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BACKGROUND 
     1. Field of Invention 
     The present disclosure relates in general to a method of detecting subterranean bed boundaries using interferometric processing. More specifically, the present disclosure relates to interferometric processing of low frequency resistivity log data to locate a subterranean bed boundary during earth boring procedures. 
     2. Description of Prior Art 
     A resistivity measurement is one typical subterranean formation evaluation procedure where a log of the resistivity adjacent a wellbore is measured. Formation resistivity is a function of any fluids trapped within the subterranean formation. Thus resistivity is often measured to identify where water and/or hydrocarbon are present in the formation. Changes in resistivity in a subterranean formation can be abrupt and define a bed boundary. Resistivity can be measured with a wireline tool or a logging while drilling (LWD) tool. Measuring resistivity with a galvanic (DC) resistivity device typically involves forming an electrical potential in the formation and measuring a voltage between voltage measuring electrodes of the device. In an induction measurement device, magnetic flux/magnetic field is induced in the formation by the current in the transmitter; which induces a measured voltage in a receiver of the tool spaced axially from the transmitter. However, during LWD operations, there is a desire to “look ahead” so as to avoid drilling across bed boundaries or faults, as well as any subterranean geological hazard. 
     SUMMARY OF THE INVENTION 
     Disclosed herein is an example of a method of interferometric processing for looking ahead of a tool to measure distance to a bed boundary, and resistivity of a formation beyond the bed boundary. In an example, a method of investigating a subterranean formation using interferometric processing includes providing a tool string in a borehole that intersects the subterranean formation, providing a first current at a first location in the tool string that has a frequency of up to about 50 kHz and that induces a magnetic field in the formation, and measuring a first voltage along a receiver antenna at a second location in the tool string that is induced by the magnetic field in the formation, measuring a second voltage along a receiver antenna that is induced by the magnetic field in the formation and that is at a third location in the tool string which is spaced a distance from the first location that exceeds a distance from the first location to an end of the tool string proximate a bottom of the borehole. The method further includes estimating voltages based on the measured first and second voltages and identifying a bed boundary that is spaced away from a bottom of the borehole where a difference in the estimated voltages exceeds a threshold value. The method can also estimate a distance to the bed boundary as well as the resistivity of the formation beyond the bed boundary. In an example, the first voltage is measured by a first receiver in the tool string and the second voltage is measured by a second receiver in the tool string. Alternatively, the first current has a frequency of up to about 20 kHz. The second and third locations can be equidistantly spaced and on opposite sides of the first location. In this example, the distance from the second and third locations from the first location ranges up to around 50 feet. The tool string can further include a drill bit for forming the wellbore. In this example, the method can also include steering the drill bit in the formation based on the step of identifying the bed boundary. 
     Also provided herein is a method of investigating a subterranean formation that involves providing a tool string in a borehole that intersects the subterranean formation, providing a current in the tool string at a first location in the tool string and that has a frequency of up to about 50 kHz and that induces a current in the formation, and estimating one of a complex voltage with amplitude (magnitude) and the phase induced by the current in the formation at upper and lower locations in the tool string disposed on opposing sides of the first location and that are spaced apart from the first location at substantially the same distance. The method further includes identifying a bed boundary that is spaced away from a bottom of the borehole based on one of a difference (or a ratio of the voltage amplitudes) between voltage amplitudes estimated at the upper and lower locations, and a difference between phases estimated at the upper and lower locations. The upper and lower locations can be axially spaced from the first location at distances substantially equal to one another. In one example, the borehole has a deviated section. A bit can optionally be provided on a bottom of the tool string for forming the borehole. This example of the method further includes steering the bit based on the step of identifying the bed boundary. In one embodiment, voltage is measured at the lower location with a lower receiver and measuring voltage at the upper location with an upper receiver, and wherein when voltages measured by the upper and lower voltages begin to differ, a lower end of the tool string is spaced away from the bed boundary. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
       Some of the features and benefits of the present invention having been stated, others will become apparent as the description proceeds when taken in conjunction with the accompanying drawings, in which: 
         FIG. 1  is a side partial sectional view of an example embodiment of a logging while drilling (LWD) system on a drill string forming a borehole and in accordance with the present invention. 
         FIG. 2A  is a side partial sectional view of an example of the LWD system and drill string of  FIG. 1  shown approaching a bed boundary in accordance with the present invention. 
         FIG. 2B  is a side partial sectional view of an example of an embodiment of LWD system and drill string shown approaching a bed boundary in accordance with the present invention. 
         FIGS. 3A and 3B  are interferometric graphical examples of a voltage response measured in a borehole with the LWD system of  FIG. 2  in accordance with the present invention. 
         FIGS. 4A and 4B  are interferometric graphical examples of a phase response measured in a borehole with the LWD system of  FIG. 2  in accordance with the present invention. 
         FIGS. 5A and 5B  are interferometric graphical examples of voltage responses measured using the LWD system of  FIG. 2 , with different resistivity ratios across bed boundaries in accordance with the present invention. 
         FIGS. 6A and 6B  are interferometric graphical examples of voltage responses measured using the LWD system of  FIG. 2 , with different resistivity ratios across bed boundaries in accordance with the present invention. 
         FIGS. 7-11  are interferometric graphical examples of voltage responses measured using the LWD system of  FIG. 2 , with different resistivity ratios across bed boundaries in accordance with the present invention. 
         FIGS. 12 and 13  are nomograms that provide graphical examples of correlating voltage responses measured using different tool parameters and in accordance with the present invention. 
     
    
    
     While the invention will be described in connection with the preferred embodiments, it will be understood that it is not intended to limit the invention to that embodiment. On the contrary, it is intended to cover all alternatives, modifications, and equivalents, as may be included within the spirit and scope of the invention as defined by the appended claims. 
     DETAILED DESCRIPTION OF INVENTION 
     The method and system of the present disclosure will now be described more fully hereinafter with reference to the accompanying drawings in which embodiments are shown. The method and system of the present disclosure may be in many different forms and should not be construed as limited to the illustrated embodiments set forth herein; rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey its scope to those skilled in the art. Like numbers refer to like elements throughout. 
     It is to be further understood that the scope of the present disclosure is not limited to the exact details of construction, operation, exact materials, or embodiments shown and described, as modifications and equivalents will be apparent to one skilled in the art. In the drawings and specification, there have been disclosed illustrative embodiments and, although specific terms are employed, they are used in a generic and descriptive sense only and not for the purpose of limitation. 
     Shown in a partial side sectional view in  FIG. 1  illustrates one example of a tool string or drill string  10  shown forming a borehole  12  through a subterranean formation  14 . In the example, a drill bit  16  is provided on a lower end of the drill string  10 . An optional mud motor  18  is included in the drill string  10  and above the bit  16 . Further provided on the example of the drill string  10  of  FIG. 1  are receivers  22 ,  24  for receiving electromagnetic signals induced in the formation  14 . The receivers  22 ,  24  are spaced axially apart on the drill string  10  and on opposite sides of a transmitter  26 . In an example, receiver  22  is spaced a distance X 1  upward from transmitter  26  and receiver  24  is spaced a distance X 2  downward from transmitter  26 . Examples exist were X 1  is substantially equal to X 2 . The transmitter  26  of  FIG. 1  includes a coil or coils (not shown); that when an electrical current is provided that flows through the coil(s), a magnetic field is induced in the formation  14 . In the example of  FIG. 1 , flux lines  28  are illustrated in the formation  14  that represent the magnetic field induced in the formation  14  by transmitter  26 . 
     The receivers  22 ,  24  can sense the current in the formation  14 , e.g. the flux lines  28 . In an example, similar to the transmitter  26 , the receivers  22 ,  24  include a coil or coils (not shown) in which a voltage is induced in response to the magnetic field in the formation  14 . Measuring the voltage induced along the coils of receivers  22 ,  24  can yield information about the formation. An interferometric comparison, which can involve comparing measurements taken by receivers  22 ,  24 , is one example of a processing technique for assessing the formation  14 . In an embodiment, an interferometric comparison includes obtaining a difference of measurements taken by receivers  22 ,  24 , and in another embodiment can be a natural log of a quotient of measurements taken by receivers  22 ,  24 . In an example when the receivers  22 ,  24  are equidistant from the transmitter  26 , and the resistivity of the formation  14  intersected by the flux lines  28  is substantially homogeneous and distal from a bed boundary, an interferometric comparison will yield a value close or equal to zero. 
     Referring now to  FIG. 2A , a lower end of the borehole  12  is shown proximate a bed boundary  30  in the formation  14 . As indicated above, the bed boundary  30  can be defined along changes in the physical characteristics of the formation  14 , such as a change in the amount or type of fluid content. Thus in the example of  FIG. 2A , a layer  32  in the formation  14  on one side of the bed boundary  30  can have physical characteristics that are measurably different from a layer  34  of the formation  14  on an opposing side of the bed boundary  30 . As is known, the type and amount of fluid content in a formation can affect its resistivity. Further illustrated in  FIG. 2A , is that the flux lines  28  emitted from the drill string  10  intersect with the bed boundary  30 , meaning some of the flux lines  28  intersect with the layers  32 ,  34  on both sides of the bed boundary  30 . In this example, an inteferometric comparison of signals measured by the receivers  22 ,  24  can yield a finite value. Moreover, the nature of the flux lines  28  generated in the formation  14  are such that the presence of the bed boundary  30  can be identified before the drill string  10  contacts the bed boundary  30 . Depending on the circumstances, the drilling operator can cease drilling upon identification of the bed boundary  30 . Optionally, the operator can steer the drill string  10  so the borehole  12  does not intersect the bed boundary  30 . It should be pointed out that the interferometric comparison discussed herein can be used in boreholes that are vertical, horizontal, or otherwise deviated.  FIG. 2B  an alternate embodiment of a downhole tool  20 A is shown having a pair of transmitters  36 ,  38  and a single receiver  40 . In this example, transmitter  36  is disposed at a portion of the tool  20 A proximate the mud motor  18  or bit  16 , and transmitter  38  is on tool  20 A and distal from mud motor  18  and bit  16 . In an example, receiver  40  is on tool  20 A at or about a midpoint between transmitters  36 ,  38 . In the example of  FIG. 2B , transmitter  36  includes a coil or coils that when energized with a current flow induces a magnetic field in the formation  14  represented by flux lines  41 A. Similarly, a current flow through coil or coils in transmitter  38  induces a magnetic field in formation  14  represented by flux lines  41 B. Flux lines  41 A,  41 B in turn induces voltage in a coil or coils in receiver  40  that generates a measurable voltage along the coil(s). In an alternative, embodiments of the tool  20  of  FIG. 2A  exist having a single one of the receivers  22 ,  24 . Optionally, embodiments exist of the tool  20 A of  FIG. 2B  having a single one of the transmitters  36 ,  38 . In these optional embodiments, voltage measurements in the coil(s) of receivers  22 ,  24 ,  40  can be taken at first and second depths, and differences of the measured voltages can then be interferometrically processed to estimate distance from the tool  20 ,  20 A to a bed boundary. 
       FIG. 3A  graphically represents an example of an interferometric processing of voltages measured by an alternate embodiment of an imaging tool  20 A. In the example of  FIG. 3A , a schematic example of alternate imaging tool  20 A is shown that includes transmitters  36 ,  38  respectively on its lower and upper ends and receiver  40  between transmitters  36 ,  38 . For the purposes of discussion herein, transmitter  36  can be referred to as a lower transmitter and transmitter  38  as an upper transmitter. In one example, the receiver  40  is disposed at substantially a midpoint between the transmitters  36 ,  38  so that the tool  20 A is symmetrical. The abscissa of the graph of  FIG. 3A  represents distance in feet from receiver  40  to a bed boundary, and the ordinate represents a measured/induced voltage at the receiver  40  when the transmitters  36 ,  38  are respectively fired. The negative values on the ordinate represent a distance above or before the bed boundary, whereas the positive values reflect a distance below or past the bed boundary. In an example, the bed boundary defines a change of resistivity in the formation from about 10 Ohm-m to about 1 Ohm-m. Curves  42 ,  44  represent voltage responses respectively from the transmitters  36 ,  38 . 
       FIG. 3B  also has an abscissa representing distance in feet from the bed boundary, and an ordinate that represents a voltage response. A difference is that  FIG. 3B  illustrates a single curve  46  which represents the difference between curves  42 ,  44 , and thus depicts a difference in the voltage responses of the transmitters  36 ,  38 . 
     Referring now to  FIGS. 4A and 4B , graphical representations of phase responses are provided for the transmitters  36 ,  38  of the tool  20 A. Each of  FIGS. 4A and 4B  have an abscissa representing distance in feet from the bed boundary to receiver  40 , and an ordinate representing the phase of the induced signal at receiver  40 .  FIG. 4A  includes curves  48 ,  50  that respectively represent phase responses from transmitters  36 ,  38 . Similar to  FIG. 3B ,  FIG. 4B  provides a single curve  52  that illustrates differences in phase responses with respect to depth in the borehole in which the tool  20 A is disposed. Accordingly, as with the interferometric processing of recorded voltage response data depicted in  FIGS. 3A and 3B , interferometric processing of recorded phase response data can indicate the presence of a bed boundary in a formation and in the path of the oncoming drill string well before the drill string encounters the bed boundary. The forward looking information thus allows evasive or corrective action on the part of the drill string operator. 
     In the example of  FIGS. 3A and 3B  and  FIGS. 4A and 4B , the distance between the transmitters  36 ,  38  is at about 100 feet, thus putting the transmitters  36 ,  38  at an offset from the receiver  40  at around 50 feet. Also, the frequency of the signal generated in the tool  20 A ranges up to and includes about 20 kHz. However, embodiments exist wherein the present method can be employed wherein the offset between the transmitters  36 ,  38  and receiver  40  is at about 40 feet, or can be up to around 100 feet, or in excess of hundreds of feet. Note that in the example graphs, the delay in detected response of the transmitters  36 ,  38  is substantially the same as the offset from the receiver  40 . Optional embodiments exist wherein the present method can be employed wherein the frequency of the signal generated in the tool  20 A ranges up to and includes about 50 kHz. From  FIG. 3A  the magnitude of curve  44 , which represents the voltage response of the lower transmitter  36 , begins to decrease starting at about 75 feet from the bed boundary. Thus the boundary may be detected when the lower transmitter  36  is about 25 feet in front of the bed boundary at the interface of adjacent formations having different values of resistivity. The voltage difference is discernible in  FIG. 3B , where the bed boundary can be detected when the receiver  40  is 80 feet from the bed boundary, or when the lower transmitter  36  is about 30 feet from the bed boundary. Further evident in  FIGS. 3A and 3B  is how the detecting distance is significantly shorter by more than 10 feet in the conductive 1 Ohm-m layer in this example. Yet further illustrated in  FIGS. 3B and 4B , is how a slope of the phase response of curve  52  is much steeper than the slope or curve of the voltage response of curve  46 . 
       FIGS. 5A and 5B  are graphical examples of voltage responses at the transmitters  36 ,  38  for multiple changes in formation resistivity along a boundary bed, where the abscissa for these graphs has units in feet and the ordinate has units in voltage. More specifically, curves  54 ,  58 , and  62  represent a voltage response at the forward or lower transmitter  36 , and curves  56 ,  60 , and  64  represent a voltage response at the rear or upper transmitter  38 . Curves  54  and  56  represent voltage responses recorded where a resistivity above the boundary layer is at about 10 Ohm-m and resistivity below the boundary layer is at about 2 Ohm-m. Curves  58  and  60  represent voltage responses recorded where a resistivity above the boundary layer is at about 10 Ohm-m and resistivity below the boundary layer is at about 1 Ohm-m. Curves  62  and  64  represent voltage responses recorded where a resistivity above the boundary layer is at about 10 Ohm-m and resistivity below the boundary layer is at about 0.5 Ohm-m. As with  FIGS. 3A-4B  above, the bed boundary is represented at a value of 0 on the abscissa. Further in the example of  FIGS. 5A and 5B , the frequency of the signal generated in the tool  20 A is about 20 kHz. 
       FIGS. 6A and 6B  include graphical examples of voltage responses that are similar to those represented in  FIGS. 5A and 5B . One difference between the responses is that the formation resistivities in  FIGS. 6A and 6B  are different from those represented in  FIGS. 5A and 5B . Specifically referring to  FIG. 6A , curves  72 ,  76 , and  80  represent a voltage response at the forward or lower transmitter  36 , and curves  74 ,  78 , and  82  represent a voltage response at the rear or upper transmitter  38 . Curves  72  and  74  represent voltage responses recorded where a resistivity above the boundary layer is at about 20 Ohm-m and resistivity below the boundary layer is at about 1 Ohm-m. Curves  76  and  78  represent voltage responses recorded where a resistivity above the boundary layer is at about 10 Ohm-m and resistivity below the boundary layer is at about 1 Ohm-m. Curves  80  and  82  represent voltage responses recorded where a resistivity above the boundary layer is at about 5 Ohm-m and resistivity below the boundary layer is at about 1 Ohm-m. The examples of  FIGS. 5A and 6A  indicate that magnitudes of differences in voltage responses between the transmitters  36 ,  38  increase with larger resistivity contrasts across the bed boundary. Moreover, as depicted in the example of  FIG. 6A , the bed boundary can be detected at a greater distance when resistivity ratios are greater. Similar to the examples of  FIGS. 5A and 5B , the frequency of the signal generated in the tool  20 A is about 20 kHz. 
     In  FIGS. 7-10 , curves are provided graphically illustrate examples of differences in voltage responses (also referred to herein optionally as attenuation) between the lower transmitter  36  and upper transmitter  38  on the tool  20 A. Referring to  FIG. 7 , curves  92 ,  94 ,  96  represent differences in voltage responses of the transmitters  36 ,  38  where the resistivity ratios across the boundary layer are 10:0.5, 10:1, and 10:2 respectively. In  FIG. 7 , the formation above the boundary bed is 10 Ohm-m. Curves  98 ,  100 ,  102  of  FIG. 8  represent differences in voltage responses of the transmitters  36 ,  38  where the resistivity ratios across the boundary layer are 20:1, 10:1, and 5:1 respectively. In  FIG. 8 , the formation below the boundary bed is 1 Ohm-m. The frequency of the signal generated in the tool  20 A of  FIGS. 7 and 8  is about 50 kHz and the distance between the receiver  40  and transmitters  36 ,  38  is about 50 feet. 
       FIGS. 9 and 10  show differences in voltage response of transmitters  36 ,  38 , where the frequency is at about 20 kHz and the offset between the receiver  40  and transmitters  36 ,  38  is about 40 feet. Specifically with regard to  FIG. 9 , curves  104 ,  106 ,  108  represent differences in voltage responses of the transmitters  36 ,  38  where the resistivity ratios across the boundary layer are 10:0.5, 10:1, and 10:2 respectively. In  FIG. 10 , curves  110 ,  112 ,  114  represent differences in voltage responses of the transmitters  36 ,  38  where the resistivity ratios across the boundary layer are 20:1, 10:1, and 5:1 respectively. 
       FIGS. 12 and 13  include nomograms with curves generated from values of correlated measured voltage differences of some of the above described figures. More specifically, curve  122  of  FIG. 12  represents corresponding voltages differences of curve  66  of  FIG. 5B  (on the abscissa) and curve  92  of  FIG. 7  (on the ordinate), The corresponding abscissa and ordinate values for generating curve  122  are taken from the same depth. In an example, from  FIG. 5B  at about a distance of 20 feet above the bed boundary, curve  66  has a corresponding voltage of about 1.8E-9 volts; curve  92  of  FIG. 7  has a corresponding voltage of about 3.2E-9 volts at distance of 20 feet above the bed boundary. As shown in  FIG. 12 , curve  122  passes through point 1.8E-9, 3.2E-9. In similar fashion, curves  124 ,  126  represent corresponding voltage values for respectively for curves  68 ,  94  and curves  70 ,  96 . Knowing the distances from the bed boundary that correspond to the voltage values of curves  122 ,  124 ,  126 , lines  128 ,  130 ,  132 ,  134  can be generated on the nomogram, where in the example of  FIG. 12 , lines  128 ,  130 ,  132 ,  134  respectively represent corresponding voltage differences at 20, 30, 40, and 50 feet from the bed boundary. As noted above, voltage differences plotted in  FIG. 5B , and on the abscissa of  FIG. 12 , represent voltage values measured by a tool having an offset of 50 feet and generating signals of 20 kHz; voltage differences provided on the ordinate of  FIG. 12 , represent voltage values measured by a tool having an offset of 50 feet and generating signals of 50 kHz.  FIG. 13 , a nomogram like  FIG. 12 , is a plot that combines the voltage differences of  FIG. 5B  and  FIG. 9 . Curve  136  represents corresponding voltage values of curve  66  and curve  104 , curve  138  represents corresponding voltage values of curve  68  and  106 , and curve  140  represents corresponding voltage values of curve  70  and  108 , Lines  142 ,  144 ,  146 ,  148  respectively represent corresponding voltage differences at 20, 30, 40, and 50 feet from the bed boundary. Voltage differences on the abscissa of  FIG. 13  represent voltage values measured by a tool having an offset of 50 feet and generating signals of 20 kHz; voltage differences provided on the ordinate of  FIG. 13  represent voltage values measured by a tool having an offset of 40 feet and generating signals of 20 kHz. Thus by changing a tool parameter, a nomogram can be generated and used to look past the boundary and estimate formation properties, such as resistivity, under or on an opposite side of the bed boundary. In an example, the nomograms of  FIGS. 12 and 13  were generated from simultaneously solving equations having more than a single unknown. 
     The present invention described herein, therefore, is well adapted to carry out the objects and attain the ends and advantages mentioned, as well as others inherent therein. While a presently preferred embodiment of the invention has been given for purposes of disclosure, numerous changes exist in the details of procedures for accomplishing the desired results. These and other similar modifications will readily suggest themselves to those skilled in the art, and are intended to be encompassed within the spirit of the present invention disclosed herein and the scope of the appended claims.

Summary:
A method of identifying a bed boundary in a subterranean formation by processing data measured by an induction logging tool. An interferometric method compares recorded voltages and/or phases recorded at axially spaced apart receivers on the logging tool. A transmitter is on the logging tool and set between the receivers, where the receivers are equally spaced apart from the transmitter. The transmitter emits a signal having frequencies up to around 50 kHz.