Patent Publication Number: US-11035981-B2

Title: Air-hang calibration for resistivity-logging tool

Description:
TECHNICAL FIELD 
     The present disclosure relates to calibrating a wellbore tool used to measure a characteristic of a wellbore. More specifically, this disclosure relates to an air-hang calibration of a resistivity-logging tool. 
     BACKGROUND 
     In oilfield services operations, resistivity-logging tools can be used to produce resistivity measurements of a formation surrounding a borehole, among other uses. The resistivity measurements may provide an operator of the resistivity-logging tool with information relating to formation characteristics. The formation characteristics may generally alert an operator to the presence of hydrocarbon-bearing formations. 
     The resistivity-logging tools may produce measurement irregularities when drilling without calibration. Calibration techniques can be used for the resistivity-logging tools. But, existing calibration techniques suffer from disruption at singularity conditions and from systematic bias due to the receivers of the resistivity-logging tool not being collocated. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a cross-sectional view of an example of a well logging system that includes a resistivity-logging tool along a drill string according to some aspects of the present disclosure. 
         FIG. 2  is a cross-sectional view of an example of a well logging system that includes the resistivity-logging tool on a wireline logging tool according to some aspects of the present disclosure. 
         FIG. 3  is a side view of an example of a resistivity-logging tool that is calibrated using an air-hang technique according to some aspects of the present disclosure. 
         FIG. 4  is a flow chart of a process for air-hang calibrating formation measurement signals according to some aspects of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     Certain aspects and examples of the disclosure relate to calibrating a resistivity-logging tool using an air-hang calibration technique. The resistivity-logging tool is used to identify formation characteristics surrounding a wellbore. The air-hang calibration technique can involve firing the resistivity-logging tool while the resistivity-logging tool is located sufficiently far from any conductive materials or partially conductive materials (e.g., 20 feet above ground level) to generate air-hang measurements. Using the air-hang measurements, the resistivity-logging tool can be calibrated for subsurface measurements within the wellbore. By calibrating the resistivity-logging tool using the air-hang calibration technique, the resistivity-logging tool may take resistivity or conductivity measurements of a formation while avoiding disruptions at singularity conditions and reducing or eliminating a systematic bias associated with the resistivity-logging tool. 
     The singularity conditions may arise in certain calibration techniques at an offset azimuth angle of a receiver of the resistivity-logging tool. The offset azimuth angle may be a difference in azimuth angle between a transmitter and a receiver. The result is a calibration technique that is unable to calibrate a subsurface signal at the offset azimuth angle. Further, the systematic bias associated with certain calibration techniques may introduce error to a measurement signal of the resistivity-logging tool. The error may be a result of approximations in equations associated with calibration techniques. Certain examples using the presently disclosed air-hang calibration technique can avoid signal disruptions associated with both the singularity conditions and the systematic bias. 
     Resistivity-logging tools, according to some examples, may be used to help steer directional drilling operations. In other examples, the resistivity-logging tools may be used to identify pay zones within a formation surrounding a wellbore. The resistivity-logging tool can transmit and receive electromagnetic signals. The transmitted and received signals may travel through a formation surrounding a wellbore, and the resistivity measurements of the formation resulting from the interaction of the formation with the electromagnetic signals may provide an indication of formation characteristics. 
     A processing device may receive the measured signals to perform a calibration operation and to perform an inversion operation. The calibration operation can avoid irregularities from the measured signals associated with the performance of the resistivity-logging sensor or performance of a calibration technique, and the inversion operation can use calibrated signals to output quantitative formation properties. Thus, calibrating the measured signals using the air-hang calibration technique improves the field of resistivity-logging tool calibration. 
     These illustrative examples are given to introduce the reader to the general subject matter discussed here and are not intended to limit the scope of the disclosed concepts. The following sections describe various additional features and examples with reference to the drawings in which like numerals indicate like elements, and directional descriptions are used to describe the illustrative aspects but, like the illustrative aspects, should not be used to limit the present disclosure. 
       FIG. 1  is a cross-sectional view of an example of a well system  100  that includes a downhole logging tool according to some aspects. The well system  100  includes a wellbore  102  extending through a hydrocarbon-bearing subterranean formation  104 . A drilling platform  106  is equipped with a derrick  108  that supports a hoist  110 . The hoist  110  suspends a top drive  112  suitable for rotating a drill string  114  and lowering the drill string  114  through a wellhead  116 . Connected to a downhole end of the drill string  114  is a drill bit  118 . As the drill bit  118  rotates, it creates the wellbore  102  that pass through the layers of the formation  104 . A pump  120  circulates drilling fluid through a supply pipe  122  to the top drive  112 , down through an interior of the drill string  114 , through orifices in the drill bit  118 , back to the surface via an annulus around the drill string  114 , and into a retention pit  124 . The drilling fluid transports cuttings from the wellbore  102  into the pit  124  and also cools and lubricates the drill bit  118 . Various materials may be used for the drilling fluid, including, but not limited to, a salt-water based conductive mud. 
     A resistivity-logging tool  126  is integrated into a bottom-hole assembly near the drill bit  118 . In the illustrated embodiment, the resistivity-logging tool  126  is a logging while drilling (LWD) tool (e.g., usable during an LWD operation); however, in other embodiments, the resistivity-logging tool  126  may be used in a wireline or tubing-conveyed logging application. The resistivity-logging tool  126  may be, for example, a very deep resistivity (VDR) logging tool (e.g., with accurate readings at greater than 6 meters from the wellbore  102 ). Other types of resistivity-logging tools may also or alternatively be used. Additionally, the resistivity-logging tool  126  may be adapted to perform logging operations in both open and cased wellbore environments. 
     As the drill bit  118  extends the wellbore  102  through the formation  104 , the resistivity-logging tool  126  collects resistivity measurement signals relating to various formation properties. A telemetry sub  128  may be included on the drill string  114  to transfer images and measurement data/signals to a surface receiver  130  and to receive commands from the surface of the well system  100 . In some embodiments, the telemetry sub  128  does not communicate with the surface, but rather stores logging data for later retrieval at the surface when telemetry sub  128  is recovered. 
     The resistivity-logging tool  126  may include a tool control system (not shown), along with processing, storage, and communication hardware, that is communicatively coupled to one or more sensors (not shown) of the resistivity-logging tool  126 . The one or more sensors of the resistivity-logging tool  126  are used to acquire formation measurement signals that represent formation characteristics. When the formation measurement signals are acquired, the tool control system decouples the measurements and uses an air-hang calibration technique to provide a final formation profile based on the measurements of the formation  104  received by the resistivity-logging tool  126 , as discussed below with respect to  FIG. 4 . Upon obtaining the final formation profile, the data is sent uphole or to other assembly components by way of the telemetry sub  128 . In another embodiment, the tool control system may be located at a remote location away from the resistivity-logging tool  126 , such as at the surface or in a different borehole. 
     The logging sensors used by the resistivity-logging tool  126  may be resistivity sensors, such as magnetic or electric sensors. The magnetic sensors may include coil windings and solenoid windings that use induction to sense resistivity or conductivity of the formation  104 . The electric sensors may include electrodes, linear wire antennas, or toroidal antennas that rely on Ohm&#39;s law to perform the measurements of the formation  104 . Additionally, the sensors may be realizations of dipoles with an azimuthal moment direction and directionality, such as tilted coil antennas. 
     The resistivity-logging tool  126  may be a very deep resistivity-logging tool. Such tools may include one or more transmitter and receiver coils that are axially separated along the drill string  114 . The transmitter coils may generate alternating displacement currents in the formation  104 . The alternating displacement currents generate voltage at the one or more receiver coils. Because of the systematic bias associated with the resistivity-logging tool  126 , an air-hang calibration may be implemented, as discussed in detail below with respect to  FIG. 4 , to eliminate the systematic bias in the readings of the resistivity-logging tool  126 . 
       FIG. 2  is a cross-sectional view of an example of a well logging system  200  that includes the resistivity-logging tool  126  on a wireline logging tool  202  according to some aspects. At various times during the drilling process, the drill string  114  may be removed from the wellbore  102 , as shown in  FIG. 2 . During these times, logging operations of the formation  104  may continue using the resistivity-logging tool  126  positioned along the wireline logging tool  202 . The wireline logging tool  202  may include a probe suspended by a cable  204  having conductors that transport power to the wireline logging tool  202  and that transport telemetry data from the wireline logging tool  202  to the surface of the wellbore  102 . A logging facility  206  collects measurements from the wireline logging tool  202 , and the logging facility  206  includes a computer system  208  that processes and stores the measurements gathered by the sensors of the wireline logging tool  202 . 
     The logging facility  206  and the computer system  208  may include one or more memory devices and one or more processing devices. The memory devices may include a non-transitory computer-readable medium capable of storing instructions for operation of the resistivity-logging tool  126  that are executable by the processing devices. Additionally, the memory devices may store the measurements gathered by the resistivity-logging tool  126 . In one or more embodiments, the memory devices and the processing devices are located at the resistivity-logging tool  126  as an integrated resistivity-logging system. In other embodiments, the memory devices and the processing devices are located remotely from the resistivity-logging tool  126  and the wireline logging tool  202 . Further, the memory devices and the processing devices may be located at the resistivity-logging tool  126  during an LWD operation, as discussed above with respect to  FIG. 1 . Similar to the resistivity-logging tool  126  positioned on the drill string  114 , the one or more sensors of the resistivity-logging tool  126  positioned on the wireline logging tool  202  are used to acquire formation measurement signals that represent formation characteristics. When the formation measurement signals are acquired, the tool control system decouples the measurements and uses an air-hang calibration technique to provide a final formation profile based on the measurements of the formation  104  received by the resistivity-logging tool  126 , as discussed below with respect to  FIG. 4 . 
       FIG. 3  is a side view of an example of the resistivity-logging tool  126  that is calibrated using an air-hang technique according to some aspects. The resistivity-logging tool  126  includes a transmitter  302  and three receivers  304 ,  306 , and  308 . The receivers  304 ,  306 , and  308  may be resistivity sensors, such as magnetic or electric sensors. The transmitter  302  may include one or more coils that generate alternating displacement currents in the formation  104 . The alternating displacement currents generate voltage at the one or more receivers  304 ,  306 , and  308 . The receivers  304 ,  306 , and  308  may include coil windings and solenoid windings that use induction to sense resistivity or conductivity of the formation  104 . In an embodiment, the receivers  304 ,  306 , and  308  may include electrodes, linear wire antennas, or toroidal antennas that rely on Ohm&#39;s law to perform the measurements of the formation  104 . Additionally, the receivers  304 ,  306 , and  308  may be realizations of dipoles with an azimuthal moment direction and directionality, such as tilted coil antennas. 
     As illustrated, the first receiver  304  is spaced a distance S 1  from the transmitter  302 , the second receiver  306  is spaced a distance S 2  from the transmitter  302 , and the third receiver  308  is spaced a distance S 3  from the transmitter  302 . The receivers  304 ,  306 , and  308  each include different azimuthal angles (i.e., rotational or tool face angles) in relation to an azimuthal angle of the transmitter  302 . Further, the transmitter  302  and each of the receivers  304 ,  306 , and  308  include tilt angles (i.e., an angular deviation from an axis  310  of the resistivity-logging tool  126 ). The tilt angle and the azimuthal angles of the receivers  304 ,  306 , and  308  provide the resistivity-logging tool  126  with azimuthal sensitivity (i.e., directional sensitivity). For example, the tilt angles and the azimuthal angles of the illustrated resistivity-logging tool  126  enable directional resistivity measurements of the formation  104 , which may aid in steering the drill bit  118  or gathering data about the formation  104  around the wellbore  102 . Further, the tilt angles, the azimuthal angles, and the distances S 1 -S 3  may be referred to as parameters of the receivers  304 ,  306 , and  308  and the transmitter  302 . 
     The transmitter  302  may generate an electromagnetic signal that travels into and through the formation  104  surrounding the wellbore  102 . The receivers  304 ,  306 , and  308  each receive the electromagnetic signal after the electromagnetic signal traverses the respective distances S 1 , S 2 , and S 3 . The formation  104  may interact with the electromagnetic signal in different ways depending on the composition of the formation. For example, a porous formation that includes salty water in the pores may have a minimal effect on the electromagnetic signal due to a low resistivity of the porous formation and high conductivity of the salty water. Alternatively, a dense formation rich with hydrocarbons may have a significant impact on the electromagnetic signal due to a very high resistivity of the dense formation and the hydrocarbons. 
     A comparison of the electromagnetic signals received at the receivers  304 ,  306 , and  308  with the corresponding electromagnetic signals generated by the transmitter  302  may provide an indication of the resistivity of the formation  104 . The resistivity of the formation  104  may provide an operator with an indication of the characteristics of the formation  104  (e.g., a low resistivity may indicate that no hydrocarbons are present in the formation  104 ). The characteristics of the formation  104  may provide a steering system of a drilling operation with steering input to maintain the wellbore  102  within a pay zone. In another embodiment, the characteristics of the formation  104  may provide an operator with an indication of a portion of the wellbore  102  to isolate for a production operation. 
       FIG. 4  is a flowchart of a process  400  for air-hang calibrating formation measurement signals according to some aspects. Measurements taken by the resistivity-logging tool  126  are calibrated based on an air-hang calibration of the resistivity-logging tool  126  that removes irregularities in the measurements recorded by the resistivity-logging tool  126 . For example, the resistivity-logging tool  126  may include a systematic bias due to the receivers  304 ,  306 , and  308  not being collocated along the drill string  114  or the wireline tool  202 . 
     At block  402 , the process  400  involves performing an air-hang measurement using the resistivity-logging tool  126 . The air-hang measurement involves taking a resistivity measurement with the resistivity-logging tool  126  when the resistivity-logging tool  126  is located sufficiently far from any conductive materials or partially conductive materials (e.g., 20 feet above ground level). For example, the resistivity-logging tool  126  may be suspended in the air on a stand while performing a baseline resistivity measurement. 
     At block  404 , the process  400  involves a computing system, such as the computer system  208 , performing measurement decoupling on the resistivity measurement taken by the resistivity-logging tool  126 . The measurement decoupling may include extracting useful components from a multi-component signal received by one or more of the receivers  304 ,  306 , and  308 . For example, the receivers  304 ,  306 , and  308  read measurements V 1 , V 2 , and V 3  from the transmitter  302 , respectively. From the measurements V 1 , V 2 , and V 3 , Green&#39;s tensor 
                   [           Z   xx           Z   xy           Z   xz               Z   yx           Z   yy           Z   yz               Z   zx           Z   zy           Z   zz           ]           
is decoupled. In the matrix that represents Green&#39;s tensor, a first subscript denotes a component of the transmitter  302 , and a second subscript denotes a component of the receivers  304 ,  306 , and  308 . For example, Z xy  is a response of a receiver  304 ,  306 , and  308  oriented in a y-direction as a result of the transmitter  302  oriented in an x-direction. That is, the value of the Z elements in the Green&#39;s tensor matrix is a measurement of a receiver  304 ,  306 , or  308  when the receiver  304 ,  306 , or  308  and the transmitter  302  are oriented in directions specified by the subscript. The x-direction, the y-direction, and the z-direction are defined by a coordinate system of the resistivity-logging tool  126 , where an origin is at a center of the transmitter. For example, the z-direction aligns with a direction from a center of the receivers  304 ,  306 , and  308  to a center of the transmitter  302  (i.e., along the axis  310 ), the x-direction is perpendicular to the z-direction, and the y-direction is perpendicular to both the x-direction and the z-direction.
 
     As used during the measurement decoupling of block  404 , Green&#39;s tensor may indicate a response at a location of the second receiver  306  caused by the transmitter  302 . Because the tool is operating in air, which has the same effect on the signal from the transmitter  302  in any direction perpendicular to the z-direction, Z xx   air  represents both Z xx   air  and Z yy   air . Therefore, Z xx   air  and Z zz   air  are displayed in the Green&#39;s tensor matrix while the remaining values are all zero. Further, because the calibration technique involves measurements taken in air (i.e., an air-hang calibration), the superscript “air” may be removed in Green&#39;s tensor, which results in the matrix 
               [           Z   xx         0       0           0         Z   xx         0           0       0         Z   zz           ]     .         
The matrix
 
                   [           Z   xx         0       0           0         Z   xx         0           0       0         Z   zz           ]           
may be referred to as an air-hang calibration matrix.
 
     The goal of measurement decoupling involves calculating values for Z xx   R2  and Z zz   R2  to use in calibrating the resistivity-logging tool  126  to correct any irregularities of the measurements received by the receivers  304 ,  306 , and  308 . To calculate the values for Z xx   R2  and Z zz   R2 , a measured response by the receivers  304 ,  306 , and  308  is given by the following equation: 
                     V   ⁡     (       θ   T     ,     β   T     ,     θ   R     ,     β   R       )       =       [           sin   ⁢           ⁢     θ   T     ⁢   cos   ⁢           ⁢     β   T             sin   ⁢           ⁢     θ   T     ⁢   sin   ⁢           ⁢     β   T             cos   ⁢           ⁢     θ   T             ]     ⁢               [           Z   xx         0       0           0         Z   xx         0           0       0         Z   zz           ]     ⁡     [           sin   ⁢           ⁢     θ   R     ⁢   cos   ⁢           ⁢     β   R                 sin   ⁢           ⁢     θ   R     ⁢   sin   ⁢           ⁢     β   R                 cos   ⁢           ⁢     θ   R             ]       =         (       sin   ⁢           ⁢     θ   T     ⁢   cos   ⁢           ⁢     β   T     ⁢   sin   ⁢           ⁢     θ   R     ⁢   cos   ⁢           ⁢     β   R       +     sin   ⁢           ⁢     θ   T     ⁢   sin   ⁢           ⁢     β   T     ⁢   sin   ⁢           ⁢     θ   R     ⁢   sin   ⁢           ⁢     β   R         )     ·     Z   xx       +     cos   ⁢           ⁢     θ   T     ⁢   cos   ⁢           ⁢       β   R     ·     Z   zz                         (     Equation   ⁢           ⁢   1     )               
where θ T  is the tilted angle of the transmitter  302 , β T  is the azimuth angle of the transmitter  302 , θ R  is the tilted angle of the receiver  304 ,  306 , or  308 , and β R  is the azimuth angle of the receiver  304 ,  306 , or  308 .
 
     Applying Equation 1 to each of the receivers  304 ,  306 , and  308 , the following equations are provided:
 
 V   1 =(sin θ T  cos β T  sin θ R1  cos β R1 +sin θ T  sin β T  sin θ R1  sin β R1 )· Z   xx   R1 +cos θ T  cos β R1   ·Z   zz   R1   =A   1   ·Z   xx   R1   +B   2   ·Z   zz   R2   (Equation 2)
 
 V   2 =(sin θ T  cos β T  sin θ R2  cos β R2 +sin θ T  sin β T  sin θ R2  sin β R2 )· Z   xx   R1 +cos θ T  cos β R2   ·Z   zz   R2   =A   2   ·Z   xx   R2   +B   2   ·Z   zz   R2   (Equation 3)
 
 V   3 =(sin θ T  cos β T  sin θ R3  cos β R3 +sin θ T  sin β T  sin θ R3  sin β R3 )· Z   xx   R3 +cos θ T  cos β R3   ·Z   zz   R3   =A   3   ·Z   xx   R3   +B   3   ·Z   zz   R3   (Equation 4)
 
where V 1 , V 2 , and V 3  are measurements at the receivers  304 ,  306 , and  308  respectively. Values for θ T , θ R1 , θ R2 , θ R3  may all be 45 degrees, values for β R1  may be β T +β off −120°, values for β R2  may be β T +β off , and values for β R3  may be β T +β off +120°. Additionally, β off  may represent an offset azimuth angle of the receiver  306  from the azimuth angle of the transmitter  302 . Other tilt angles and azimuth angles are also contemplated within the scope of this disclosure.
 
     V 1 , V 2 , and V 3  may be derived using the values of Z xx   R2  and Z zz   R2 , as in the following equation: 
                     [             C     1   ⁢   2       ⁢     V   1                 V   2                 C     3   ⁢   2       ⁢     V   3             ]     =         [           A   1           B   1               A   2           B   2               A   3           B   3           ]     .     [           Z     x   ⁢   x       R   ⁢           ⁢   2                 Z   zz     R   ⁢           ⁢   2             ]       =         T     _   _       4     ·     [           Z     x   ⁢   x       R   ⁢   2                 Z   zz     R   ⁢           ⁢   2             ]                 (     Equation   ⁢           ⁢   5     )               
where C 12  and C 32  are correcting factors introduced to remove a dependence of V 1  on Z zz   R1  and Z xx   R1  and to remove a dependence of V 3  on Z zz   R3  and Z xx   R3 . An equivalent representation of Equation 5 is as follows:
 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             V 
                             2 
                           
                         
                       
                       
                         
                           
                             
                               
                                 C 
                                 
                                   1 
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                                   2 
                                 
                               
                               ⁢ 
                               
                                 V 
                                 1 
                               
                             
                             + 
                             
                               
                                 C 
                                 
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                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
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                         ] 
                       
                       · 
                       
                         [ 
                         
                           
                             
                               
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                                   ⁢ 
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                       · 
                       
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                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     6 
                   
                   ) 
                 
               
             
           
         
       
     
     Solving for Z xx   R2  and Z zz   R2  using Equation 5 provides: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             Z 
                             
                               x 
                               ⁢ 
                               x 
                             
                             
                               R 
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                             Z 
                             zz 
                             
                               R 
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                     ] 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               
                                 T 
                                 _ 
                               
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                             4 
                             H 
                           
                           · 
                           
                             
                               
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                         - 
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                     · 
                     
                       
                         
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                       4 
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                                 C 
                                 
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                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     7 
                   
                   ) 
                 
               
             
           
         
       
     
     And solving for Z xx   R2  and Z zz   R2  using Equation 6 provides: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             Z 
                             
                               x 
                               ⁢ 
                               x 
                             
                             
                               R 
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                             Z 
                             zz 
                             
                               R 
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                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               
                                 T 
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                             5 
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                     · 
                     
                       
                         
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                       5 
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                     · 
                     
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                               V 
                               2 
                             
                           
                         
                         
                           
                             
                               
                                 
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                                     1 
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                                   V 
                                   1 
                                 
                               
                               + 
                               
                                 
                                   C 
                                   
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                                     2 
                                   
                                 
                                 ⁢ 
                                 
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                                   3 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   C 
                                   
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                                   1 
                                 
                               
                               - 
                               
                                 
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                                     ⁢ 
                                     2 
                                   
                                 
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                                   3 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     8 
                   
                   ) 
                 
               
             
           
         
       
     
     To determine the values of C 12  and C 32  that are used to solve for Z xx   R2  and Z zz   R2 , a physical model based on Green&#39;s theorem of electromagnetism provides: 
                         H   _     _     ⁡     (   r   )       =     =         e   ikr       4   ⁢           ⁢   π   ⁢           ⁢     r   3         ⁡     [               k   2     ⁢     r   2       +   ikr   -   1         0       0           0             k   2     ⁢     r   2       +   ikr   -   1         0           0       0         2   -     2   ⁢           ⁢   ikr             ]                 (     Equation   ⁢           ⁢   9     )               
where k is a wavenumber in air. Using Equation 9, the values of C 12  and C 32  are calculated with the following equations:
 
                     C   12     =               [           sin   ⁢           ⁢     θ   T     ⁢   cos   ⁢           ⁢     β   T             sin   ⁢           ⁢     θ   T     ⁢   sin   ⁢           ⁢     β   T             cos   ⁢           ⁢     θ   T             ]     ⁢         H   _     _     ⁡     (     S   2     )                   [           sin   ⁢           ⁢     θ     R   ⁢           ⁢   1       ⁢   cos   ⁢           ⁢     β     R   ⁢           ⁢   1                   sin   ⁢           ⁢     θ     R   ⁢           ⁢   1       ⁢   sin   ⁢           ⁢     β     R   ⁢           ⁢   1                   cos   ⁢           ⁢     θ     R   ⁢           ⁢   1               ]                     [           sin   ⁢           ⁢     θ   T     ⁢   cos   ⁢           ⁢     β   T             sin   ⁢           ⁢     θ   T     ⁢   sin   ⁢           ⁢     β   T             cos   ⁢           ⁢     θ   T             ]     ⁢         H   _     _     ⁡     (     S   1     )                   [           sin   ⁢           ⁢     θ     R   ⁢           ⁢   1       ⁢   cos   ⁢           ⁢     β     R   ⁢           ⁢   1                   sin   ⁢           ⁢     θ     R   ⁢           ⁢   1       ⁢   sin   ⁢           ⁢     β     R   ⁢           ⁢   1                   cos   ⁢           ⁢     θ     R   ⁢           ⁢   1               ]                     (     Equation   ⁢           ⁢   10     )                 C   32     =               [           sin   ⁢           ⁢     θ   T     ⁢   cos   ⁢           ⁢     β   T             sin   ⁢           ⁢     θ   T     ⁢   sin   ⁢           ⁢     β   T             cos   ⁢           ⁢     θ   T             ]     ⁢         H   _     _     ⁡     (     S   2     )                   [           sin   ⁢           ⁢     θ     R   ⁢           ⁢   3       ⁢   cos   ⁢           ⁢     β     R   ⁢           ⁢   3                   sin   ⁢           ⁢     θ     R   ⁢           ⁢   3       ⁢   sin   ⁢           ⁢     β     R   ⁢           ⁢   3                   cos   ⁢           ⁢     θ     R   ⁢           ⁢   3               ]                     [           sin   ⁢           ⁢     θ   T     ⁢   cos   ⁢           ⁢     β   T             sin   ⁢           ⁢     θ   T     ⁢   sin   ⁢           ⁢     β   T             cos   ⁢           ⁢     θ   T             ]     ⁢         H   _     _     ⁡     (     S   3     )                   [           sin   ⁢           ⁢     θ     R   ⁢           ⁢   3       ⁢   cos   ⁢           ⁢     β     R   ⁢           ⁢   3                   sin   ⁢           ⁢     θ     R   ⁢           ⁢   3       ⁢   sin   ⁢           ⁢     β     R   ⁢           ⁢   3                   cos   ⁢           ⁢     θ     R   ⁢           ⁢   3               ]                     (     Equation   ⁢           ⁢   11     )               
where S 1  is a distance between the receiver  304  and the transmitter  302 , S 2  is a distance between the receiver  306  and the transmitter  302 , and S 3  is a distance between the receiver  308  and the transmitter  302 . In this manner, the value of C 12  is based, in part, on a distance between the receiver  304  and the receiver  306  (i.e., taking into account the values of S 1  and S 2 ), and the value of C 32  is based, in part, on a distance between the receiver  306  and the receiver  308  (i.e., taking into account the values of S 2  and S 3 ). Further, each of C 12  and C 32  are based on Green&#39;s theorem of electromagnetism.
 
     In an embodiment where a downhole measurement is corrupted by noise, the Tikhonov regularization may be applied to add robustness to the calculation of Z xx   R2  and Z zz   R2 . Applying the Tikhonov regularization to Equation 7 and Equation 8 results in the following equations: 
                     [           Z   xx     R   ⁢   2                 Z   zz     R   ⁢   2             ]     =         (           T     ¯   _       4   H     ·         T   ¯     ¯     4       +     λ   ⁢       I   ¯     ¯         )       -   1       ·         T   ¯     ¯     4   H     ·     [             C     1   ⁢   2       ⁢     V   1                 V   2                 C     3   ⁢   2       ⁢     V   3             ]               (     Equation   ⁢           ⁢   12     )                 [           Z   xx     R   ⁢   2                 Z   zz     R   ⁢   2             ]     =         (             T   ¯     ¯     5   H     ·         T   ¯     ¯     5       +     λ   ⁢       I   ¯     ¯         )       -   1       ·         T   ¯     ¯     5   H     ·     [           V   2                   C   12     ⁢     V   1       +       C   32     ⁢     V   3                       C   12     ⁢     V   1       -       C   32     ⁢     V   3               ]               (     Equation   ⁢           ⁢   13     )               
where a regularization factor λ is determined using an L-curve method. Using Equations 7, 8, 12, or 13 as a basis for the calculation of Z xx   R2  and Z zz   R2  and the air-hang calibration matrix results in continuity at a singularity condition and removal of systematic bias of the resistivity-logging tool  126  from the measurements.
 
     Turning now to operation of the resistivity-logging tool  126  in a downhole environment within the wellbore  102 , at block  406 , the process  400  involves the resistivity-logging tool  126  performing a formation resistivity measurement on the formation  104 . As opposed to the air-hang measurement described above at block  402 , the formation resistivity measurement is performed within the wellbore  102 . For example, the resistivity-logging tool  126  may operate on the drill string  114  or the wireline logging tool  202  to take the formation resistivity measurement. 
     At block  408 , the process  400  involves the computing system, such as the computer system  208 , performing measurement decoupling on the formation resistivity measurements taken by the resistivity-logging tool  126 . Similar to the air-hang decoupling operation at block  404 , the formation measurement decoupling involves extracting useful components from a multi-component signal. For example, the receivers  304 ,  306 , and  308  read measurements V 1 , V 2 , and V 3  from the transmitter  302 , respectively. From the measurements V 1 , V 2 , and V 3 , Green&#39;s tensor, 
               [           Z   xx   DH         0       0           0         Z   yy   DH         0           0       0         Z   zz   DH           ]     ,         
is decoupled. The superscript DH represents that the response occurs in a downhole location within the wellbore  102 . The resulting matrix may be referred to as a decoupled formation measurement matrix.
 
     At block  410 , the process  400  involves the computing system, such as the computer system  208 , calibrating the decoupled measurement components identified at block  408  using air-hang calibration. The values Z xx   R2  and Z zz   R2  of the air-hang calibration matrix found at block  404  are used to offset irregularities associated with the receivers  304 ,  306 , and  308 . For example, Z xx   R2  may be subtracted from Z xx   DH  and Z yy   DH . Additionally, Z xx   R2  may be subtracted from Z zz   DH  to remove signal irregularities associated with the resistivity-logging tool  126  in a manner that does not fail at a singularity condition and removes systematic bias associated with the varying distances S 1 -S 3  of the receivers  304 ,  306 , and  308  from the transmitter  302 . The resulting matrix of the calibration process may be referred to as a calibrated formation measurement matrix. 
     At block  412 , the process  400  involves the computing system, such as the computer system  208 , performing an inversion operation on the calibrated measurement components to generate quantitative formation properties. The inversion operation may involve generating a model of Earth layers of the formation  104  based on the calibrated measurements of the resistivity-logging tool  126 . The computer system  208  may perform one or more inversion techniques to generate the quantitative formation properties. In an embodiment, the inversion operation may involve generating a two-dimensional representation of a location of the layers of the formation  104  surrounding the wellbore  102 . In other embodiments, the inversion operation may involve generating a three-dimensional representation of the layers of the formation  104  surrounding the wellbore  102 . Other inversion techniques are also contemplated within the scope of this disclosure that make use of the calibrated measurements of the resistivity-logging tool  126 . 
     Additionally, the quantitative formation properties identified by the inversion operation may guide steering operations for the drill bit  118  in a directional drilling environment. For example, when the inversion operation on the calibrated measurements indicates that the drill bit  118  is presently within the pay zone of the formation  104 , a steering system for the drill bit  118  may be controlled by the computing system to drill in a direction that stays within the pay zone. In this manner, the wellbore  102  may be generated with a focus on increasing surface area of the wellbore  102  within the pay zone of the formation  104 . 
     In some aspects, systems, devices, and methods for using a resistivity-logging tool using air-hang calibration are provided according to one or more of the following examples: 
     As used below, any reference to a series of examples is to be understood as a reference to each of those examples disjunctively (e.g., “Examples 1-4” is to be understood as “Examples 1, 2, 3, or 4”). 
     Example 1 is a method, comprising: performing an air-hang measurement using a resistivity-logging tool; performing a first measurement decoupling operation on the air-hang measurement, wherein the first measurement decoupling operation comprises: generating correcting factors based on a physical model; and calculating an air-hang calibration matrix based on the correcting factors; performing a formation measurement using the resistivity-logging tool; performing a second measurement decoupling operation on the formation measurement to generate a decoupled formation measurement matrix; and calibrating the resistivity-logging tool to generate a calibrated formation measurement matrix by subtracting the air-hang calibration matrix from the decoupled formation measurement matrix. 
     Example 2 is the method of example 1, further comprising: performing an inversion operation on the calibrated formation measurement matrix to generate quantitative formation properties; and steering a drill bit within a wellbore based on the quantitative formation properties. 
     Example 3 is the method of examples 1 or 2, wherein the physical model is based on Green&#39;s theorem of electromagnetism. 
     Example 4 is the method of examples 1-3, wherein the air-hang calibration matrix comprises a response of a receiver oriented in an x-direction as a result of a transmitter oriented in the x-direction, and a response of the receiver oriented in a z-direction as a result of the transmitter oriented in the z-direction. 
     Example 5 is the method of examples 1-4, wherein the resistivity-logging tool is a very deep resistivity-logging tool, and the very deep resistivity-logging tool takes measurements of greater than 6 meters from a borehole. 
     Example 6 is the method of examples 1-5, wherein generating the correcting factors comprises: generating a first correcting factor based on a first distance between a first receiver and a second receiver of the resistivity-logging tool; and generating a second correcting factor based on a second distance between the second receiver and a third receiver of the resistivity-logging tool. 
     Example 7 is the method of examples 1-6, wherein generating the calibrated formation measurement matrix comprises removing systematic bias of the resistivity-logging tool and removing disruptions at singularity conditions. 
     Example 8 is the method of examples 1-7, wherein calculating the air-hang calibration matrix is further based on a Tikhonov regularization. 
     Example 9 is a non-transitory computer-readable medium that includes instructions that are executable by a processing device to perform operations comprising: receiving results of an air-hang measurement using a resistivity-logging tool; performing a first measurement decoupling operation on the air-hang measurement, wherein the first measurement decoupling operation comprises: generating correcting factors based on a physical model, distances between a set of receivers of the resistivity-logging tool, and device parameters of the set of receivers and a transmitter of the resistivity-logging tool; and calculating an air-hang calibration matrix based on the correcting factors; receiving results of a formation measurement using the resistivity-logging tool; performing a second measurement decoupling operation on the formation measurement to generate a decoupled formation measurement matrix; and calibrating the decoupled formation measurement matrix to generate a calibrated formation measurement matrix by subtracting the air-hang calibration matrix from the decoupled formation measurement matrix. 
     Example 10 is the non-transitory computer-readable medium of examples 9, wherein the device parameters of the set of receivers and the transmitter comprise tilt angles and azimuth angles of the set of receivers and the transmitter. 
     Example 11 is the non-transitory computer-readable medium of examples 9 or 10, wherein the set of receivers comprises three receivers, and each of the three receivers is positioned at a different distance from the transmitter. 
     Example 12 is the non-transitory computer-readable medium of examples 9-11, wherein the air-hang measurement is performed at a location remote from any conductive materials or partially conductive materials. 
     Example 13 is the non-transitory computer-readable medium of examples 9-12, wherein the physical model is based on Green&#39;s theorem of electromagnetism. 
     Example 14 is the non-transitory computer-readable medium of examples 9-13, wherein the formation measurement is performed during a logging while drilling operation, and the instructions are executable by the processing device to perform operations comprising: performing an inversion operation on the calibrated formation measurement matrix to generate quantitative formation properties; and steering a drill bit based on the quantitative formation properties. 
     Example 15 is the non-transitory computer-readable medium of example 14, wherein the quantitative formation properties comprise a representation of a location of formation layers surrounding a wellbore. 
     Example 16 is a resistivity-logging system, the resistivity-logging system comprising: a resistivity-logging tool to measure formation resistivity within a wellbore, wherein the resistivity-logging tool comprises an electromagnetic signal transmitter and a set of receivers; a processing device; and a memory device in which instructions are stored that are executable by the processing device for causing the processing device to: perform an air-hang measurement using the resistivity-logging tool; perform a first measurement decoupling operation on the air-hang measurement, wherein the first measurement decoupling operation comprises: generating correcting factors based on a physical model; and calculating an air-hang calibration matrix based on the correcting factors; perform a formation measurement using the resistivity-logging tool; perform a second measurement decoupling operation on the formation measurement to generate a decoupled formation measurement matrix; and calibrate the decoupled formation measurement matrix to generate a calibrated formation measurement matrix by subtracting the air-hang calibration matrix from the decoupled formation measurement matrix. 
     Example 17 is the resistivity-logging system of example 16, wherein the instructions stored in the memory device are executable by the processing device for causing the processing device to: perform an inversion operation on the calibrated formation measurement matrix to generate quantitative formation properties; and control a drill bit steering operation based on the quantitative formation properties. 
     Example 18 is the resistivity-logging system of example 17, wherein the quantitative formation properties comprise a three-dimensional representation of layers of a formation surrounding the wellbore. 
     Example 19 is the resistivity-logging system of examples 16-18, wherein generating the correcting factors is based on a distance of each receiver of the set of receivers from the electromagnetic signal transmitter. 
     Example 20 is the resistivity-logging system of examples 16-19, wherein the resistivity-logging tool is positioned along a wireline logging tool. 
     The foregoing description of certain examples, including illustrated examples, has been presented only for the purpose of illustration and description and is not intended to be exhaustive or to limit the disclosure to the precise forms disclosed. Numerous modifications, adaptations, and uses thereof will be apparent to those skilled in the art without departing from the scope of the disclosure.