transceiver, the pulsed sinusoidal acoustic signal;
• generating, with the one or more processors, a time-reversed signal by time reversing the pulsed sinusoidal acoustic signal received by the second transceiver;
• transmitting, with the second transceiver, a time-reversed acoustic signal in accordance with the time-reversed signal, whereby the time-reversed acoustic signal forms a focal volume centered on the first transceiver;
• receiving, with the first transceiver, acoustic signals within the focal volume, the acoustic signals received within the focal volume by the first transceiver including second and third harmonic signals of the frequency of the pulsed sinusoidal acoustic signal, the second and third harmonic signals having amplitudes;
• and
• determining, with the one or more processors, nonlinear elastic parameters alpha, beta, and delta of materials within the focal volume based on the amplitudes of the second and third harmonic signals of the acoustic signals, wherein the nonlinear elastic parameter alpha relates to a strength of hysteresis and the nonlinear elastic parameters beta and delta relate to acoustoelasticity.

An apparatus for measuring pore pressure in a formation, comprising:

• a first signal generator configured to generate a pulsed sinusoidal signal, the pulsed sinusoidal signal;
• a first transceiver disposed in a borehole in the formation, the first transceiver configured to receive the pulsed sinusoidal signal and to transmit a pulsed sinusoidal acoustic signal in accordance with the pulsed sinusoidal signal, the pulsed sinusoidal acoustic signal having a frequency;
• a second transceiver disposed in the borehole, the second transceiver configured to receive the pulsed sinusoidal acoustic signal and to generate a first electrical signal therefrom;
• one or more processors configured to receive the first electrical signal and to time reverse the first electrical signal to generate a time-reversed electrical signal;
• a second signal generator configured to receive the time-reversed electrical signal, and to direct the time-reversed electrical signal to the second transceiver, the second transceiver being further configured to transmit a second acoustic signal in accordance with the time-reversed electrical signal;
• whereby:
• the second acoustic signal forms a focal volume centered on the first transceiver, the second acoustic signal including second and third harmonics of the frequency of the pulsed sinusoidal acoustic signal, the first transceiver being further configured to receive the second acoustic signal and to produce a second electrical signal based the second acoustic signal, the second electrical signal reflecting amplitudes of the second and third harmonics of the second acoustic signal;
• and
• nonlinear elastic parameters alpha, beta, and delta of materials within the focal volume are determined based on the amplitudes of the second and third harmonics of the second acoustic signal, wherein the nonlinear elastic parameter alpha relates to a strength of hysteresis and the nonlinear elastic parameters beta and delta relate to acoustoelasticity.