Abstract:
An apparatus for investigating a subsurface volume may include an acoustic transducer disposed in an enclosure and generating acoustical signals, an electronics assembly disposed in the enclosure and controlling the acoustic transducer, and a lens assembly. The lens assembly may be disposed in the enclosure and next to the acoustic transducer. The lens assembly may be formed of a plurality of cells. Each cell may be formed as a column oriented transverse to a direction of travel of the acoustical signals. Each cell may have a hub, a plurality of spokes radiating from the hub, and a plurality of fingers circumferentially distributed around the hub. The hub, spokes, and fingers may be oriented to cause the acoustic waves to travel at a different speed in each of three orthogonal directions. A related method uses the apparatus in a wellbore.

Description:
FIELD OF THE DISCLOSURE 
     This disclosure relates to acoustic sensors and methods of making and using such acoustic sensors in various tools, including acoustic logging tools. 
     BACKGROUND OF THE DISCLOSURE 
     Downhole acoustic logging tools, which are used to investigate subsurface features, can include one or more acoustic transmitters or sources and a number of acoustic receivers. These tools may also include a transceiver that can function as a transmitter and a receiver. In either case, the transmitters emit acoustic energy into the subsurface environment surrounding the wellbore. The acoustic signals are reflected by interfaces associated with the wellbore, well structures, and/or the formation. The reflected acoustic signals are detected by the receivers in the logging tool and processed to provide estimates of one or more properties of the wellbore, well structures, and/or the formation. The present disclosure provides acoustic sensors that utilize a metamaterial lens to manipulate such acoustic waves. 
     SUMMARY OF THE DISCLOSURE 
     In aspects, the present disclosure provides a cell for manipulating an acoustic wave. The cell may include a plurality of spokes radiating from a hub. Each spoke may have a plurality of junctures and a plurality of fingers may be circumferentially distributed around the hub. At least one finger is connected at each juncture. 
     In still further aspects, the present disclosure provides an acoustic tool. The acoustic tool may include a transducer configured to generate and detect an acoustic wave and a lens. The lens may be spaced-apart from the acoustic source. The lens may have a planar surface bounded by a plurality of edges. The acoustic wave enters the lens via at least one the edges and travels along an axis that is parallel to the planar surface. The lens may include at least one lens element having a plurality of spokes radiating from a hub. Each spoke may include a plurality of fingers that are circumferentially distributed around the hub. The hub and the plurality of spokes may be planar members arranged parallel to the planar surface. 
     In aspects, the present disclosure provides an apparatus for investigating a subsurface volume. The apparatus may include an enclosure configured to be conveyed along a wellbore, an acoustic transducer disposed in the enclosure and generating acoustical signals, an electronics assembly disposed in the enclosure and controlling the acoustic transducer, and a lens assembly. The lens assembly may be disposed in the enclosure and next to the acoustic transducer. The lens assembly may be formed of a plurality of cells. Each cell may be formed as a column oriented transverse to a direction of travel of the acoustical signals. Each cell may have a hub, a plurality of spokes radiating from the hub, and a plurality of fingers circumferentially distributed around the hub. The hub, spokes, and fingers may be oriented to cause the acoustic waves to travel at a different speed in each of three orthogonal directions. 
     In aspects, the present disclosure provides a method for investigating a subsurface volume. The method may include positioning an acoustic tool in a wellbore. The acoustic tool may include an enclosure configured to be conveyed along a wellbore, an acoustic transducer disposed in the enclosure and generating acoustical signals, an electronics assembly disposed in the enclosure and controlling the acoustic transducer, and a lens assembly. The lens assembly may be disposed in the enclosure and next to the acoustic transducer. The lens assembly may be formed of a plurality of cells. Each cell may be formed as a column oriented transverse to a direction of travel of the acoustical signals. Each cell may have a hub, a plurality of spokes radiating from the hub, and a plurality of fingers circumferentially distributed around the hub. The hub, spokes, and fingers may be oriented to cause the acoustic waves to travel at a different speed in each of three orthogonal directions. The method may include directing the acoustic waves through an adjacent aberrating media that at least partially blocks the direction of travel of the acoustic waves and into the volume of interest. 
     Example features of the disclosure have been summarized rather broadly in order that the detailed description thereof that follows may be better understood and in order that the contributions they represent to the art may be appreciated. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       For a detailed understanding of the present disclosure, reference should be made to the following detailed description of the embodiments, taken in conjunction with the accompanying drawings, in which like elements have been given like numerals, wherein: 
         FIG. 1  is a schematic diagram of a prior art acoustic tool having an emitted acoustic signal distorted by an aberrating media; 
         FIG. 2  is a schematic diagram of an acoustic tool in accordance with the present disclosure that emits an emitted acoustic signal that is distorted less by an aberrating media vis-à-vis the  FIG. 1  acoustic tool; 
         FIG. 3  is a schematic diagram of an exemplary circular cell according to embodiments of the disclosure; 
         FIG. 4  is a schematic diagram of an exemplary octogonal cell according to embodiments of the disclosure; 
         FIG. 5  is a schematic diagram of an exemplary cell according to embodiments of the disclosure that uses fingers with projections; 
         FIG. 6  is a schematic diagram of an exemplary octogonal cell resulting after a shaping transformation of the  FIG. 3  embodiment; 
         FIG. 7  illustrates a lens that includes cells made in accordance with embodiments of the present disclosure; 
         FIG. 8  illustrates an acoustic tool using cells made in accordance with the present disclosure and disposed in a borehole intersecting an earth formation in which an aberrating media obstructs a travel path to a volume of interest; 
         FIGS. 9A-B  illustrate a lens assembly made in accordance with the present disclosure; 
         FIG. 10  illustrates an end view of the  FIG. 8  embodiment; 
         FIG. 11  is a theoretical plot of thru-casing intensity transmission of acoustic signals using and not using the teachings of the present disclosure; 
         FIGS. 12  A-B illustrate acoustic intensities obtained using a transducer without and with a lens according to the present disclosure; 
         FIG. 13A  illustrates an expected received signal from an anomaly reflection at the casing-cement bond interface when using embodiments of the present disclosure; and 
         FIG. 13B  illustrates an expected normalized net transmitter voltage spectrum due to the bond reflection when using embodiments of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     In aspects, the present disclosure relates to acoustic devices and related methods for investigating a volume of interest when a signal transmission path to that volume of interested is partially or completely obstructed by an aberrating media. In aspects, the present disclosure also relates to acoustic devices and related methods for investigating a volume of interest when a signal transmission path from that volume of interested is partially or completely obstructed by an aberrating media. Referring to  FIG. 1 , there is shown a source  200  for generating a signal  202 , which may be sonic or ultrasonic. An aberrating media  204  may partially or completely obstruct the path of the signal  202  to a volume of interest  206 . By obstruct, it is meant that the media  204  has one or more properties or characteristics that can block, distort, refract, reflect or otherwise undesirably affect the signal  202 . By way of example, the signal  202  is shown as having a reflection  208 . 
     Referring to  FIG. 2 , embodiments of the present disclosure position a lens  210  formed of an acoustic metamaterial between the source  200  and the aberrating media  204 . The lens  210  is shaped and configured to manipulate the signal  202  in a manner that minimizes undesirable effects in the signal  202  due to the aberrating media  204 . Specifically, the lens  210  may have anisotropic properties and may transmit longitudinal acoustic waves at a different speed in each of three orthogonal directions. The metamaterial making up the lens  210  may also deform with a different bulk moduli in each of the same three orthogonal directions. As a result, the lens  210  may be characterized by an anisotropic density tensor (diagonalized) and an anisotropic bulk modulus tensor having terms that are highly frequency dependent. In certain instances, these characteristics may generate an unnatural manipulation of incident acoustic energy that enhances evanescent, or rapidly disappearing, wave vector components of incident energy. This manipulation of acoustic energy can lead to the formation of complementary or quasi-complementary wave vectors that transmit the acoustic energy through an adjacent aberrating media with minimal loss or distortion relative to the incident energy. 
     The characteristics of such metamaterials derive from the basic dynamic mechanism of resonances, occurring in aggregate, to affect the creation of material properties beyond the bounds of Nature. In the particular case of acoustic metamaterials, these characteristics begin directly from the frequency response behavior of the two basic material parameters: effective mass density and effective bulk modulus. Both of these material parameters can take on negative attributes and even anisotropic behavior in the presence of particular types of resonance transition zones, specifically within proximity of anti-resonances. These anisotropic and negative properties can give rise to a wide range of spectral features within certain frequency bands, including negative refraction and hyperlensing, which can open possibilities for beam focusing and amplification in flat lenses. Actions such as focusing and amplification will be generally referred to as manipulating an acoustic wave. 
     Negative index properties arising from unit cell resonance within the frequency band may occur over a very narrow spectral bandwidth, the range of which is an effect of the resonance characteristics of the unit cell and the properties of the matrix and/or background fluid. In order to affect these negative properties over a broader frequency range, a multitude of compatible resonances in the elemental cell must couple constructively. This can be achieved through the metamaterial unit cell designs of the present disclosure. 
     There are several factors determining whether a particular unit cell design can affect the dispersion characteristics of a propagating wave to the extent of exhibiting anisotropic and negative index properties behavior necessary for hyperlensing. One particularly relevant factor is the creation of an aggregate of resonances in the transmission coefficient spectra that will couple to form a wide band of wave manipulation in the frequency range of interest. Whether the frequency band formed from the aggregate will give rise to anisotropic and negative index responses is a function of other properties of the resonance couplings. The extent to which a unit cell design exhibits significant acoustic impedance mismatch, absorption loss, and/or significant magnitude of bulk modulus are dominant influences, since any one of these can negate the hyperlensing effect in the frequency band. Cells according to the present disclosure may be referred to as elemental components in the superlens or hyperlens, depending on their characteristics. 
     Referring to  FIG. 3 , there is shown an exemplary metamaterial cell  10  for manipulating an acoustic wave in accordance with the present disclosure. Generally speaking, the cell  10  is a platen and disc like member. The cell  10  has two opposing planar surfaces that are parallel. As illustrated, the visible planar surface  13  is parallel with the paper. The distance between the two surfaces, or thickness, may be in the range of 1 millimeter to 100 millimeter. The diameter of a circle enclosing the cell  10  may be in the range of 1 millimeter to 5 millimeter. These dimensions are generally selected to allow phenomena such as resonances to have a measurable influence on the behavior of the cell  10  and affect wave manipulation in the particular frequency ranges of interest. The cells, such as cell  10 , of the present disclosure may be formed of metals or non-metals. Suitable metals include, but are not limited to, steel, platinum, tungsten, gold, and exotic options such as iridium, with the important material property for acoustic wave manipulation being the mass density of the metal. 
     One non-limiting embodiment of a cell  10  may include a hub  12 , a plurality of spokes  14  radiating from the hub  12 , and a plurality of concentrically arranged leaves  16 . 
     The hub  12  acts as a central support structure for the spokes  14 . In the embodiment shown, the hub  12  is formed of four separate segments, an illustrative segment being labeled with numeral  18 . The hub  12  may be circular or have any other suitable geometric shape. Also, while four segments  18  are shown, the hub  12  may be formed as a single integral body or have two or more segments  18 . Each segment  18  is physically connected to one or more spokes  14 . 
     The spokes  14  provide the structure for supporting the leaves  16 . An illustrative spoke is labeled with numeral  20 . The spoke  20  may be formed as an elongated bar having one or more necks  22 . A neck  22  is a section of the spoke  20  that has a cross-sectional area that is smaller than the immediately adjacent cross-sectional areas. Thus, the spoke  20  is more flexible at the necks  22  and can bend, twist, or otherwise deform more easily at the necks  22  than at other locations along the spoke  20 . In one arrangement, the necks  22  are formed immediately adjacent to and radially inward of each juncture  24  between the spoke  20  and the leaf  16 . Also, a neck  22  may be immediately radially outward of a juncture  26  between the spoke  20  and the hub  12 . 
     Each of the leaves  16  may include a circumferentially distributed array of fingers  28  arranged in the form of a circle. Each finger  28  is cantilevered from the juncture  24 . While two fingers  28  are shown at each juncture  24  with an orientation transverse to the spoke  20 , greater or fewer fingers  28  and different orientations may be used. The finger  28  may be a curved member that includes one or more enlarged sections  30 . An enlarged section  30  has more mass than an immediately adjacent section of the finger  28 . The increased mass is formed by providing the enlarged section  30  with a larger width than other sections of the finger  28 . Thus, there may be an asymmetric mass distribution along the length of the finger  28 . To interleave the fingers  28 , the radial positions of the finger  28  are staggered for each successive spoke  14 . Thus, a finger  28  of one spoke  14  may nest radially between two fingers  28  of an adjacent spoke  14 . While six leaves  16  are shown, greater or fewer leaves  16  may be used. 
     The cell  10  of  FIG. 3  has four segments  11   a - d . Each segment  11   a - d  has two spokes  14  radiating from a hub segment  18 . Each spoke  14  has three sets of fingers  28 . Each set has two fingers  28   a,b . The fingers  28  are circumferentially distributed to form six leaves  16 . It should be noted that the four segments  11   a - d  are structurally independent with one another. While the elements making up each of the segments  11   a - d  have been described separately, it should be understood that each segment  11   a - d  may be manufactured as a unitary body as opposed to being assembled from discrete components. It should also be noted that while four segments are depicted, the cell  10  may use fewer or greater number of segments. 
     The resonances and anti-resonances within the cell  10  are affected by the interaction of the various structural features through the matrix or background fluid medium described above. Thus, the number, size, shape, and orientation of features such as the spokes  14 , leaves  16 , fingers  28 , and enlarged sections  30  influence where and to what extent resonances and anti-resonances occur and how they complement or negate one another in affecting manipulation and control of the incident acoustic wave. 
     Referring to  FIG. 4 , there is shown another embodiment of a cell  40  for manipulating an acoustic wave. The general shape and dimensions of the cell  40  is similar to those of cell  10  ( FIG. 3 ). The cell  40  may include a hub  42 , a plurality of spokes  44  radiating from the hub  42 , and a plurality of concentrically arranged leaves  46 . The cell  40  is similar to the cell  10  of  FIG. 1  in many aspects. The variations of the cell  40  are discussed below. 
     As before, the spokes  44  provide the structure for supporting the leaves  46 . An illustrative spoke is labeled with numeral  50 . In this embodiment, the spoke  50  may be formed as an elongated bar that does not include any reduced cross-sectional areas. Also, the leaves  46  may include a circumferentially distributed array of fingers  52 , with each finger  52  being cantilevered from a juncture  54 . In this arrangement, the fingers  52  are straight members that include one or more enlarged sections  56 . 
     In  FIG. 4 , the leaves  46  are arranged as a plurality of concentric polygons. In the illustrated arrangement, the each of the leaves  46  has an octagon shape. However, other polygon shapes may be used. As before, the fingers  52  are interleaved by staggering the radial positions of the fingers  52  for each successive spoke  44 . The cell  40  of  FIG. 2  has four segments configured in the same manner as the cell  10  of  FIG. 1 . However, any number of segments may be used. 
     Referring to  FIG. 5 , there is shown another embodiment of a cell  70  for manipulating an acoustic wave. In a manner previously discussed, the fingers  72  of each leaf  74  are cantilevered from a spoke  76 . In this embodiment, the fingers  72  have projections  78  that are oriented transverse to the fingers  72 . Each of the projections  78  may be tab or bar like elements that can move independently relative to one another. By moving, it is meant bending, twisting, vibration, etc. While the projections  78  are shown projecting radially inward to a hub  80 , it should be appreciated that the fingers  72  may be arranged to have the projections  78  project radially outward, or both. 
     The shape, size, number, and orientation of the projections  78  within each leaf  74  and between the leaves  74  may be varied in order to influence the resonant behavior of the cell  70 . Thus, for instance, the projections  78  may be of different sizes along a finger  72  and each of the fingers  72  may have a different number of projections  78 . 
     Further, the cells according to the present disclosure need not be symmetric or quasi-isotropic as shown in  FIGS. 3-5 . 
     Certain embodiment of the present disclosure may incorporate anisotropy in the shape factor to influence the bandwidth and hyperlensing effect of negative index resonant bandwidths formed by a cell. In one methodology, anisotropy may be applied by invoking geometry shaping transformations that maintain the invariance of the Helmholtz wave equation; e.g., a Joukowsky shaping transformation. For instance, the transformation may be described by the relation: 
             ξ   =     Z   +       b   2     Z             
with the original cell coordinates:
 
 Z=x+iy  
 
and the transformed (shaped) cell coordinates:
 
ξ= u+iv  
 
A Joukowsky shaping may transform the  FIG. 4  octagonal cell boundary that can be circumscribed by a circle of radius R into a shaped cell geometry that can be circumscribed by the ellipse with shaping factor S=a/b where a and b are the elliptical dimensions collinear with the x and y axes, respectively. All remaining coordinates in the original interleaf cell geometry transform according to the same shaping factor S. Therefore, the Joukowsky transformation for an octagonal interleaf cell is:
 
     
       
         
           
             
               [ 
               
                 u 
                 + 
                 
                   i 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   v 
                 
               
               ] 
             
             = 
             
               
                 x 
                 ⁡ 
                 
                   [ 
                   
                     
                       2 
                       ⁢ 
                       S 
                     
                     
                       S 
                       + 
                       1 
                     
                   
                   ] 
                 
               
               + 
               
                 i 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   y 
                   ⁡ 
                   
                     [ 
                     
                       2 
                       
                         S 
                         + 
                         1 
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     These equations may be used to transform the 2D geometrical [x,y] coordinates of the cross-section for the quasi-isotropic cell illustrated previously in  FIG. 4  with an anisotropic Joukowsky transformation with shape factor S=2 anisotropy in the cell shape. The resulting anisotropic cell  90  is shown in  FIG. 6 . The S=2 anisotropy is merely one illustrative value. One feature of the anisotropic shaping is that a thickness of at least two fingers varies non-linearly along the plane on which a cell lies. For example, the thickness of fingers,  92 ,  94 , and  96  are different and the difference is mathematically non-linear. The comparison of thickness may be done by selecting the same feature (e.g., an enlarged section) and measuring a distance along the same location along the same axis. For instance, the edges  93 ,  95 , and  97  may be used as a measure of the thickness of the transformed sections of the fingers. An anisotropic shaping may also be applied to the cells of  FIGS. 3 and 5  or other cell configurations according to the present disclosure. 
     Referring to  FIG. 7 , there is shown a lens  100  having a plurality of cells  102 . While the cells  102  are depicted as the same shape of the cells of  FIG. 4 , the lens  100  may include any cell configuration according to the present disclosure. The cells  102  are arranged edge-to-edge and have planar surfaces aligned co-planar with one another; e.g., each has a planar surface parallel with the paper. An acoustic wave  104  enters the lens  100  principally through an edge  106 . The acoustic wave  104  travels through the lens  100  in a direction that is parallel with the plane along which all of the cells  102  lie. The cells  102  of the lens  100  manipulate the acoustic wave  104  such that a manipulated wave  108  exits from an edge  110  of the lens  100 . In one non-limiting embodiment, a substrate  101  may be used as a support from which each of the cells  102  project. For example, the cells  102  may be grown from the substrate  101  such that the cells  102  and substrate  101  are integral. For instance, the substrate  101  may be a wafer or a plate. The cells  102  and the structurally independent features making up each cell  102  may project from the substrate  101  in a cantilever fashion. 
     Referring to  FIG. 8 , there is shown one embodiment of an acoustic tool  120  according to the present disclosure. The tool  120  may be conveyed by a suitable conveyance device (not shown) along a borehole  124  drilled in an earthen formation  126 . The conveyance device may be a non-rigid carrier such as a wireline, e-line, slick-line, or coiled tubing, a rigid carrier such as drill pipe, a drop tool, or an autonomous device. In one non-limiting embodiment, the tool  120  includes an enclosure  128  that has an acoustic source cavity  130  that receives an acoustic source assembly  132  and an electronics cavity  134  that receives an electronics assembly  136 . A window  138  seals and encloses the acoustic source assembly  132  in the acoustic source cavity  130 . The window  138  may be constructed and formed to have similar acoustic impedance with the fluid filling the lens cavity  142 ; e.g., fabricating the window from Teflon material and using pure water as the lens cavity fluid. A cover  140  seals and encloses the electronics assembly  136  within the electronics cavity  134 . The acoustic source cavity  130  may have a lens section  142  and a source section  144 . Pressure compensation chambers  146  and  148  equalize the pressure between the exterior of the enclosure  128  and the sections  142  and  144 , respectively. The pressure compensation fluid in source section  144  will in general have acoustic impedance properties different than those of the fluid in lens section  142 . The fluid properties in lens section  142  are variables dictated by the cell acoustic properties and the frequency range desired. 
     The acoustic source assembly  132  generates and emits acoustic energy that can pass through an aberrating media with reduced distortion. In some situations, the aberrating media may be the metal making up a well tubular, such as a casing  150 . In one embodiment, the acoustic source assembly  120  includes a transducer  152  and a lens  154 . The transducer  152  may be any device configured to generate and receive sonic or ultrasonic signals. One illustrative non-limiting source may include piezoelectric elements. 
     The lens  154 , which is better illustrated in  FIGS. 9A and 9B , includes a plurality of cells  160  arranged in a grid-type manner. Each cell  160  may be shaped as a column that extends orthogonally/transversely to a direction of signal propagation, which is shown with arrow  162 . The illustrated embodiment includes two sections  170 ,  172 , each of which has a base  176  from which the cells  160  project in a cantilever-type fashion. The sections  170 ,  172  are arranged to mirror one another. Further, the cells  160  are aligned such that two facing cells  160  form effectively one column-like structure between the two bases  176 . The facing cells  160  may be separated by a gap, contact one another, or be fixed to one another. In embodiments, one end of the cell  160  may be fixed or both ends may be fixed. The cells  160  may have any of the cross-sectional shape and structures that were discussed above and shown in  FIGS. 3-5 . It should be understood that the lens  154  is not limited to any particular distribution of cells  160  or that such a distribution be symmetric or conform to a particular geometric shape. It should also be understood that the lens  154  may include only one section, e.g., section  170 , from which the cells  160  project. Also, the lens  154  may include an arrangement wherein two bases  148  are spanned by one cell  160  as opposed to two facing cells. 
     The electronics assembly  136  may include suitable electronics, microprocessors, memory modules, algorithms, power supplies, and circuitry in order to drive and sense the acoustic transducer  152 . The electronics assembly  136  may also include bi-directional communication hardware in order to transmit and/or receive data signals. 
     Referring now to  FIGS. 8-10 , an illustrative mode of operation of the acoustic tool  120  involves evaluating a cement body  180  ( FIG. 8 ), i.e., a volume of interest, that surrounds a well casing  150 . The well casing may be formed of a metal, such as steel. The evaluation may include estimating a quality of the contact or bond between the cement and the well casing  150 . During use, the electronics assembly  136  activates the acoustic transducer  152 . In response, the acoustic transducer  152  emits acoustical waves through the lens  154  along the arrow  162 . The acoustic waves may be sonic or ultrasonic and may have a narrow or wide frequency band. It should be noted that the waves enter the cells  120  along a surface facing toward the acoustic transducer surface and exit the cells  120  along a surface facing away from the acoustic transducer  152 . Thereafter, the acoustical waves pass through the well case  150  and into the cement body  180 . As discussed previously, the lens  154  manipulates the acoustic waves in a manner that allows these waves to pass through the metal of the well casing  150  with reduced distortion. A reflected wave  130  returns from the formation and enters the lens  154 . After being manipulated by the lens  154 , the wave  120  enters the transducer  152  and is processed. 
     Unexpectedly, the inventor discovered that the lens  154  can reduce distortion in acoustic signals that have already traveled through the aberrating media  204  as well as for acoustic signals travelling into the aberrating media  204 . That is, the lens  154  can manipulate an acoustic signal emitted into the aberrating media  204  and also manipulate a reflected signal from the zone of interest  206  that has traveled through the aberrating media  204 . Thus, the transducer  152  can act as a signal emitter and a signal detector. 
     Generally, it is desirable to evaluate a parameter or characteristic, such as a cement bond, along a complete circumference at a specified depth in the well. Thus, embodiments of the present disclosure may mount the tool  120  on a platform that is rotated by a suitable rotary device such as an electric or hydraulic motor. In some instances, the conveyance device on which the tool  120  is mounted, e.g., a drill string, may be rotated. In still other embodiments, a stationary array of two or more tools  120  may be circumferentially distributed along a plane in order to obtain full circumferential coverage. 
     Referring now to  FIG. 11 , there is shown an illustrative theoretical plot  190  of thru-casing intensity transmission. Line  192  illustrates the intensity of an acoustic transmission over a range of frequencies for a signal through one-half inch metal casing. This signal is transmitted directly into the one-half inch casing. Line  194  illustrates the intensity of an acoustic transmission over a range of frequencies for a signal through one-half inch metal casing. However, this signal is first manipulated by a lens as described above that has a cell with geometries as discussed above before entering the one-half inch casing. It should be noted that the line  194  demonstrates an increased signal intensity over a relatively broad frequency range. A peak  196  may occur as shown at a particular frequency. It should be appreciated, that the increased signal intensity is obtained without increasing the amplitude of the voltage signal applied to the transducer. 
     Referring now to  FIG. 12A ,B, there are shown graphs  230 ,  232 , respectively, illustrating the acoustic intensity of acoustic waves emitted by a transducer  152  in a casing  234  fixed in a borehole  236  that is filled by a borehole fluid  237 . The graphs  230 ,  232  depict an end view or top view; i.e., along a longitudinal axis of a borehole  236 . Also, for simplicity, the tool modeling is done using a symmetric half-section. The dark blue areas show regions of low acoustic intensity and the dark red regions show areas of high acoustic intensity. In  FIG. 12A , the transducer  152  emits a signal directly into the casing  234 . The acoustic intensity in the region  238  along the radial direction from the transducer has a diffuse acoustic intensity of low magnitude, which is generally considered undesirable for acoustic imaging. In  FIG. 12B , the transducer  152  emits a signal through a lens  154 . As can be seen, the lens  154  creates a relatively focused zone  240  of acoustic intensity along a radial direction from the transducer  152 , which is generally considered desirable for acoustic imaging. Numerical modeling suggests that the acoustic intensity in the region  240  may be an order of magnitude higher than the acoustic intensity in the region  238 . 
     Referring now to  FIG. 13A , there is shown a graph  260  illustrating a received signal  262  from an anomaly reflection at the casing-cement bond interface  264 . As before, there is shown a transducer  152  in a casing  234  fixed in a borehole  236  that is filled by a borehole fluid  237 . The graph  260  depicts an end view or top view; i.e., along a longitudinal axis of a borehole  236 . Graph  260  is an example of the reciprocal nature of the acoustic wave manipulation with the lens  154 , The right-hand side shows a contour plot of the received acoustic intensity distribution with dark blue color-coding indicating regions  266  of low intensity and dark red indicating regions  268  of high intensity. 
     Referring to  FIG. 13B , there is shown a normalized net transmitter voltage spectrum  270  due to the bond reflection. Line  272  represents the signal intensity of a thru-casing receiver with a lens according to the present disclosure and line  274  represents a signal intensity of a thru-casing receiver without such a lens. As can be seen in the region  276  of line  272 , about 10% of the voltage magnitude applied to the transducer  152  ( FIG. 13A ) during transmission is sensed by the transducer  152  ( FIG. 13A ) as a received signal from the cement bond reflection  264  ( FIG. 13A ). This ratio of received signal to applied voltage is generally considered desirable for acoustic imaging. 
     While the present disclosure is discussed in the context of a hydrocarbon producing well, it should be understood that the present disclosure may be used in any borehole environment (e.g., a water or geothermal well). Also, embodiments may be used in acoustic tools used at the surface or in bodies or water. 
     The present disclosure is susceptible to embodiments of different forms. There are shown in the drawings, and herein are described in detail, specific embodiments of the present disclosure with the understanding that the present disclosure is to be considered an exemplification of the principles of the disclosure and is not intended to limit the disclosure to that illustrated and described herein. While the foregoing disclosure is directed to the one mode embodiments of the disclosure, various modifications will be apparent to those skilled in the art. It is intended that all variations be embraced by the foregoing disclosure.