Abstract:
A compact array of transducers is employed as a downhole instrument for acoustic investigation of the surrounding rock formation. The array is operable to generate simultaneously a first acoustic beam signal at a first frequency and a second acoustic beam signal at a second frequency different than the first frequency. These two signals can be oriented through an azimuthal rotation of the array and an inclination rotation using control of the relative phases of the signals from the transmitter elements or electromechanical linkage. Due to the non-linearity of the formation, the first and the second acoustic beam signal mix into the rock formation where they combine into a collimated third signal that propagates in the formation along the same direction than the first and second signals and has a frequency equal to the difference of the first and the second acoustic signals. The third signal is received either within the same borehole, after reflection, or another borehole, after transmission, and analyzed to determine information about rock formation. Recording of the third signal generated along several azimuthal and inclination directions also provides 3D images of the formation, information about 3D distribution of rock formation and fluid properties and an indication of the dynamic acoustic non-linearity of the formation.

Description:
GOVERNMENT RIGHTS 
       [0001]    This invention was made with Government support under Cooperative Research and Development Agreement (CRADA) Contract Number DE-AC52-06NA25396 awarded by the United States Department of Energy. The Government may have certain rights in this invention. 
     
    
     FIELD 
       [0002]    The present invention relates generally to acoustic interrogation of rock formations around a borehole and more particularly to using the combination of an acoustic source including a single transducer or an array of transducers in the wellbore and the acoustically non-linear near wellbore rock formation as the source of an acoustic beam. 
       BACKGROUND 
       [0003]    Acoustic interrogation of subsurface features tend to be limited by the size and power of practical sources, and in practice, the output of downhole acoustic transducers is limited by the power transmission capabilities of the wire line cable. High frequency signals have a relatively short penetration distance, while low frequency signals generally require large sources, clamped to the borehole wall, to maximize energy transfer to the formation and minimize unwanted signals within the well bore. Attempts have been made to produce such a low frequency signal, but each has had its own shortcomings. For example, U.S. Pat. No. 5,521,882 by D&#39;Angelo et al. discloses an acoustic tool designed to record non-linear waves that propagate along the borehole wall and refract back into the well, with limited penetration onto the surrounding rock formation. U.S. Pat. No. 7,301,852 by Leggett, III et al. discloses using a bottom hole assembly having an array of transducers located in opposite ends of the assembly to produce the signals that converge at an angle, as reported by Johnson et al (1987). The signals from non-linear mixing at an angle are extremely weak and diffuse, and would not penetrate deep into the surrounding rock formation. In light of these prior attempts, there is a need for an acoustic signal source that can produce low frequency energy from the borehole in the subsurface environment. 
       SUMMARY 
       [0004]    In accordance with an aspect of the invention, a method of generating a beam of acoustic energy in a rock formation penetrated by a borehole is provided. The method comprises generating a first acoustic signal at a first frequency; generating a second acoustic signal at a second frequency different from the first frequency, wherein the first acoustic signal and second acoustic signal are generated by an array of transducers located within the borehole; transmitting the first and the second acoustic signals into an acoustically non-linear portion of the rock formation outside of the borehole; combining the first and the second acoustic signals by a three wave mixing process to generate a collimated third signal in the rock formation, wherein the collimated third signal propagates through the rock formation in a same direction as an initial direction of the first and the second acoustic signals and has a frequency equal to a difference of the first and the second acoustic signals and an amplitude dependent on a non-linear property of the rock formation; receiving the third signal at one or more receivers after it has reflected or backscattered from an inhomogeneity in the formation; and analyzing the received signal to determine information about the rock formation. 
         [0005]    In accordance with an aspect of the invention, a system for determining information about a rock formation penetrated by a borehole is provided. The system comprises one or more transducers configured to generate a first acoustic signal at a first frequency and generate a second acoustic signal at a second frequency different from the first frequency; one or more receivers in communication with the one or more transducers, the receivers configured to detect through the rock formation a third signal that results from the interaction of the first and the second acoustic signals in region of the rock formation outside the borehole, wherein the detected third signal is detected after passing through a portion of the rock formation and a processor programmable with machine executable instructions for analyzing the received signal to determine information about the rock formation. 
         [0006]    In accordance with an aspect of the invention, a system for characterizing a non-linear property of a rock formation penetrated by a first borehole is provided. The system comprises one or more transducers configured to generate a first acoustic signal at a first frequency and generate a second acoustic signal at a second frequency different from the first frequency, wherein the array is arranged in the first borehole; one or more receivers, arranged in a second borehole, and in communication with the one or more transducers, the receivers configured to detect through the rock formation a third signal that results from the interaction of the first and the second acoustic signals in region of the rock formation outside the first borehole, wherein the detected third signal is detected after traveling through a portion of the rock formation and reflecting from an inhomogeneity; and a processor programmable with machine executable instructions for analyzing the received signal to determine information about the rock formation in the non-linear interaction region between the first and the second boreholes. 
         [0007]    These and other objects, features, and characteristics of the present invention, as well as the methods of operation and functions of the related elements of structure and the combination of parts and economies of manufacture, will become more apparent upon consideration of the following description and the appended claims with reference to the accompanying drawings, all of which form a part of this specification, wherein like reference numerals designate corresponding parts in the various Figures. It is to be expressly understood, however, that the drawings are for the purpose of illustration and description only and are not intended as a definition of the limits of the invention. As used in the specification and in the claims, the singular form of “a”, “an”, and “the” include plural referents unless the context clearly dictates otherwise. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0008]      FIG. 1  generally shows how the three wave mixing phenomenon for P-waves in the presence of a borehole occurs, in which two incident waves at two different frequencies, f 1  and f 2  are emitted in the formation and, as a result of the non-linear properties of the formation, lead to the generation of a third wave with a low frequency of f_=f 2 −f 1 . 
           [0009]      FIG. 2  shows an experimental test design set-up in accordance with an aspect of the invention. 
           [0010]      FIGS. 3   a ,  3   b  and  3   c  show some results from the experimental test design set-up of  FIG. 2 , illustrating the clear coherent 5 kHz beam signal generated by mixing two primary beams with frequencies 38 and 43 kHz. 
           [0011]      FIG. 4  shows the geometry of the source located at a borehole and the mixing region located outside the borehole where the signals from the sources mix. 
           [0012]      FIG. 5  is a schematic illustration of a transducer array along with the definition of the phase shift Δφ and the angle of steering θ. 
           [0013]      FIG. 6  is a schematic illustration illustrating an example of how control of the azimuth and inclination of the plane of the array may be achieved by rotation around the borehole axis, and rocking in a plane perpendicular to the borehole axis. 
           [0014]      FIG. 7  shows the distribution of the sources of the array used for the experimental set-up in  FIGS. 2 ,  3 ,  4  and  8 . 
           [0015]      FIGS. 8   a  and  8   b  shows experimental results obtained from the distribution of sources in the transducer array of  FIG. 7 , showing that the 5 kHz difference frequency beam can be steered by steering the primary beams. The black circle indicates the position of the transducer array on the opposite side of the sandstone block.  FIG. 8   b  shows how steering the primaries results in a steered beam at the difference frequency. 
           [0016]      FIGS. 9   a  and  9   b  show an aspect of an embodiment useable to create images, including three-dimensional images, around the borehole. 
           [0017]      FIG. 10  shows an aspect of an embodiment for performing cross-well tomography. 
           [0018]      FIGS. 11   a  and  11   b  show an aspect of an embodiment wherein the source in the first borehole is moved axially and received at a plurality of locations within a second borehole in a configuration designed for cross-well formation property and bedding interpretation. 
           [0019]      FIG. 12  shows system components for producing the third signal in accordance with an embodiment of the invention. 
       
    
    
     DETAILED DESCRIPTION 
       [0020]    In an embodiment, an array of transducers is used to generate an acoustic beam in the non-linear rock formation around the borehole via a three-wave mixing phenomenon by mixing two collimated primary acoustic signals of higher frequency, as illustrated in  FIG. 1 . The two collimated primary acoustic signals  110 ,  120  can be generated over a wide range of higher frequencies by a compact array since high frequency sources can readily fit inside a borehole  130 . The non-linearity of the rock formation  140  around the borehole leads to the generation of a collimated acoustic beam  150 , commonly referred to as a parametric array beam, having a frequency equal to the difference between the primary acoustic signals and with sufficient strength to propagate a considerable distance into the rock formation. As shown, the high frequency sources are distributed in a plane parallel to the axis of the borehole. The primary sources can be of several types, depending on the application, such as a monopole source (i.e., isotropic radiation) or a dipole source (i.e., equivalent to point forces), or a collection of monopole baffled sources clamped to the borehole wall, or fixed to a tool located within the borehole. 
         [0021]    The non-linear behavior of a formation may be characterized through the analysis of the properties of P-waves resulting from the three wave mixing phenomenon in which two incident waves at two different frequencies, f 1  and f 2 , mix to generate third frequency components at the harmonics and intermodulation frequencies f 2 −f 1 , f 2 +f 1 , 2f 1  and 2f 2 , etc. In an aspect of the invention, the three wave collinear mixing phenomenon is designed to occur in the vicinity of a wellbore. Only the resulting third wave of difference frequency f 2 −f 1  is of interest to this application. The third wave may be recorded by one or more receivers located in the same or another wellbore. This arrangement is illustrated in  FIG. 1  where the generation of the third wave having the difference frequency is shown produced by two primary sources. 
         [0022]    In the instance where the primary sources are arranged as a monopole baffled sources, the sources are rigidly baffled in the plane of the array and can be in contact with the borehole wall. Within the transducer array, some sources or transducers are driven at one frequency f 1 , while others are simultaneously driven at another frequency f 2 . Adjacent to the wellbore, a low frequency (f_=f 2 −f 1 ) wave is created by nonlinear mixing in the formation of the two high frequency primaries. β is the nonlinear parameter that characterizes the nonlinear relationship between the stress and the strain and, for example, in one dimension, appears in the equation σ=Kε(1+βε+ . . . ), where σ is the applied stress, K the bulk modulus and ε the strain. β typically ranges from 200 to 1000 for rocks. The volume in which the third wave is created is given by V_=πα 2 L T , where a is a radius encompassing the sources and L T  is a length related to the attenuation of these frequencies. It should be noted that the attenuation (or absorption) length L α , of a wave characterizes a decrease to 1/e of the amplitude, is well known by those versed in the art and is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       L 
                       α 
                     
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         [0000]    where f is the frequency (Hz), Q is the quality factor and c is the wave velocity (m/s). L T  is the total attenuation length expressed as L T   −1 =L 1   −1 +L 2   −1 ≈2L 2   −1 . 
         [0023]    The difference frequency signal is generated and propagates from the volume V_, to a receiver or receiver array in the same borehole by reflection from interfaces with contrasting acoustic impedance, or by backscatter from inhomogeneities, or by transmission to a second borehole. The propagation occurs in a collimated beam, the width of which is determined by k_L T , where k is the wave number of the third wave: k_=k 2 −k 1 =2π(f 2 −f 1 )/c. 
         [0024]    Another length of interest is the so called Rayleigh distance, L R , that defines the limit between the near field and the far field regions of the beam: 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                       R 
                     
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         [0025]    where k is the wave number, i.e. k=2πf p /c and α is the radius of the source. The frequency f p  refers to the primaries. The near field region corresponds to a distance L x  less than L p . In this region, the beam is unchanged in size. In the far field region, i.e. for L x &gt;L p , the beam amplitude decays as L p /L x , which is the beam spreading factor. 
         [0026]      FIG. 2  shows an experimental test design set-up in accordance with an aspect of the invention. In the set-up, the dimensions of a block of Berea sandstone  210  were 1.8 m×0.4 m×0.4 m. The individual sources  220  that comprise the array  230  are 2.5 cm in diameter, arranged within a circle of 6 cm radius. Half of the sources were excited with a frequency f 2 =43 kHz and the other half with a frequency f 1 =38 kHz. The third wave then had a frequency f — =43 kHz-38 kHz=5 kHz. Each of the sources  220  having a frequency f 1  are driven by a signal generator  240  that is configured to generate a signal at a frequency f 1  and an amplifier  250 . Similarly, each of the sources  220  having a frequency f 2  are driven by a signal generator  260  that is configured to generate a signal at a frequency f 2  and an amplifier  270 . The motions at the difference frequency were detected with a laser vibrometer  280  located on the opposite side of the block of Berea sandstone  210 , ensuring that there was no possible electronic interference in the strain measurement. 
         [0027]      FIGS. 3   a ,  3   b  and  3   c  are a series of strain maps at three frequencies for the set-up of  FIG. 2 , corresponding to the primary frequencies and the difference frequency, recorded with the laser vibrometer  280 . In particular,  FIG. 3   a  shows a primary signal at 43 kHz and  FIG. 3   b  shows another primary signal at 38 kHz.  FIG. 3   c  shows the third signal of frequency 5 kHz resulting from parametric array effect of the non-linear mixing of the two primaries. As is seen in  FIG. 3   c , this third signal has inherited the collimated properties of the primaries. For the laboratory experiment, using a quality factor of 50 and a wave velocity of 2000 m/s, the attenuation length for the two primaries is 0.37 m and 0.42 m (respectively for 43 kHz and 38 kHz), while it is 3.2 m for the third wave. The Rayleigh distance, with a source radius of 6 cm, is roughly equal to 0.2 m. This demonstrates that the strain measurements were made in the far field region. 
         [0028]      FIG. 4  shows the geometry of the source located within a borehole and the mixing region outside the borehole where the signals from the sources are coupled. The strength of the difference (f 2 −f 1 ) beam source and the focusing effect of the beam depend on the strengths of the primary sources f 1  and f 2 , the non-linear properties of the rock formation surrounding the borehole and the wave numbers of the primary source and resulting difference frequency. In  FIG. 4 , k 1  and k 2  are the wavenumbers associated with the primary frequencies f 1  and f 2 , respectively. k_ is the wavenumber associated with the third wave of frequency (f 2 −f 1 ) generated by nonlinear mixing in the formation. The angle θ is counted from the horizontal which corresponds to θ=0 and corresponds to the azimuth direction relative to the borehole. The primary source are considered to be distributed in a circle of radius α. 
         [0029]    A dimensionless equation for the strain field at the receiver borehole associated with the third wave, denoted by ε_, can be determined from 
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         [0030]    wherein β is the nonlinear parameter of the formation, ζ is equal to k_L, where k_=k 2 −k 1  and L is the borehole separation. ε 1  and ε 2  are the strains at frequencies f 1  and f 2 , respectively. The functions D A  and D W  (of the angle theta measured from the beam axis) are directivities that arise from the finite width a, and length L T  of the source respectively. The directivity D A  is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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         [0031]    where J 1  is the Bessel function of order 1 with D A (0)=1. The directivity D W  due to Westervelt) is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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         [0032]    Usually, L source  is the attenuation length L T  so that 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                       W 
                     
                     = 
                     
                       
                         
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         [0033]    Note that similarly to D A , D W (0)=1, (namely the directivity factor is a maximum on-axis of the beams). 
         [0034]      FIG. 5  shows a schematic array along with the definition of the phase shift Δφ and the angle of steering θ: 
         [0000]    
       
         
           
             
               
                 
                   
                     Δϕ 
                     = 
                     
                       
                         
                           ω 
                            
                           
                               
                           
                            
                           d 
                         
                         c 
                       
                        
                       sin 
                        
                       
                           
                       
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                   , 
                 
               
               
                 
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         [0035]    where ω is the frequency of the primary, f 1  or f 2  that is to be steered, d is the distance between the sources and c is the P-wave velocity of the formation. In the illustration, the primary sources having the frequency of f 1  and f 2  alternate along the x-axis and the steering angle θ is measured away from the z-axis and towards the x-axis, wherein the x-axis and the z-axis are mutually perpendicular. 
         [0036]    In some embodiments, the direction of the beam is steered by selectively controlling the azimuth of the array by rotation around the tool axis, and the inclination, the angle between the plane of the front of the array and the tool axis. The corresponding axes of rotation are illustrated in  FIG. 6 . By use of actuators (not shown) the plane of the array can be effectively controlled in azimuth and inclination. The actuators can thus be used to steering or changing the direction of the primary frequencies f 1  and f 2 . 
         [0037]    In some embodiments, the inclination of the primary signals f 1  and f 2  is achieved by controlling the relative phase of some of the elements in the source array, noting that the third signal within the formation will inherit the direction of the primaries. An example of such control realized in the laboratory is shown in  FIG. 7 . By way of a non-limiting example, the array used in the laboratory was divided into five regions or several arrays referred to as “columns”. In column  1 , two transducers are shown both labeled “1” that are configured to produce a frequency of f 1  without any phase delay. In column  2 , four transducers are shown wherein the transducers labeled “2” are configured to produce frequency f 1  having been offset by a phase of Δφ and the transducers labeled “3” are configured to produce a frequency f 2  having a zero phase offset. In column  3 , five transducers are shown wherein the transducers labeled “4” are configured to produce frequency f 1  having been offset by a phase of 2Δφ and the transducers labeled “5” are configured to produce a frequency f 2  having been offset by a phase of Δφ. In column  4 , four transducers are shown wherein the transducers labeled “6” are configured to produce frequency f 1  having been offset by a phase of 3Δφ and the transducers labeled “7” are configured to produce a frequency f 2  having been offset by a phase of 2Δφ. In column  5 , two transducers are shown wherein the transducers labeled “8” are configured to produce a frequency f 2  having been offset by a phase of 3Δφ. In the laboratory demonstration, the phase difference between each of these “columns” chosen to steer the primaries 15 degrees from the perpendicular to the plane of the transducers. In this non-limiting example, the transducers were attached to the sandstone block as shown in  FIG. 2 . Other configurations are also possible without diverting from the scope of invention. For example, more or fewer transducers may be used, the transducers may be arranged in alternative patterns, and the relative phase of the signals may differ considerably from the example described, so long as phase differences among the primary acoustic signals result in steering or control of their direction of propagation. 
         [0038]      FIG. 8   a  shows the experimental results obtained with no steering (in which case no phase shift is applied between the “columns”) while  FIG. 8   b  shows the results when steering is applied for the set-up shown in  FIG. 7 . As is shown in  FIG. 8   a , a set of transducers operating at a frequency of 60 kHz and a second set operating at 65 kHz, neither of which has any phase offset applied to their respective transducers, produces a third signal with a frequency of f 2 −f 1 =5 kHz that is focused as indicated by the circle. In contrast, when steering is applied by using the phase control described in relation to  FIG. 7 , the location of each of the first and the second frequencies, as well as, the difference signal, are offset from the center of the array on the opposite face of the sandstone block indicated by the circle in the figure. 
         [0039]    In some embodiments, the third signal is encoded with a time-varying code, which can be introduced into either the first or the second signal, or both. The time-varying code may include one or more of a variation in amplitude, a variation in frequency, and/or a variation in phase of the first, the second, or both the first and the second signals. The received time-varying code of the third signal can be used to measure a time-of-flight of the third signal. Additionally, in some embodiments, the collimated difference beam can be broad band if one of the primary frequencies is swept through a range of frequencies while the other is fixed on a particular frequency. Thus, the resulting third beam f 2 −f 1  will be swept across a wide frequency range. 
         [0040]      FIGS. 9   a  and  9   b  show an aspect of the invention used to create images, including three-dimensional images, around the borehole. The collimated difference beam (f 2 −f 1 ) generated by the method described in this invention is suitable for creating an image of the rock formation around the wellbore. The difference beam can be aimed in a particular direction to investigate a particular region of the rock formation. Inhomogeneities of the formations localized along the beam will generate reflection, linear or scattering of the seismic waves. The reflected and scattered waves are recorded by receivers arrays in the same borehole (for the case of single well imaging).  FIGS. 9   a  and  9   b  illustrate examples of the many possible measurement and tool configurations. 
         [0041]    In  FIG. 9   a , an array of transducers is arranged on a circular array  905  with a borehole  910 . Two primary frequencies f 1  and f 2  are produced by the array and mix in a non-linear region outside the wellbore  925  to produce the difference frequency f 3 . The difference frequency beam is reflected or scattered from an adjacent bed or other body and this reflected beam  930  is received by the receivers  915  in the borehole  910 . The array on the circular array  905  including the receivers  915  can be moved up and down the length of the borehole to image a particular formation near the borehole. Moreover, the array can be rotated around the axis of the borehole to image formation in any direction around the borehole.  FIG. 9   b  shows an alternative linear array configuration  920 .  FIG. 9   b  is similar to  FIG. 9   a , but shows a linear array configured along the axis of the borehole. Either array may be rotated around the tool axis to achieve imaging of formations around the borehole. 
         [0042]    The recordings of received waveforms are processed to generate an image of the reflection or transmission characteristics of the formation. The directivity of the beam and the time of flight may fix the locations where scattered waves are generated, distinguishing this device from normal sonic imaging techniques using conventional non-directional monopole and dipole sources. An associated effect of using a beam compared with conventional sources is that the computation of an image of formation acoustic properties may not require a detailed specification of a velocity field for the rock formation. The directivity of the beam and the time of flight measurement simplify and improve the ability to identify the location where the waves are reflected or scattered. In particular, the directivity of the array beam source localizes the sources of recorded scattered wave along the beam direction and the time delay localizes the position of the scattered source along the beam path. Thus the borehole imaging with a beam source presents a simplification and reduction in uncertainty of the final image in contrast to conventional (not beam) sources which require an accurate detailed velocity model for focusing computation of the 3D image. Furthermore, because the beam is focused and steerable, the array beam has azimuthal and inclination directivity with respect to the borehole, and the imaging would tend to have higher resolution than a conventional source with little directivity and consequent ambiguity as to the origin of the reflection. The anticipated difference beam frequency for the invention ranges from 500 Hz to 10 kHz. The low frequency part of the range is at the very low end of the range used by conventional sonic logging tools, which implies a greater depth of penetration, up to several hundreds of feet, due to less attenuation. It is important to note that, since the beam is being generated in the formation outside the borehole, this low frequency beam generation mechanism will not lead to the generation of borehole modes, such as the Stoneley wave, that capture a large part of the energy generated in the borehole by conventional sources. Moreover, the backscatter of acoustic energy can be used to image around the borehole by rotation in azimuth and inclination, and linear movement of the array  920  along the borehole. 
         [0043]      FIG. 10  shows an aspect of the invention for performing cross-well tomography. The frequency range of 500 Hz to 10 kHz, with a range of investigation of several hundreds of feet from the borehole, can be applied in a cross-well imaging and tomography application. Furthermore, the beam nature of the source permits distinct processing methodologies for cross-well imaging and cross-well tomography that differ from approaches using non-focused sources. In this aspect, the beam source  1010  is generated from one borehole  1020  and detected with receivers  1030 ,  1040 ,  1050  in a second borehole  1060 . In cross-well tomography, the travel times and amplitudes of direct arrivals between various source and receivers between wells are processed with various tomography inversion methods to obtain a cross-section of velocity and attenuation strength of the rock formation between the wells. With a steerable parametric beam source, the beam can be scanned and detected by an array of detector Y 1 , Y 2 , Y 3  etc. in the second borehole. In this configuration, the maximum signal would systematically sweep sequentially across the receivers making the detection of the direct arrival easier and more robust. 
         [0044]      FIGS. 11   a  and  11   b  show an aspect of the invention wherein the source in the first borehole is moved axially and received at a plurality of locations within a second borehole. In this aspect, the parametric array beam source generated by mixing within the rocks around the borehole can be configured in tools to measure and create a log of the non-linear properties of the near wellbore region where three wave mixing occurs. In this configuration, the parametric array beam is generated from a location X 1  in one borehole and steered toward many receivers at location Y i  in a second borehole. Let the measured signal be M(X i , Y i ). The source is moved up hole to a location X 2  and similar data are collected to give a measured signal of M(X 2 , Y i ). From Equation (4), it is evident that the sum over many Y i  of the ratio of M(X 1 , Y i )/M(X 2 , Y i ) would be proportional to the ratio β(X 1 )/β(X 2 ), where β are non-linear properties within a mixing distance (a few feet) of the source borehole. By repeating the measurements over many locations X i , a log of relative strengths of non-linear properties close to the source borehole and within a chosen depth interval can be obtained from Equation (4) as a weighted average of a signal ratio such as the formula 
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         [0045]    It is understood that the above relation applies to the signal whose dominant frequency is the difference frequency, i.e. (f 2 −f 1 ). Another relative measurement of the non-linearity of the formation near the borehole can be made by taking the amplitude ratio of the received signal at the frequency (f 2 −f 1 ) to the one emitted at the source. 
         [0046]    If there are strong reflections surrounding a borehole, the source can be configured in tools to measure and create a log of non-linear properties very close to the borehole. Because the statistical distribution of the reflection strengths of the earth, the earth reflectivity response would be a slowly varying function of the source location while the non-linear properties of the rock just around the borehole would vary more rapidly with respect to source location. Therefore, using a single well, a high spatial frequency log of the β function can be also generated using equation (9). 
         [0047]      FIG. 12  shows system components for producing the third signal. A first generator  1210  configured to generate a first acoustic signal at a frequency f 1  and a second generator  1215  configured to generate a second acoustic signal at a frequency f 2  are coupled to a transducer assembly  1220  arranged within a borehole. A beam steering control system  1225  and a beam azimuth and inclination control mechanism  1230  are coupled to the transducer assembly  1220  and are configured to steer the first, the second, and/or both the first and the second signals. The azimuth and inclination control mechanism  1230  can be configured to steer by either modulating the phase of the first, the second, and/or both the first and the second signals or activating actuators on the transducer assembly to effect control of the beams. A receiver assembly  1235  can be located in either the borehole containing the transducer assembly  1220  or in another borehole. The receiver assembly  1235  includes one or more receivers configured to receive the third signal. The received third signal can be stored in a raw data recorder  1240  and processed using a signal processor  1245 . The processed data can be stored and displayed by a computer  1250 . The receiver assembly can be connected to the data recorder  1240  and signal processor  1245  though a wired or wireless connection. The computer  1250  can include machine readable instruction configured to process and manipulate the processed date as would be apparent. 
         [0048]    Although the invention has been described in detail for the purpose of illustration based on what is currently considered to be the most practical and preferred embodiments, it is to be understood that such detail is solely for that purpose and that the invention is not limited to the disclosed embodiments, but, on the contrary, is intended to cover modifications and equivalent arrangements that are within the spirit and scope of the appended claims. For example, though reference is made herein to a computer, this may include a general purpose computer, a purpose-built computer, an ASIC including machine executable instructions and programmed to execute the methods, a computer array or network, or other appropriate computing device. As shown in  FIG. 12 , the data collected by the receivers would undergo some processing and are either stored in memory in the tool, or transmitted up hole for further processing and storage. As a further example, it is to be understood that the present invention contemplates that, to the extent possible, one or more features of any embodiment can be combined with one or more features of any other embodiment.