Time-reversed nonlinear acoustics for downhole pressure measurements

Abstract

Apparatus (10) and methods for combining time reversal and elastic nonlinearity of formation materials for qualtitatively probing for over-pressured regions down hole in advance of a well drilling bit, to determine the distance to the over-pressured region, and for accurately measuring pore pressure downhole in a formation, are described. Classical and reciprocal time reversal methods may be utilized to achieve these measurements.

Claims

1. An apparatus for detecting regions of high pressure in a formation ahead of a down-hole well drilling bit, comprising: a first signal generator that generates a pulsed sinusoidal acoustic signal, the pulsed sinusoidal acoustic signal having a frequency; a first transceiver disposed downhole above said drilling bit and being configured to transmit the pulsed sinusoidal acoustic signal; a second transceiver disposed downhole, the second transceiver being configured to receive the pulsed sinusoidal acoustic signal transmitted by the first transceiver, and to produce 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, to generate a second acoustic signal therefrom, and to direct the second acoustic signal onto the second transceiver, whereby the second acoustic signal is transmitted by the second transceiver; whereby: the second acoustic signal forms a focal volume centered on the first transceiver, the second acoustic signal including harmonic signals of the frequency of the pulsed sinusoidal acoustic signal, the first transceiver being configured to receive the second acoustic signal and produce a second electrical signal reflecting amplitudes of the harmonic signals 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 harmonic signals 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.

2. The apparatus of claim 1, wherein pore pressure in the formation is determined based on the nonlinear elastic parameters alpha, beta, and delta.

3. 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.

4. The method of claim 3, further comprising the step of determining, with the one or more processors, the pore pressure within the formation based on the nonlinear elastic parameters alpha, beta, and delta.

5. The method of claim 3, further comprising the step of varying the frequency of the pulsed sinusoidal acoustic signal to vary the size of the focal volume formed by the time-reversed acoustic signal.

6. 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.

7. The apparatus of claim 6, within the one or more processors are further configured to determine the pore pressure within the formation based on the nonlinear elastic parameters alpha, beta, and delta.

8. The apparatus of claim 6, wherein the frequency of the pulsed sinusoidal acoustic signal is varied to vary the size of the focal volume formed by the second acoustic signal.

9. The apparatus of claim 1, wherein the nonlinear elastic parameter alpha is determined based on an amplitude of a third harmonic signal of the second acoustic signal.

10. The apparatus of claim 1, wherein the nonlinear elastic parameter beta is determined based on an amplitude of a second harmonic signal of the second acoustic signal.

11. The apparatus of claim 1, wherein the nonlinear elastic parameter delta is determined based on an amplitude of a third harmonic signal of the second acoustic signal.

12. The method of claim 3, wherein the nonlinear elastic parameter alpha is determined based on the amplitude of the third harmonic signal of the acoustic signals.

13. The method of claim 3, wherein the nonlinear elastic parameter beta is determined based on the amplitude of the second harmonic signal of the acoustic signals.

14. The method of claim 3, wherein the nonlinear elastic parameter delta is determined based on the amplitude of the third harmonic signal of the acoustic signals.

15. The apparatus of claim 6, wherein the nonlinear elastic parameter alpha is determined based on the amplitude of the third harmonic of the second acoustic signal.

16. The apparatus of claim 6, wherein the nonlinear elastic parameter beta is determined based on the amplitude of the second harmonic of the second acoustic signal.

17. The apparatus of claim 6, wherein the nonlinear elastic parameter delta is determined based on the amplitude of the third harmonic of the second acoustic signal.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The accompanying drawings, which are incorporated in and form a part of the specification, illustrate the embodiments of the present invention and, together with the description, serve to explain the principles of the invention. In the drawings:

(2) FIGS. 1A and 1B provide graphs of the nonlinear parameters β and α respectively, as a function of pore pressure.

(3) FIG. 2A is a schematic representation of an embodiment of the present apparatus for practicing look-ahead drilling based on Time Reversal Nonlinear Elastic Wave Spectroscopy, illustrating an acoustical source located above, and in the vicinity of a drill bit transmitting an acoustic wave that is swept in frequency as a function of time, the signals propagating both into the well bore and into the surrounding formation, and received and recorded by transceivers disposed in the wellbore on a transceiver mount disposed near the drilling apparatus, while FIG. 2B illustrates the signals detected by the transceivers being digitally reversed-in-time by microprocessors located at the transceivers or at the surface, amplified, and rebroadcast into the bore hole and into the surrounding formation, and focusing the space distribution of elastic wave energy in the region of the original acoustic source.

(4) FIG. 3A is a schematic representation of an embodiment of the transceiver mount illustrated in FIGS. 2A and 2B showing a possible arrangement for the low-frequency transceivers, and FIGS. 3B and 3C show a possible arrangement for the mid-frequency and high-frequency transceivers, respectively, while FIG. 3D illustrates a transceiver mount having a transceiver at the focus of the other transceivers for use in completed holes.

(5) FIG. 4A illustrates the forward or calibration steps to achieve the classical time reversed process using the apparatus of FIGS. 1A and 2B, and 3, while FIG. 4B illustrates reverse or focusing steps for the time reversed process.

(6) FIG. 5 illustrates the ‘training’ process for reciprocal time reversal, where a predefined calibration signal is sequentially imposed on transceivers, received by a transceiver at the focal position, and stored for use for time reversal processing once the stored signals are time reversed.

(7) FIG. 6 is a schematic representation of a laboratory apparatus for carrying out reciprocal time reversal measurements for a simulated down hole environment.

(8) FIG. 7A is a graph of the detected pulse propagation down the pipe in the apparatus shown in FIG. 6, hereof, from a single transducer without time reversal, FIG. 7B is a graph of the detected pulse propagation from all 10 transducers without time reversal, FIG. 7C is a graph of the detected focused pulse propagation down the pipe from a single transducer using classical time reversal, while FIG. 7D is a graph of the detected focused pulse propagation from all 10 transducers using classical time reversal.

(9) FIG. 8A is a graph of the delay in the arrival time of a detected pulse in the pipe of the apparatus of FIG. 6 as the amplitude of the signal pulse increases from (a) to (c), while FIG. 8B is a graph of the detected pulse as a function of frequency as the amplitude of the signal pulse increases from (a) to (c).

(10) FIG. 9 is a graph of the pore pressure as a function of depth measured by conventional techniques for an actual formation, permitting the prediction of gas/water contacts for locating hydrocarbon deposits.

(11) FIG. 10 is a schematic representation of an embodiment of the apparatus used to determine α, and to verify that the determined value of α is meaningful.

(12) FIG. 11A is a graph of the time dependence of the particle velocity, while FIG. 11B is an expanded time period between 0.32 ms and 0.37 ms, both being measured at 500 mm from the source of vibrational excitation for 20 source amplitudes.

(13) FIG. 12A is a graph of the relative change in elasticity as a function of strain estimated from the propagation of a pulse centered at 22.4 kHz and measured 500 mm from the source at 20 source amplitudes, while FIG. 12B shows a sample applied pulse, the “star”, “circle” and “plus” symbols corresponding to the first five peaks the 9.sup.th-21.sup.st peak, and the final five peaks, respectively.

(14) FIG. 13A is a graph of the vibrational spectra measured on a sample of Berea sandstone at 22 source amplitudes, the first 12 modes of longitudinal vibration L.sub.1 through L.sub.12 being indicated, the star symbols denoting the location of the resonances for all source amplitudes, while FIG. 13B clearly shows the shift in frequency for L.sub.10 as an example.

(15) FIG. 14 is a graph illustrating the relative frequency shift as a function of the strain component ε.sub.xx for the longitudinal modes L.sub.4 through L.sub.12.

DETAILED DESCRIPTION OF THE INVENTION

(16) As a result of the need for accurate pore pressure prediction for drilling operators to reduce borehole trouble time and avoid drilling incidents, oil companies and oil service companies have been seeking methods for detecting high pressures ahead drilling bits as they penetrate the earth, such that corrective action can be taken before the region is breached.

(17) Overpressure rock has a signature elastic response that can be detected by combining Time Reversal techniques with Elastic Nonlinearity in a technique which is known as Time Reversal Nonlinear Elastic Wave Spectroscopy (TR NEWS). The nonlinear elastic wave response is directly related to the effective pressure (hydrostatic load minus the pore pressure). Time reversal is a method for focusing acoustic waves such that large wave amplitudes are obtained in a localized region of space. As a result of the large acoustic wave amplitudes at the focus and the nonlinearity of the material, harmonics may be generated (and sum and difference frequencies if two waves are present). These harmonic frequencies are detected at the focus and, as will be discussed in more detail below, changes in the amplitude of the detected harmonics indicate that high pressure may be present.

(18) Nonlinear materials exhibit a nonlinear stress-strain relation which can be probed by acoustic waves, leading to pressure-specific acoustic signatures. Harmonics of the incident acoustic frequencies are created when the acoustic waves are focused. The effective pressure in a formation may be written as,
P.sub.eff=σ−bP  (1)
where α is the confining pressure, P is the pore pressure, and b is the Biot coefficient (typically 0.4-0.9 in rock). The effective pressure can also be described by a nonlinear stress-strain relationship,

(19) $\begin{array}{cc}{P}_{\mathrm{eff}}=K\left[1-\mathrm{\beta .Math.}-{\mathrm{\delta .Math.}}^2\right]\mathrm{.Math.}+K\frac{\alpha }{2}\left[\left({\left(\mathrm{\Delta .Math.}\right)}^2-{\mathrm{.Math.}}^2\right)\mathrm{sign}\left(\stackrel{.}{\mathrm{.Math.}}\right)-2\left(\mathrm{\Delta .Math.}\right)\mathrm{.Math.}\right]& \left(2\right)\end{array}$

(20) where K is the linear stiffness constant, ε is the strain, Δε is the strain amplitude, {dot over (ε)} denotes the partial derivative with respect to time, sign is a function returning the sign (positive or negative) of the argument, β and δ are combinations of third- and fourth-order elastic constants representing the acoustoelasticity (quadratic and cubic classical nonlinearity), and the parameter α relates to the strength of the hysteresis, according to the Preisach-Mayergoyz model of elasticity. See, e.g., K. R. McCall et al., “A new theoretical paradigm to describe hysteresis, discrete memory and nonlinear elastic wave propagation in rock,” Nonlin. Proc. Geophys. 3, 89-101 (1996), R. A. Guyer et al., “Quantitative implementation of Preisach-Mayergoyz space to find static and dynamic elastic moduli in rock,” J. Geophys. Res. 102(B3), 5281-5293 (1997), and G. Douglas Meegan, Jr. et al., “Observations Of Nonlinear Elastic Wave Behavior In Sandstone,” J. Acoust. Soc. Am. 94, (1993) 3387-3391. Combining Eqs. (1) and (2) leads to an expression of the pore pressure as a function of confining pressure and nonlinear elastic parameters of the material,

(21) $\begin{array}{cc}P=\frac{1}{b}\left(\sigma -K\left[1-\mathrm{\beta .Math.}-{\mathrm{\delta .Math.}}^2\right]\mathrm{.Math.}+K\frac{\alpha }{2}\left[\left({\left(\mathrm{\Delta .Math.}\right)}^2-{\mathrm{.Math.}}^2\right)\mathrm{sign}\left(\stackrel{.}{\mathrm{.Math.}}\right)-2\left(\mathrm{\Delta .Math.}\right)\mathrm{.Math.}\right]\right).& \left(3\right)\end{array}$
The parameters α, β, and δ may be obtained from the time reversal signal, with α being obtained from the velocity change of the focused signal as a function of strain amplitude. The velocity change may be also measured using cross correlation or another standard technique on a low amplitude (linear) wave at the time reversal focus, and the progressive delays caused by using progressively larger amplitude excitation waves. Cross correlation is a commonly applied method for measuring time delays between a reference signal and a signal that has experienced a velocity change. β is obtained from the amplitude dependence of the second harmonic of a pulsed pure sinusoid or the amplitude dependence of sum (ω.sub.1+ω.sub.2) and difference (ω.sub.1−ω.sub.2) frequencies if two waves are employed. See, also, TenCate, J. A. et al. (1996) “Laboratory Study Of Linear And Nonlinear Elastic Pulse Propagation In Sandstone,” J. Acoust. Soc. Am. 100(3), 1383-1391. δ is obtained from the amplitude dependence of the third harmonic of the fundamental drive amplitude at small, but still nonlinear amplitudes and, in general, can be ignored. At larger amplitudes, however, α dominates and δ becomes overwhelmed and can be ignored.

(22) α is given by:

(23) $\begin{array}{cc}\alpha =\frac{\Delta C}{C_0}\frac{1}{\mathrm{.Math.}}\mathrm{.Math.}=\frac{\stackrel{\mathrm{.Math.}}{u}}{2\pi {\mathrm{fC}}_0},& \left(4\right)\end{array}$
where C.sub.0 is the linear velocity and C the perturbed velocity. The second derivative of u with respect to t is the particle acceleration measured in the frequency domain, f is the wave fundamental frequency, and ε is the strain measured at frequency fin the focal region as the signal source amplitude is increased. By plotting the change in wave speed as a function of strain, alpha can be obtained.

(24) Alternatively, alpha can be obtained from the third harmonic amplitude also when wave amplitudes are large. In the following alpha, beta and delta are shown.

(25) $\begin{array}{cc}\alpha =\frac{c_0^2}{L}\frac{{\stackrel{\mathrm{.Math.}}{u}}_{3f}}{{\stackrel{\mathrm{.Math.}}{u}}_{1f}^2}\beta =\frac{c_0^2}{L}\frac{{\stackrel{\mathrm{.Math.}}{u}}_{2f}}{{\stackrel{\mathrm{.Math.}}{u}}_{1f}^2}\delta =\frac{\omega c_0^3}{L}\frac{{\stackrel{\mathrm{.Math.}}{u}}_{3f}}{{\stackrel{\mathrm{.Math.}}{u}}_{1f}^3}& \left(5\right)\end{array}$
where L is the wavelength of the fundamental frequency divided by two, equivalent to the radius of the focal region, the second derivative of u with respect to time, 3f, is the third harmonic acceleration amplitude, the second derivative of u with respect to time, 2f, is the second harmonic acceleration amplitude, the second derivative of u with respect to time, 1f, is the fundamental harmonic acceleration amplitude, and ω=2πf, where f is the fundamental frequency.

(26) FIG. 1A is a graph of β as a function of pore pressure, while FIG. 1B is a graph of α as a function of pore pressure.

(27) Time reversal permits the generation of focused, intense (non-damaging) sound in a region to induce local nonlinearities if high pressure is present, by taking advantage of the above relation for u.sub.2f, thereby permitting detection and imaging of overpressure regions. As an example, waves may be introduced into a specimen using a piezoelectric transducer. The waves are recorded on another transducer located elsewhere on the sample surface. The recorded waves are then reversed in time, and emitted from the detecting transducers, where they follow their forward wave paths backwards-in-space, and coalesce, focusing at the original source transducer, since the elastic wave equation is symmetric with respect to time. That is, the wave equation may be evaluated either forward or backward in time, the physics being identical. Amplitudes at the time-reversed focus are large due to conservation of energy, since all of the energy contained in the long-duration scattered-signal is collapsed onto the focal point in space and time. Since wave amplitudes are largest at the focus, the local response may be nonlinear, but only at the focus.

(28) Further, by measuring α and β for a formation using time reversal techniques, one can obtain accurate values for the pore pressure in a formation, using Equations 2 and 3, above. Among the uses for the gradient of the pore pressure are the prediction of gas/water contacts, which permit more accurate location of hydrocarbons in the formation.

(29) Reference will now be made in detail to the present embodiments of the invention, examples of which are illustrated in the accompanying drawings. In the FIGURES, similar structure will be identified using identical reference characters. Turning now to FIG. 2A, a schematic representation of an embodiment of the present apparatus, 10, for practicing look-ahead drilling based on TR NEWS is illustrated. Acoustical source, 12, located above, and in the vicinity of drilling bit, 14, turned by shaft, 15, transmits a sine wave that is swept in frequency as a function of time. Other signal types may be employed, including pulsed signals. Signals, 16, propagate both into well bore, 18, and into formation, 20, surrounding the borehole, and are received and recorded by transceivers, 22, also disposed in the wellbore on transceiver mount, 24, disposed near the drilling apparatus. One end of transceiver mount 24 may be placed between just behind drilling bit 14 and 10-20 m therefrom. Apparatus, 25, is employed to change the direction of drilling bit 14 in response to pore pressure measurements in accordance with embodiments of the present invention. As shown in FIG. 2B, the detected signals are digitally reversed-in-time, by microprocessors located at detectors (transceivers) 22 or at the surface, amplified (not shown in FIGS. 2A and 2B), and rebroadcast, 26, into the bore hole and into the surrounding formation. In accordance with the time-reversal process, the signals follow their forward propagation directions in reverse, and focus at source 12 that also acts as a detector. The phase relationships among the returning waves permit the constructive interference thereof resulting in space and time focusing. The focused signal is large in amplitude and is effective for inducing an elastic nonlinear response in focal volume, 28. If a portion of focal volume 28 encompasses a high fluid or gas pressured region, the nonlinear response of generated harmonic frequencies (and potentially sum and difference frequencies), and time delays due to wave speed decreases, will be significantly greater than at an established baseline thereof. This nonlinear response is detected and interpreted at the surface or by microprocessors located behind the drilling string (not shown in FIGS. 2A and 2B). By varying the frequency of the swept sine signal, the distance to the over-pressured region can be estimated from the frequency, if the sound velocity in formation 20 is known (as is generally the situation) using the relationship for the wavelength, λ=velocity/frequency. The diameter of the focal spot measured at the half maximum value is equal to one-half of the dominant wavelength. See, e.g., “Depth Profile Of A Time-Reversal Focus In An Elastic Solid,” by Marcel C. Remillieux et al., Ultrasonics 58 (2015) 60-66. Corrective action can then be taken by the drillers (placing blow-out stops etc.).

(30) A schematic representation of an embodiment of transceiver mount 24 is shown in FIGS. 3A-3C. FIG. 3A shows a possible arrangement for low-frequency transceivers, 22a, while FIGS. 3B and 3C show an arrangement for mid-frequency, 22b, and high-frequency transceivers, 22c, respectively. The transducer mount includes a long metallic portion, 30, adapted to fit in a cased borehole having an inner diameter of 6 in. Transducer mount 24 may also be constructed of sturdy plastics capable of withstanding high temperatures and caustic environments. Advantageously, the long dimension of mount 24 is equal to or larger than five times the largest wavelength of the elastic waves propagating in the formation. If mount 24 is made of steel (Young modulus E=200 GPa and mass density ρ=8500 kg/m.sup.3), and is operated at a center frequency f.sub.c=2 kHz, then the elastic wavelength is λ˜√(E/ρ)/f.sub.c=2.6 m, and the minimum length of the mount would be about 12 m. The actual shape of the mount can be optimized to improve the transfer of energy from the tool to the formation.

(31) FIG. 3D is a schematic representation of another embodiment of transceiver mount 24 where transceiver 12 is located approximately in the center of the mount. This design is suitable for measuring pore pressure in formations through well casings.

(32) Independently controlled low-frequency transceivers 22a mid-frequency transceivers 22b and high-frequency transceivers 22c controlled by digital synthesizers, 32, 34, and 36, respectively, which are directed by microcontroller and digital signal processor, 38, are affixed along mount 24 to provide the required excitation signals. Transducers vary in size and relative spacing depending on the center frequency of excitation signal that is intended to be generated. For low frequency excitation, large transducers are distributed over the entire length of the tool. For high frequency excitation, smaller transducers are centered with a smaller span around the point where focus should be achieved (at transceiver 12).

(33) Source 12 generates a swept sine wave that encompasses frequencies f.sub.i . . . I that provide the spatial resolution λ.sub.i . . . I of interest in a given group of strata. For example, given a typical formation velocity c of 2000 m/s, and a desired probe distance of I=10 m in advance of the drill bit, the time-reversed focal diameter would be d=20 m, and the center frequency would therefore be f.sub.j=100 Hz. Using a swept sine wave f.sub.i . . . I, spatial wavelengths above and below this value may be probed. The spatial wavelength may be reduced by increasing the frequency until the large nonlinear response disappears. In this manner the distance to the over-pressured region can be determined.

(34) FIG. 4A illustrates the forward or calibration steps to achieve the classical time reversed process using the apparatus of FIGS. 2A and 2B, and 3, while FIG. 4B illustrates that reverse or focusing steps for the time reversed process. In Step 40, of FIG. 4A, a predetermined calibration signal, P, from memory storage is directed to an arbitrary waveform generator, which drives, Step 42, source transducer 12 at the focal position, S. Signals generated from source transducer 12 are received by all of the transceivers 22, Step 44, and stored in memory, Step 46. Step 48 in FIG. 4B simultaneously directs the time reversed signals stored in memory in step 46 to an arbitrary waveform generator which drives transceivers 22, Step 50, the transmitted signals being focused onto transducer 12 in Step 52. Nonlinear signals, F, generated, Step 54, in the focal area are stored in memory for later processing. This calibration process would be undertaken every time measurements of α and β are to be made, since external conditions, such as increasing pressures of drilling mud, may change the calibration.

(35) Noise from impulsive elastic waves generated from the action of drilling bit 14 on the materials in a formation can be used as a source for the classical time reversal measurements in place of acoustic source 12 in accordance with embodiments of the present invention. In this situation, the drilling bit would be stopped when the amplified time-reversed signals generated by transceivers 22 are employed to generate harmonics in front of drilling bit 14, the harmonic signals being correlated with the time-reversed signals from the drilling bit.

(36) The method described in FIG. 4 is based on ‘classical’ time reversal. As will now be described, ‘reciprocal’ time reversal can also be applied. Here, transceivers 22 located on transceiver mount 24 are caused to individually broadcast a signal for a ‘training’ step. Transceiver 12 detects these signals that are broadcast one at a time. The detected signals are time reversed, and amplified and reemitted from transceivers 22. They again focus on the detector 12. This process works due to the reciprocity of the wave equation, since the transfer function in one direction is the same as that in the other direction. FIG. 5 illustrates the ‘training’ process. A predefined calibration signal, P, in memory storage is directed, Step 56, to an arbitrary waveform generator, which drives transceivers 22, Step 58, sequentially, the generated signals being focused onto transceiver 12, Step 60, and the received signals stored in memory for time reversal processing, Step 62, after each stored signal is time reversed. See, e.g., T. J. Ulrich et al., “Time reversal and non-linear elastic wave spectroscopy (TR NEWS) techniques,” Intl. J. Non-Linear Mech. 43 (2008) 209-216.

(37) Having generally described embodiments of the present invention, the following EXAMPLES provides additional details.

Example 1

(38) FIG. 6 is a schematic representation of an apparatus for carrying out reciprocal time reversal measurements described above in the laboratory for a simulated down hole environment. Pipe, 100, was embedded in block, 102, of Berea Sandstone. Ten acoustic transceivers 22 were affixed to portion, 104, of pipe 100 emerging from block 102. A reference signal, for example, a pulsed (10-20 kHz) sinusoidal waveform having a 50 kHz bandwidth, is directed into at least one arbitrary waveform generator, 106, by computer 38. After amplification by at least one power amplifier, 108, each generator signal is directed to a single transceiver 22, one generator signal at a time. The signal traveling through pipe 100 and block 102 is recorded by laser vibrometer, 110, after being received by fiber optic attachment, 112, disposed inside pipe 100. The signal received for each emission is digitized by digitizer, 114, and directed to computer 38, which time reverses each of the received signals and programs the arbitrary waveform generators 106 with the time-reversed signals. Reciprocal time reversal, in its basic form, includes first sending the last digitized element, then the second to last digitized element and so forth until the entire waveform has been inverted. Focusing in pipe 100 may be improved (typically, to sharpen the focus), by filtering the time-reversed signal (not shown in FIG. 6). Signal averaging may also be performed on the detected signals. The final step is for all of the arbitrary waveform generator to simultaneously direct the time-reversed signals to the transducers through the power amplifiers. During that phase, all the generators are synchronized via pulser, 116. While all the transceivers are emitting, laser vibrometer 110 records the signals generated in the focal volume that are digitized and analyzed for nonlinear components by computer 38.

(39) The time-reversed signals may be broadcasted successively at different amplitudes to assist in the detection of the nonlinear signals. As discussed above, the size of the region probed by focused waves in the formation depends of the wavelength used for the first reference signal.

(40) FIG. 7A is a graph of the detected pulse propagation down pipe 100 from a single transducer 22 without time reversal, while FIG. 7B is a graph of the detected pulse propagation from all 10 transducers 22 without time reversal. FIG. 7C is a graph of the detected focused pulse propagation down pipe 100 from a single transducer 22 using reciprocal time reversal, while FIG. 7D is a graph of the detected focused pulse propagation down pipe 100 from all 10 transducers using classical time reversal.

(41) The signal strength increases by a factor of 10 when using reciprocal time reversal over that resulting from the use of conventional sources. This is clear example of an apparatus capable of transmitting elastic wave energy to a formation in a simulated borehole/casing/rock system using the method in accordance with embodiments of the present invention.

(42) FIG. 8A is a graph of the delay in the arrival time of a detected pulse in the pipe of the apparatus of FIG. 6 as the amplitude of the signal pulse increases from (a) to (c). As mentioned above, this shift is related to a. FIG. 8B is a graph of the detected pulse as a function of frequency as the amplitude of the signal pulse increases from (a) to (c). The fundamental as well as the second and third harmonics are readily observable. As discussed above, 0 may be obtained by monitoring the second harmonic. Monitoring the third harmonic is an alternative method for obtaining α. When the bandwidth of the fundamental is broader, however, the third harmonic can partially overlap with the second harmonic, which may make third harmonic measurements more difficult. Additionally, the third harmonic may be influenced by δ (see Equations 5 above). Due to the relative sizes of α and δ, this latter issue is generally not a serious problem, but having an alternative way of measuring α, such as by using the time delay, is advantageous.

(43) FIG. 9 is a graph of the pore pressure as a function of depth measured by conventional techniques for an actual formation, permitting the prediction of gas/water contacts (GWC) for locating hydrocarbon deposits.

(44) Once these contacts are located, drilling can be redirected using apparatus 25 in FIG. 2A to accomplish the change in direction.

Example 2

(45) Propagation of an Impulsive Elastic Waveform in a Long, Thin Bar

(46) As discussed above, with the aid of time reversal, elastic wave energy is focused at a point in space and an impulsive waveform will be generated. Since this process involves waves traveling through materials, and material properties may be strain dependent, the arrival time of the impulsive waveform may be dependent on the amplitude of the excitation. The term of hysteretic nonlinearity a in the equation of state (Equ. 3) governs this effect.

(47) To verify α can be quantified by monitoring the propagation speed of an elastic wave as a function of the strain amplitude, laboratory experiments were performed. Although the propagation of impulsive elastic waves remains the principal measurement, time reversal is not required to generate the strain since the measurements are restricted to a one-dimensional waveguide over a known propagation distance. The hysteretic nonlinearity parameter has never been measured in this manner, so the determination is validated using nonlinear resonant ultrasound spectroscopy.

(48) FIG. 10 is a schematic representation of the apparatus, 118, used to determine α in sample, 120, of Berea sandstone (Cleveland Quarries, Amherst, Ohio) having a length of 1794 mm (70.63 in) and a diameter of 39.6 mm (1.56 in). The sample was supported by a foam pad, not shown in FIG. 10, in order to simulate free (unconstrained) boundary conditions. Elastic waves were generated using a piezoelectric transducer, 122, epoxied onto one flat end of sample 120. Impulsive elastic waveforms generated in signal generator, 124, and amplified by voltage amplifier, 126, were propagated in sample 120 at different amplitudes. The vibrational response of bar 120 was recorded on the surface of the sample using 3D Laser Doppler vibrometer, 128, and received by data acquisition apparatus, 130.

(49) Returning to FIGS. 2A, 2B, and 6, if one places a vibrational motion sensor, such as a piezoelectric transducer, 132, on wall, 134 of wellbore, 18, in FIGS. 2A and 2B, corresponding to pipe 100 in FIG. 6, one can measure the particle velocity as a function of excitation intensity of the waves generated in the formation by time reversal, from which the peak strain (EQU. 8) can be evaluated in focal volume 28. As described in this EXAMPLE 2 for laboratory experiments, from the time history of the particle velocity and the peak strain, a can be evaluated. With β being available from measurement of the intensity of harmonics, the pore pressure can be determined. Note that since the metal walls or casings of wellbores exhibit linear response to vibrations, the particle velocity, which is the motion excited in the focal region is accurately measured through the wall.

(50) FIGS. 11A and 11B show the time histories of the axial component of the particle velocity measured at 500 mm from the source 122 for 20 source amplitudes ranging from 1 to 20 Vpp in steps of 1 Vpp (before amplification), with FIG. 11B illustrating an expanded abscissa between 0.32 ms and 0.37 ms. The original sinusoidal waveform observed at the lowest amplitudes progressively evolves into a triangular wave, as a result of hysteretic nonlinearity, with additional distortion caused by classical nonlinearity. It is observed that the waveforms experience a significant delay in arrival time as the source amplitude increases. This delay is observed not only at the extrema (peaks of amplitude) but also at the zero crossings. Classical nonlinearity would not induce any delay at the zero crossings (where the strain is equal to zero), because it produces an instantaneous variation of the modulus without time constants involved. Therefore, the observed delay is a direct consequence of hysteretic nonlinearity.

(51) The data shown in FIGS. 11A and 11B can be further reduced to quantify hysteretic nonlinearity. For a signal measured at the source amplitude i, the signal measured at the source amplitude i−1 is taken as a reference to compute a relative time delay, Δt.sup.i/i−1/t.sub.0. The relative time delay between two signals is estimated by cross-correlation. The time delay between the lowest source amplitude (i=0) and the source amplitude i is then obtained by summation as,

(52) $\begin{array}{cc}\Delta t^{i\text{/}0}\text{/}{t}_0=\left(\underset{n=1}{\overset{i}{.Math.}}\Delta t^{n\text{/}n-1}\right)\text{/}{t}_0& \left(6\right)\end{array}$
The relative time delay between the signals is also equal to the relative change in speed of the longitudinal wave, Δc.sup.i/0/c.sub.0. Further, at the perturbation level, the relative change in the Young's modulus E (the modulus involved in the propagation of a longitudinal wave in a long thin bar) is related to the relative change in the speed of the longitudinal wave as,
ΔE.sup.i/0/E.sub.0=2Δc.sup.i/0/c.sub.0  (7)
The relative changes in the elastic modulus over the propagation path of the waveform can be followed as a function of the maximum strain amplitude at the measurement point. The strain component of interest is ε.sub.xx, where x is axial direction. The strain component ε.sub.xx can be expressed analytically as a function of the axial component of the particle velocity v.sub.x as,

(53) $\begin{array}{cc}{\mathrm{.Math.}}_{\mathrm{xx}}=\frac{v_x}{c}& \left(8\right)\end{array}$
Recall that the particle velocity is obtained from the vibrational motion measured by 3D Scanning Laser Doppler Vibrometer 110.

(54) Reduced data from FIGS. 11A and 11B are shown in FIGS. 12A and 12B. Three distinct regimes may be identified in the waveform: (i) the initial portion of the waveform, where the dynamic strain at the measurement point transitions from zero to steady state, and labeled with a “star”; (ii) the center portion of the waveform, where the absolute value of the peak amplitude of the dynamic strain is quasi-constant (that is, steady state), and labeled with a “0”; and (iii) the final portion of the waveform, where the dynamic strain at the measurement point transitions from steady state back to zero, that is, a path similar to the earlier portion of the waveform but in reverse, and also labeled with a “+”. In the initial portion of the waveform, the elastic modulus decreases progressively from its undisturbed equilibrium value to a smaller value (ultimately up to 5.3% smaller at the largest source amplitude employed as the steady state is approached). This is the portion typically referred to as conditioning. In the center portion of the waveform, the evolution of the material softening with the strain amplitude is essentially independent of the wave cycle selected for the analysis (that is, similar results are obtained from the 9.sup.th to 21.sup.st peaks of the waveforms). In this regime, the slope of the curve is 10480±600 (dimensionless). This value quantifies the hysteretic nonlinearity, but it will be determined in EXAMPLE 3 below how this value compares to the parameter α.

Example 3

(55) Nonlinear Resonant Ultrasound Spectroscopy:

(56) Returning to FIG. 10, with minor modifications, the apparatus may be utilized for resonant ultrasound spectroscopy. Sample, 120, of Berea sandstone (Cleveland Quarries, Amherst, Ohio) having a length of 1794 mm (70.63 in) and a diameter of 39.6 mm (1.56 in) is shown. The sample was supported by a foam pad, not shown in FIG. 10. Piezoelectric transducer 122 was driven with a sequence of harmonic voltage signals. Each harmonic signal was applied for 55 ms, and the transient vibrational response was recorded during the last 40 ms of the source signal, to ensure that steady state conditions had been reached. Vibrational spectra were constructed from the harmonic responses. Frequencies ranging between 0.3 kHz and 7 kHz in steps of 2.5 Hz were employed, and at 22 excitation amplitudes ranging between 0.25 to 10 Vpp (before amplification). The axial component of the acceleration was measured on the flat end opposite to the source by an accelerometer using transducer, 132, the output of which was processed by signal conditioner, 134, and analyzed using data acquisition apparatus 136. The resonance frequencies are not computed for the first three modes because of the poor signal-to-noise ratio.

(57) Any resonance mode can be selected to quantify hysteretic nonlinearity as long as the mode type is purely longitudinal. The vibrational spectra for this experiment are shown in FIGS. 13A and 13B. Material softening, a drop in the effective elastic moduli resulting from the fact that elastic moduli are dependent on the strain amplitude and strain rate, is observed when the drive amplitude of the source becomes sufficiently large (Equ. 3). For a given drive amplitude, such softening can be quantified by plotting the relative frequency shift as a function of the maximum strain in the sample, the slope of which is the nonlinear parameter α. The maximum strain in the sample may be inferred analytically from the measured vibrational response. For the longitudinal modes, the strain component of interest is ea. For a system having a one-dimensional geometry and unconstrained boundaries, the expression given in Eq. (8) can also be used in the context of a resonance experiment to relate the maximum amplitude of the axial component of the particle velocity at the free end of the sample (where data is acquired) to the maximum amplitude of the axial component of the strain in the sample.

(58) As depicted in FIG. 14, the relative frequency shift varies almost linearly with the maximum strain beyond 4 microstrains. When a mode has sufficient data beyond this strain value, the points may be linearly fitted and the slope of this fit calculated. It appears that material softening converges to a single value (all curves superimpose) of α=5260±160 for resonance modes L.sub.6 through L.sub.12. Below the 6.sup.th resonance mode, the elastic response does not have a sufficiently large amplitude to reach the threshold strain value of about 4 microstrain.

(59) The slope of the relative change of the resonance frequency is approximately twice the value of the relative change of the Young's modulus observed in the pulse propagation experiment, which is consistent with the analytical relationship between Young's modulus and resonance frequency of a longitudinal mode at the perturbation level,
ΔE/E.sub.0=2Δf/f.sub.0  (9)
Therefore, the quantification of hysteretic nonlinearity in the pulse propagation experiment and with nonlinear resonant ultrasound spectroscopy are equivalent.

(60) In the pulse propagation experiments, the Young's modulus is approximately constant below 4 microstrains in the conditioning phase and varies linearly with strain above this value, with a sharp transition between the two regimes (see FIGS. 11A and 11B). In the resonance experiment, the resonance frequency also varies linearly with strain beyond 4 microstrains but experiences a smooth transition to this regime. During this transition, classical nonlinearity plays a substantial role. Classical nonlinearity can induce a frequency shift in the resonance experiment, but cannot induce a time delay in the pulse propagation experiment; hence, the sharp transitions in FIGS. 11A and 11B. In this sense, it is possible to decouple the contributions from classical nonlinearity and nonequilibrium dynamics in the pulse propagation experiment.

(61) In summary, application of a method that combines time reversal and elastic nonlinearity (TR NEWS) provides the means to quantitatively probe for over pressured regions in advance of the drilling bit, and to determine the distance to an over pressured region. Moreover, gas/water contacts may be located in accordance with the teachings of the present invention, and drilling directed to more successfully locate hydrocarbons.

(62) The foregoing description of the invention has been presented for purposes of illustration and description and is not intended to be exhaustive or to limit the invention to the precise form disclosed, and obviously many modifications and variations are possible in light of the above teaching. The embodiments were chosen and described in order to best explain the principles of the invention and its practical application to thereby enable others skilled in the art to best utilize the invention in various embodiments and with various modifications as are suited to the particular use contemplated. It is intended that the scope of the invention be defined by the claims appended hereto.