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

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.