Abstract:
Signal measurements collected by azimuthally sensitive electromagnetic logging tool as a function of position in a borehole are obtained. The signal measurements are used to identify a type of formation model. An inversion technique appropriate to the type of formation model is applied to the set of signal measurements to determine a geophysical property. The geophysical property is used to make a decision regarding drilling a well.

Description:
BACKGROUND 
       [0001]    The basic principles and techniques for electromagnetic (EM) logging for earth formations are well known. For example, induction logging to determine the resistivity (or its inverse, conductivity) of earth formations adjacent a borehole has long been a standard and important technique in the search for and recovery of subterranean petroleum deposits. In brief, a transmitter transmits an electromagnetic signal that passes through formation materials around the borehole and induces a signal in one or more receivers. The amplitude and/or phase of the receiver signals are influenced by the formation resistivity, enabling resistivity measurements to be made. The measured signal characteristics and/or formation properties calculated therefrom are recorded as a function of the tool&#39;s depth or position in the borehole, yielding a formation log that can be used by analysts. 
         [0002]    EM techniques are used in logging-while-drilling (LWD) and wireline logging. Such techniques are used to evaluate formation parameters, including formation resistivity, formation anisotropy, boundary position, etc. 
         [0003]    A parameter (such as resistivity) of a given formation may be isotropic (equal in all directions) or anisotropic (unequal in different directions). Measuring such a parameter in an unconventional formation, containing for example fractures and/or folds, is a challenge. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0004]      FIG. 1  shows an illustrative logging while drilling environment including dipping formation beds. 
           [0005]      FIG. 2  shows an illustrative wireline logging environment including dipping formation beds. 
           [0006]      FIG. 3  shows a relationship between the orientation of a borehole and a dipping formation bed. 
           [0007]      FIG. 4  shows a hypothetical antenna arrangement for a tool having an orthogonal triaxial transmitter and two orthogonal triaxial receivers. 
           [0008]      FIG. 5  shows angles for defining the orientation of a tilted antenna. 
           [0009]      FIG. 6  is a block diagram of an illustrative electronics module for an electromagnetic resistivity tool. 
           [0010]      FIG. 7  shows an illustrative electromagnetic resistivity logging tool having tilted transmitter and receiver antennas. 
           [0011]      FIG. 8  illustrates an antenna arrangement. 
           [0012]      FIGS. 9 and 10  illustrate conventional formation models. 
           [0013]      FIG. 11A  illustrates bins. 
           [0014]      FIG. 11B  illustrates mathematical relationships between the bins and the formation from which data is being collected. 
           [0015]      FIGS. 12-15 , and  18 - 19  illustrate modeling responses. 
           [0016]      FIGS. 16 and 17  illustrate unconventional formation models. 
           [0017]      FIGS. 20-23  illustrate antenna arrangements. 
           [0018]      FIG. 24  is a flow chart. 
           [0019]      FIG. 25  illustrates an environment. 
       
    
    
     DETAILED DESCRIPTION 
       [0020]    The disclosed tool configurations and operations are best understood in the context of the larger systems in which they operate. Accordingly, an illustrative logging while drilling (LWD) environment is shown in  FIG. 1 . A drilling platform  2  supports a derrick  4  having a traveling block  6  for raising and lowering a drill string  8 . A kelly  10  supports the drill string  8  as it is lowered through a rotary table  12 . A drill bit  14  is driven by a downhole motor and/or rotation of the drill string  8 . As bit  14  rotates, it creates a borehole  16  that passes through various formations  18 . A pump  20  circulates drilling fluid through a feed pipe  22  to kelly  10 , downhole through the interior of drill string  8 , through orifices in drill bit  14 , back to the surface via the annulus around drill string  8 , and into a retention pit  24 . The drilling fluid transports cuttings from the borehole into the pit  24  and aids in maintaining the borehole integrity. 
         [0021]    An electromagnetic resistivity logging tool  26  is integrated into the bottom-hole assembly near the bit  14 . As the bit extends the borehole through the formations, logging tool  26  collects measurements relating to various formation properties as well as the tool orientation and position and various other drilling conditions. The orientation measurements may be performed using an azimuthal orientation indicator, which may include magnetometers, inclinometers, and/or accelerometers, though other sensor types such as gyroscopes may be used. In some embodiments, the tool includes a 3-axis fluxgate magnetometer and a 3-axis accelerometer. The logging tool  26  may take the form of a drill collar, i.e., a thick-walled tubular that provides weight and rigidity to aid the drilling process. A telemetry sub  28  may be included to transfer tool measurements to a surface receiver  30  and to receive commands from the surface receiver. 
         [0022]    More preferably, a rotational position indicator located near the logging tool  26  or the telemetry sub  28  may contain both a 3-axis fluxgate magnetometer and a 3-axis accelerometer. As is known in the art, the combination of those two sensor systems enables the measurement of the toolface, inclination, and azimuth orientation angles of the borehole. The toolface and hole inclination angles are calculated from the accelerometer sensor output. The magnetometer sensor outputs are used to calculate the hole azimuth. With the toolface, the hole inclination, and the hole azimuth information, a tool in accordance with the present invention can be used to steer the bit to the desirable bed. Specifically, the response difference or the response ratio can be used effectively to enter a desired payzone or to stay within the payzone of interest. 
         [0023]    At various times during the drilling process, the drill string  8  may be removed from the borehole as shown in  FIG. 2 . Once the drill string has been removed, logging operations can be conducted using a wireline logging tool  34 , i.e., a sensing instrument sonde suspended by a cable  42  having conductors for transporting power to the tool and telemetry from the tool to the surface. A resistivity imaging portion of the logging tool  34  may have centralizing arms  36  that center the tool within the borehole as the tool is pulled uphole. A logging facility  44  collects measurements from the logging tool  34 , and includes computing facilities for processing and storing the measurements gathered by the logging tool. 
         [0024]    The formations  18  shown in  FIG. 1  are not perpendicular to the borehole, which may occur naturally or due to directional drilling operations. The borehole may have a coordinate system  50  defined in accordance with the borehole&#39;s long axis (the z axis) and the north side (or alternatively, the high side) of the hole (the x-axis). The formations  18 , when characterized as a plane, may have a coordinate system  51  defined in accordance with the normal to the plane (the z″ axis) and the direction of steepest descent (the x″-axis). As shown in  FIG. 3 , the two coordinate systems are related by two rotations. Beginning with the borehole&#39;s coordinate system (x, y, z), a first rotation of angle γ is made about the z axis. The resulting coordinate system is denoted (x′, y′, z′). Angle γ is the relative strike angle, which indicates the direction of the formation dip relative to the borehole&#39;s coordinate system. A second rotation of angle α is then made about the y′ axis. This aligns the borehole coordinate system with the formation coordinate system. Angle α is the relative dip angle, which is the slope angle of the beds relative to the long axis of the borehole. 
         [0025]    The vertical resistivity is generally found to be the resistivity as measured perpendicular to the plane of the formation, and the horizontal resistivity is the resistivity as measured within the plane of the formation. Determination of each of these parameters (dip angle, strike angle, vertical resistivity, and horizontal resistivity) is desirable. 
         [0026]      FIG. 4  shows a hypothetical antenna configuration for a multi-component electromagnetic resistivity logging tool. The electromagnetic resistivity logging tool may be embodied as a wireline tool and as a logging while drilling tool. A triad of transmitter coils T X , T Y  and T Z , each oriented along a respective axis, is provided. At least one triad of similarly oriented receiver coils R 1X , R 1Y , and R 1Z  is also provided. For receive signal measurements relative to the amplitude and phase of the transmit signal (sometimes called “absolute” measurements), only one receiver triad would be used. A second triad of similarly oriented receiver coils pairs R 2X , R 2Y , and R 2Z  may also provided when differential measurements are desired (e.g., a signal amplitude ratio or a phase difference between receiver coils oriented along a given axis). Differential measurements may offer increased spatial resolution. 
         [0027]    Moran and Gianzero, in “Effects of Formation Anisotropy on Resistivity Logging Measurements” Geophysics, Vol. 44, No. 7, p. 1266 (1979), noted that the magnetic field h in the receiver coils can be represented in terms of the magnetic moments m at the transmitters and a coupling matrix C: 
         [0000]        h=Cm   (1)
 
         [0028]    In express form, equation (1) is: 
         [0000]    
       
         
           
             
               
                 
                   
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         [0029]    where M X , M Y , and M Z  are the magnetic moments (proportional to transmit signal strength) created by transmitters T X , T Y , and T Z , respectively. H X , H Y , H Z  are the magnetic fields (proportional to receive signal strength) at the receiver antennas R X , R Y , and R Z , respectively. 
         [0030]    In the antenna configuration of  FIG. 4 , if each transmitter is fired in turn, and signal measurements are made at each receiver in response to each firing, nine absolute or differential measurements are obtained. These nine measurements enable the determination of a complete coupling matrix C. (C IJ =a IJ V I   J , where I is the index for receiver R X , R Y , or R Z , J is the index for transmitter T X , T Y , or T Z , a IJ  is a constant determined by the tool design, and V I   J  is a complex value representing in the signal amplitude and phase shift measured by receiver I in response to the firing of transmitter J.) Knowledge of the complete coupling matrix enables the determination of dip angle, strike angle, vertical resistivity, and horizontal resistivity. A number of techniques may be used to determine these parameters. For example, dip and strike angle may be determined from coupling matrix values as explained by Li Gao and Stanley Gianzero, U.S. Pat. No. 6,727,706 entitled “Virtual Steering of Induction Tool for Determination of Formation Dip Angle”. Given these angles, vertical and horizontal resistivity can be determined in accordance with equations provided by Michael Bittar, U.S. Pat. No. 7,019,528 entitled “Electromagnetic Wave Resistivity Tool Having a Tilted Antenna for Geosteering within a Desired Payzone”. Alternatively, a simultaneous solution for these parameters may be found as described in the Bittar reference. 
         [0031]      FIG. 5  shows two angles that may be used to specify the orientation of a coil antenna. The coil antenna may be considered as residing in a plane having a normal vector. Tilt angle θ is the angle between the longitudinal axis of the tool and the normal vector. Azimuth angle β is the angle between the projection of the normal vector in the X-Y plane and the tool scribe line. Alternatively, in the downhole context, azimuthal angle β may represent the angle between projection of the normal vector in the X-Y plane and the x-axis of the borehole coordinate system. 
         [0032]    It is noted that three transmitter antenna orientations and three receiver antenna orientations are employed in the antenna configuration of  FIG. 4 . It has been discovered, as described in Michael Bittar and Guoyu Hu, WO2008/076130, entitled “Antenna Coupling Component Measurement Tool Having Rotating Antenna Configuration,” that when tool rotation is exploited, it is possible to determine the full coupling matrix with only one transmitter and two receiver antenna orientations (or equivalently, one receiver and two transmitter antenna orientations). Moreover, with certain assumptions about the configuration of the formation, one transmitter and receiver antenna orientation may be sufficient. 
         [0033]    Before considering various tools having specific antenna configurations, the electronics common to each tool are described.  FIG. 6  shows a functional block diagram of the electronics for a resistivity tool. The electronics include a control module  602  that is coupled to an analog switch  604 . Analog switch  604  is configured to drive any one of the transmitter coils T 1 , T 2 , T 3 , T 4  with an alternating current (AC) signal from a signal source  606 . In some embodiments, the signal source provides radio frequency signals. The control module  602  preferably selects a transmitter coil, pauses long enough for transients to die out, then signals data storage/transmit module  610  to accept an amplitude and phase sample of the signals received by each of the receivers. The control module  602  preferably repeats this process sequentially for each of the transmitters. The amplitude and phase shift values are provided by amplitude and phase shift detector  608  which is coupled to each of the receiver coils R 1  and R 2  for this purpose. 
         [0034]    Control module  602  may process the amplitude and phase shift measurements to obtain compensated measurements and/or measurement averages. The raw, compensated, or averaged measurements may be transmitted to the surface for processing to determine dip and strike angles, vertical and horizontal resistivity, and other information such as (i) distance to nearest bed boundary, (ii) direction of nearest bed boundary, and (iii) resistivity of any nearby adjacent beds. The data storage/transmitter module  610  may be coupled to telemetry unit  28  ( FIG. 1 ) to transmit signal measurements to the surface. Telemetry unit  28  can use any of several known techniques for transmitting information to the surface, including but not limited to (1) mud pressure pulse; (2) hard-wire connection; (3) acoustic wave; and (4) electromagnetic waves. 
         [0035]      FIG. 7  shows an electromagnetic resistivity logging tool  702  having only two receiver antenna orientations. The tool  702  is provided with one or more regions  706  of reduced diameter. A wire coil  704  is placed in the region  706  and in some embodiments is spaced away from the surface of subassembly  702  by a constant distance. To mechanically support and protect the coil  704 , a non-conductive filler material (not shown) such as epoxy, rubber, or ceramic may be used in the reduced diameter regions  706 . Coil  704  is a transmitter coil, and coils  710  and  712  are receiving coils. In operation, transmitter coil  704  transmits an interrogating electromagnetic signal which propagates through the borehole and surrounding formation. Receiver coils  710 ,  712  detect the interrogating electromagnetic signal and provide a measure of the electromagnetic signal&#39;s amplitude attenuation and phase shift. For differential measurements additional receiver coils parallel to coils  710 ,  712  may be provided at an axially-spaced distance. From the absolute or differential amplitude attenuation and phase shift measurements, the coupling matrix components can be determined and used as the basis for determining formation parameters and as the basis for geosteering. 
         [0036]    The transmitter coil  704  may be spaced approximately 30 inches from the receiver coils  710 ,  712 . The transmitter and receiver coils may comprise as little as one loop of wire, although more loops may provide additional signal power. The distance between the coils and the tool surface is preferably in the range from 1/16 inch to ¾ inch, but may be larger. Transmitter coil  704  and receiver coil  712  may each have a tilt angle of about 45° and aligned with the same azimuth angle, while receiver coil  710  may have a tilt angle of about 45° and an azimuth 180° apart from receiver coil  712  (or equivalently, a tilt angle of minus 45° at the same azimuth angle as receiver coil  712 ). 
         [0037]    The signal measured by a tilted receiver in response to the firing of a tilted transmitter can be expressed in terms of the signals V R   T  that would be measured by the tool of  FIG. 4 . One embodiment of an antenna configuration, illustrated in  FIG. 8 , includes a tilted transmitter (T) and a tilted transmitter (R). When both the transmitter and receiver are oriented at the same azimuth angle β, the tilted receiver signal V R   T  is: 
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         [0000]    Where, θ t  is the tilt angle of the transmitter relative to the z-axis (line  802  in  FIG. 8 ), θ r  is the tilt angle of the receiver relative to the z-axis direction, v I   J  is a complex value representing the signal amplitude and phase shift measured by the receiver in I-directional dipole in response to the firing of transmitter in J-directional dipole. The tilted receiver signal V R   T  can be a measurement of a voltage or of an electromagnetic field. Equation (3) can be also expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0000]    where: 
         [0038]    C xx =v x   x  sin θ t  sin θ r ; C yx =v x   y  sin θ t  sin θ r ; C zx =v x   z  cos θ t  sin θ r    
         [0039]    C xy =v y   x  sin θ t  sin θ r ; C yy =v y   y  sin θ t  sin θ r ; C zy =v y   z  cos θ t  sin θ r    
         [0040]    C xz =v z   x  sin θ t  cos θ r ; C yz =v z   y  sin θ t  cos θ r ; C zz =v z   z  cos θ t  cos θ r    
         [0041]      FIG. 9  illustrates a conventional formation model where homogeneous isotropic or transversely isotropic (TI) formation medium is considered. For a TI medium, a formation electrical property (i.e., formation resistivity) in the horizontal plane is different than a formation electrical property in the vertical plane. As indicated in  FIG. 9 , the horizontal plane has same resistivity profile (horizontal resistivity, R h ) in all horizontal directions and a vertical resistivity profile (vertical resistivity, R v ) in the vertical direction. An isotropic formation is described if R v  equals R h , and an anisotropic formation model is defined if R v  is not equal to R h . 
         [0042]    Another conventional formation model, illustrated in  FIG. 10 , consists of inhomogeneous layered formations  1002 ,  1004 ,  1006 ,  1008  with TI media in each layer and parallel boundaries  1010 ,  1012 ,  1014  between each layer. 
         [0043]    During logging, in one embodiment, control module  602  sorts samples according to the orientation of the tool when the sample was taken into N bins, such as those illustrated in  FIG. 11A , each bin centered at β i  degrees relative to the high side of the tool. 
         [0044]    With conventional formation models such as those in  FIGS. 9 and 10 , Eq. (2) can be simplified as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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                               ( 
                               
                                 β 
                                 i 
                               
                               ) 
                             
                           
                           + 
                           
                             A 
                             const 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where: 
         [0045]    β i  is azimuth angle corresponding to the bin direction i in  FIG. 11A , 
         [0000]    
       
         
           
             
               
                 
                   A 
                   double 
                 
                  
                 
                   ( 
                   
                     β 
                     i 
                   
                   ) 
                 
               
               = 
               
                 
                   ( 
                   
                     
                       
                         C 
                         xx 
                       
                       - 
                       
                         C 
                         yy 
                       
                     
                     2 
                   
                   ) 
                 
                  
                 cos 
                  
                 
                     
                 
                  
                 2 
                  
                 
                     
                 
                  
                 
                   β 
                   i 
                 
               
             
             , 
             
               
 
             
              
             
               
                 
                   A 
                   single 
                 
                  
                 
                   ( 
                   
                     β 
                     i 
                   
                   ) 
                 
               
               = 
               
                 
                   ( 
                   
                     
                       C 
                       zx 
                     
                     + 
                     
                       C 
                       xz 
                     
                   
                   ) 
                 
                  
                 cos 
                  
                 
                     
                 
                  
                 
                   β 
                   i 
                 
               
             
             , 
             and 
           
         
       
       
         
           
             
               A 
               const 
             
             = 
             
               
                 
                   
                     2 
                      
                     
                       C 
                       zz 
                     
                   
                   + 
                   
                     C 
                     xx 
                   
                   + 
                   
                     C 
                     yy 
                   
                 
                 2 
               
               . 
             
           
         
       
     
         [0046]    In practice, β i  is a tool azimuth angle relative to a tool high side direction, as illustrated in  FIG. 11B . The tool&#39;s “high side” is normally a magnetic north direction determined by a magnetometer device or as upward direction determined by a gravity-sensing device.  FIG. 11B  also illustrates the relationship between a formation strike direction (i.e., a direction along a line representing an intersection of a formation bedding plane with a horizontal plane) and the tool high side. The cosine wave signals in equation (5) (i.e., A double  and A single ) can be used to calculate strike angle β 0  (or relative azimuth, the angle between the tool&#39;s high side and the formation strike direction) by examining the azimuth shift in the tool azimuth spectrum. Further,  FIG. 11B  illustrates the relative dip angle defined by the angle between the tool&#39;s drilling direction (i.e., the “z” direction) and a direction normal to the formation bedding plane. 
         [0047]    Equation (5) includes three types of signals with respect to tool azimuthal angle β i : 
         [0048]    (1) a cosine wave signal with double periods (A double  (β i )), 
         [0049]    (2) a cosine wave signal with a single period (A single  (β i )), and 
         [0050]    (3) a constant signal (A const ) regardless of tool azimuth. 
         [0000]    These three signals are all complex numbers consisting of a real part and an imaginary part. Consequently, Eq. (5) indicates that both real part and imaginary part of the cosine wave should have same shift in azimuth angle.  FIGS. 12 and 13  illustrate tool azimuth spectra modeling responses of an antenna structure such as that shown in  FIG. 8  (d=44 inch, θ t =45°=θ r , operation frequency of 125 kHz) in a conventional formation model (Rh=0.62 Ω·m, Rh=1.68 Ω·m, relative dip angle of 20°, and strike angle of 0°). As shown in  FIGS. 12 and 13 , the real part and imaginary part for each type of sinusoidal signals have the same shift in azimuth angle (that is, a peak in the real and imaginary parts of A double  and a peak in the real and imaginary parts of A single  occurs at 0°, indicating a strike angle of 0°).  FIGS. 14 and 15  present the same modeling responses but with a defined formation model as in  FIG. 9  or  10  with a 30° azimuth shift away from the tool. In this situation, the real part and imaginary part of each type of sinusoidal wave present the same azimuth shift (that is, a peak in the real and imaginary parts of A double  and a peak in the real and imaginary parts of A single  occurs at 30°) (described in Eq. (5)). The real and imaginary parts of A double  and A single  present the same azimuth shift in tool azimuth spectrum, indicating the true strike angle of the formation bedding structure. 
         [0051]    Equation (5) is not valid if the tool is located in an unconventional formation model such as those illustrated in  FIGS. 16 and 17 . Instead of a TI medium,  FIG. 16  shows an unconventional formation model consisting of two resistivity profiles in the horizontal plane (R hx  and R hy ). The unconventional formation  1602  illustrated in  FIG. 16  has three individual resistivity profiles in triaxial directions, where R hx  and R hy  are located in horizontal plane in x and y direction, respectively, and R v  indicating the vertical resistivity in the z direction. Such an unconventional model is defined as biaxial anisotropic formation. 
         [0052]      FIG. 17  illustrates an unconventional layered formation model with layers  1702  (having resistivity profile R hx0 , R hy0  in the horizontal plane and R v0  in the vertical plane),  1704  (having resistivity profile R hx , R hy  in the horizontal plane and R v  in the vertical plane),  1706  (having resistivity profile R hx1 , R hy1  in the horizontal plane and R v1  in the vertical plane), and  1708  (having resistivity profile R hx2 , R hy2  in the horizontal plane and R v2  in the vertical plane) and with non-parallel boundaries  1710 ,  1712 ,  1714  between the layer. 
         [0053]    With such unconventional formation models, such as shown in  FIGS. 16 and 17 , equation (5) is no long valid and equation (4) can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             V 
                             R 
                             T 
                           
                            
                           
                             ( 
                             
                               β 
                               i 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         sin 
                                          
                                         
                                             
                                         
                                          
                                         
                                           θ 
                                           t 
                                         
                                          
                                         cos 
                                          
                                         
                                             
                                         
                                          
                                         
                                           β 
                                           i 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         sin 
                                          
                                         
                                             
                                         
                                          
                                         
                                           θ 
                                           t 
                                         
                                          
                                         sin 
                                          
                                         
                                             
                                         
                                          
                                         
                                           β 
                                           i 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         cos 
                                          
                                         
                                             
                                         
                                          
                                         
                                           θ 
                                           t 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               T 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     
                                       v 
                                       x 
                                       x 
                                     
                                   
                                   
                                     
                                       v 
                                       y 
                                       x 
                                     
                                   
                                   
                                     
                                       v 
                                       z 
                                       x 
                                     
                                   
                                 
                                 
                                   
                                     
                                       v 
                                       x 
                                       y 
                                     
                                   
                                   
                                     
                                       v 
                                       y 
                                       y 
                                     
                                   
                                   
                                     
                                       v 
                                       z 
                                       y 
                                     
                                   
                                 
                                 
                                   
                                     
                                       v 
                                       x 
                                       z 
                                     
                                   
                                   
                                     
                                       v 
                                       y 
                                       z 
                                     
                                   
                                   
                                     
                                       v 
                                       z 
                                       z 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       r 
                                     
                                      
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     
                                       β 
                                       i 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       r 
                                     
                                      
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     
                                       β 
                                       i 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       r 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             [ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         C 
                                         xx 
                                       
                                       - 
                                       
                                         C 
                                         yy 
                                       
                                     
                                     2 
                                   
                                   ) 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   β 
                                   i 
                                 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         C 
                                         yx 
                                       
                                       + 
                                       
                                         C 
                                         xy 
                                       
                                     
                                     2 
                                   
                                   ) 
                                 
                                  
                                 
                                     
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   β 
                                   i 
                                 
                               
                             
                             ] 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             [ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       C 
                                       zx 
                                     
                                     + 
                                     
                                       C 
                                       xz 
                                     
                                   
                                   ) 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   β 
                                   i 
                                 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       C 
                                       zy 
                                     
                                     + 
                                     
                                       C 
                                       yz 
                                     
                                   
                                   ) 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   β 
                                   i 
                                 
                               
                             
                             ] 
                           
                           + 
                           
                             
                               
                                 2 
                                  
                                 
                                   C 
                                   zz 
                                 
                               
                               + 
                               
                                 C 
                                 xx 
                               
                               + 
                               
                                 C 
                                 yy 
                               
                             
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               A 
                               double 
                             
                              
                             
                               ( 
                               
                                 β 
                                 i 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               A 
                               single 
                             
                              
                             
                               ( 
                               
                                 β 
                                 i 
                               
                               ) 
                             
                           
                           + 
                           
                             A 
                             const 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       A 
                       double 
                     
                      
                     
                       ( 
                       
                         β 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               C 
                               xx 
                             
                             - 
                             
                               C 
                               yy 
                             
                           
                           2 
                         
                         ) 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       2 
                        
                       
                           
                       
                        
                       
                         β 
                         i 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               C 
                               yx 
                             
                             + 
                             
                               C 
                               xy 
                             
                           
                           2 
                         
                         ) 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       2 
                        
                       
                           
                       
                        
                       
                         β 
                         i 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     7 
                      
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       A 
                       single 
                     
                      
                     
                       ( 
                       
                         β 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             C 
                             zx 
                           
                           + 
                           
                             C 
                             xz 
                           
                         
                         ) 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         β 
                         i 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             C 
                             zy 
                           
                           + 
                           
                             C 
                             yz 
                           
                         
                         ) 
                       
                        
                       sin 
                        
                       
                           
                       
                        
                       
                         β 
                         i 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     7 
                      
                     b 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     A 
                     const 
                   
                   = 
                   
                     
                       
                         2 
                          
                         
                           C 
                           zz 
                         
                       
                       + 
                       
                         C 
                         xx 
                       
                       + 
                       
                         C 
                         yy 
                       
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   
                     7 
                      
                     c 
                   
                   ) 
                 
               
             
           
         
       
     
         [0054]    Equations (7a) and 7(b) can be expressed as:) 
         [0000]        A   double (β i )= A   DR  cos(2β i −β DR )+ i×A   DI  cos(2β i −β SI )  (8a)
 
         [0000]        A   single (β i )= A   SR  cos(2β i −β SR )+ i×A   SI  cos(2β i −β SI )  (8b)
 
         [0000]    where: 
         [0000]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               A 
                               DR 
                             
                             = 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       real 
                                        
                                       
                                         { 
                                         
                                           
                                             
                                               C 
                                               xx 
                                             
                                             - 
                                             
                                               C 
                                               yy 
                                             
                                           
                                           2 
                                         
                                         } 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       real 
                                        
                                       
                                         { 
                                         
                                           
                                             
                                               C 
                                               yx 
                                             
                                             + 
                                             
                                               C 
                                               xy 
                                             
                                           
                                           2 
                                         
                                         } 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                           
                           , 
                         
                       
                       
                         
                           
                             β 
                             DR 
                           
                           = 
                           
                             
                               tan 
                               
                                 - 
                                 1 
                               
                             
                              
                             
                               ( 
                               
                                 real 
                                  
                                 
                                   { 
                                   
                                     
                                       
                                         C 
                                         yx 
                                       
                                       + 
                                       
                                         C 
                                         xy 
                                       
                                     
                                     
                                       
                                         C 
                                         xx 
                                       
                                       - 
                                       
                                         C 
                                         yy 
                                       
                                     
                                   
                                   } 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               A 
                               DI 
                             
                             = 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       imag 
                                        
                                       
                                         { 
                                         
                                           
                                             
                                               C 
                                               xx 
                                             
                                             - 
                                             
                                               C 
                                               yy 
                                             
                                           
                                           2 
                                         
                                         } 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       imag 
                                        
                                       
                                         { 
                                         
                                           
                                             
                                               C 
                                               yx 
                                             
                                             + 
                                             
                                               C 
                                               xy 
                                             
                                           
                                           2 
                                         
                                         } 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                           
                           , 
                         
                       
                       
                         
                           
                             β 
                             DI 
                           
                           = 
                           
                             
                               tan 
                               
                                 - 
                                 1 
                               
                             
                              
                             
                               ( 
                               
                                 imag 
                                  
                                 
                                   { 
                                   
                                     
                                       
                                         C 
                                         yx 
                                       
                                       + 
                                       
                                         C 
                                         xy 
                                       
                                     
                                     
                                       
                                         C 
                                         xx 
                                       
                                       - 
                                       
                                         C 
                                         yy 
                                       
                                     
                                   
                                   } 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               A 
                               SR 
                             
                             = 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       real 
                                        
                                       
                                         { 
                                         
                                           
                                             C 
                                             zx 
                                           
                                           + 
                                           
                                             C 
                                             xz 
                                           
                                         
                                         } 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       real 
                                        
                                       
                                         { 
                                         
                                           
                                             C 
                                             zy 
                                           
                                           + 
                                           
                                             C 
                                             yz 
                                           
                                         
                                         } 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                           
                           , 
                         
                       
                       
                         
                           
                             β 
                             SR 
                           
                           = 
                           
                             
                               tan 
                               
                                 - 
                                 1 
                               
                             
                              
                             
                               ( 
                               
                                 real 
                                  
                                 
                                   { 
                                   
                                     
                                       
                                         C 
                                         zy 
                                       
                                       + 
                                       
                                         C 
                                         yz 
                                       
                                     
                                     
                                       
                                         C 
                                         zx 
                                       
                                       + 
                                       
                                         C 
                                         xz 
                                       
                                     
                                   
                                   } 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               A 
                               SI 
                             
                             = 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       imag 
                                        
                                       
                                         { 
                                         
                                           
                                             C 
                                             zx 
                                           
                                           + 
                                           
                                             C 
                                             xz 
                                           
                                         
                                         } 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       imag 
                                        
                                       
                                         { 
                                         
                                           
                                             C 
                                             zy 
                                           
                                           + 
                                           
                                             C 
                                             yz 
                                           
                                         
                                         } 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                           
                           , 
                         
                       
                       
                         
                           
                             β 
                             SI 
                           
                           = 
                           
                             
                               tan 
                               
                                 - 
                                 1 
                               
                             
                              
                             
                               ( 
                               
                                 imag 
                                  
                                 
                                   { 
                                   
                                     
                                       
                                         C 
                                         zy 
                                       
                                       + 
                                       
                                         C 
                                         yz 
                                       
                                     
                                     
                                       
                                         C 
                                         zx 
                                       
                                       + 
                                       
                                         C 
                                         xz 
                                       
                                     
                                   
                                   } 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0055]    As can be seen in equations (8a), (8b) and (9), the real part and imaginary parts of each type of sinusoidal wave has an additional azimuth shift (β DR , β DI , β SR , and β SI ) caused by the unconventional formation models. Further, the azimuth shift for the real part and the imaginary part vary on the basis of the unconventional formation model and measured multi-components in Eq. (6). Using the same spacing and operating frequency as  FIGS. 12 and 13 ,  FIGS. 18 and 19  illustrate tool azimuth spectra modeling responses for two different formation models.  FIG. 18  illustrates a plot for a conventional TI formation model (Rh=0.25 Ω·m, Rv=2 Ω·m, relative dip angle of 20° and strike angle of 0°).  FIG. 19  is a plot for an unconventional biaxial anisotropic formation model (Rhx=0.25 Ω·m, Rhy=1 Ω·m, Rv=2 Ω·m, relative dip angle of 20° and strike angle of 0°). As shown in  FIG. 19 , there are angle differences between azimuth shift in the real part as compared to the azimuth shift in the imaginary part for sinusoidal waves in unconventional formation model, especially for A double . The true formation strike angle can be determined as described above for the azimuth spectra shown in  FIG. 18 . In contrast, however, because of the angle differences between the azimuth shift in the real and azimuth shift in the imaginary part, it is difficult to determine the true formation strike angle for the azimuth spectra shown in  FIG. 19  using the techniques described above. 
         [0056]    In one embodiment, unconventional formation signals are decoupled and unconventional formation model parameters are determined using the techniques described above. In one embodiment, several antenna configurations enable an LWD system to capture and invert data to produce unconventional formation parameters. In one embodiment, shown in  FIG. 20 , two sets of transmitter-receive antenna systems are used to decouple multi-components in a conventional formation model. In one embodiment, shown in  FIG. 20 , the two transmitters (T 1  and T 2 ) and the receiver (R x ) are oriented at the same tool azimuth, the transmitters are tilted at two different angles relative to the z-axis  2002  (T 1  at θ t1  and T 2  at θ t2 ), and the receiver (R x ) is tilted relative to the z-axis  2002  (at θ r ). 
         [0057]    Under the reciprocity theorem, transmitter(s) and receiver(s) are exchangeable.  FIG. 21  shows one embodiment of an antenna structure with similar functionalities as  FIG. 20 . Instead of two transmitters and one receiver as in  FIG. 20 ,  FIG. 21  shows two receivers (R 1  and R 2 ) and one transmitter (T 1 ). In one embodiment, shown in  FIG. 21 , the two receivers (R 1  and R 2 ) and the transmitter (T x ) are oriented at the same tool azimuth, the receivers are tilted at two different angles relative to the z-axis  2102  (R 1  at θ r1  and R 2  at θ r2 ), and the transmitter (T x ) is tilted relative to the z-axis  2102  (at θ t ). 
         [0058]    While the systems shown in  FIGS. 20 and 21  are able to capture the unconventional formation signals, the received LWD measurements with only single frequency operation are not enough to decouple such formation parameters. Consequently, multi-frequency measurements are included with the systems in  FIGS. 20 and 21  to allow unconventional formation determination. 
         [0059]      FIGS. 22 and 23  show antenna structures capable of capturing and decoupling unconventional formation signals with measurements received at single operation frequency. In embodiments illustrated  FIGS. 22 and 23 , at least three sets of transmitter-receiver system are included where each antenna orientation is tilted at a certain angle and oriented at a certain azimuth. With additional introduction of different antenna azimuth, measurements from the embodiments of systems illustrated in  FIGS. 22 and 23  are able to capture and invert unconventional formation parameters. 
         [0060]    In one embodiment illustrated in  FIG. 22 , one transmitter (T 1 ) is oriented at a different azimuth (β t1 ) relative to strike angle  2202  compared to the orientation of the other two transmitters (i.e., T 2  at β t2  and T 3  at β t3 , β t1 ≠β t2  and β t1 ≠β t3 ). In one embodiment, T 2  and T 3  are at the same azimuth (i.e., β t2 =β t3 ) but at different tilt angles (i.e., θ t2 ≠θ t3 ). In one embodiment, the receiver (Rx) is tilted at angle θ r  and is orientated at azimuth angle β r . 
         [0061]    In one embodiment, the receivers and transmitters in  FIG. 22  are exchanged. 
         [0062]    In one embodiment illustrated in  FIG. 23 , every transmitter and receiver oriented at different azimuth angle and at least one transmitter and one receiver is tilted. 
         [0063]    In one practical embodiment using the system illustrated in  FIG. 22 , R x  is tilted at θ r =45° and oriented at azimuth of β r =0° and three receivers are all tilted at β t1 =β t2 β t3 =45° but oriented at various azimuth angles (θ t1 =0°, θ t2 =120°, and θ t3 =240°, respectively). 
         [0064]    In one embodiment of use, illustrated in  FIG. 24 , the tool  26  is rotated (block  2405 ). For example, in one embodiment the tool  26  is an LWD tool and the tool  26  is rotated as part of the drilling operation. In one embodiment, another scheme is used to rotate the tool in a wireline environment. 
         [0065]    In one embodiment, single and/or multiple-frequency azimuthal measurements are made and binned (block  2410 ). In one embodiment, the three types of signals (equations 7(a), 7(b), and 7(c)) are decoupled using equation (6) (block  2415 ). In one embodiment, A double  and/or A single  are examined for the type of phase shift described above that indicates the samples were taken from an unconventional formation (block  2420 ). If the phase shift is not present, the data is processed, for example by applying one or more inversion techniques appropriate to conventional formations to extract formation parameters)(block  2425 ). If the phase shift is present, the data is processed, for example by applying one or more inversion techniques appropriate to unconventional formations to extract formation parameters (block  2430 ). 
         [0066]    In one embodiment, shown in  FIG. 25 , the control module  602  ( FIG. 6 ) is controlled by software in the form of a computer program on a non-transitory computer readable media  2505 , such as a CD, a DVD, a USB drive, a portable hard drive or other portable memory. In one embodiment, a processor  2510 , which may be the same as or included in the control module  602 , reads the computer program from the computer readable media  2505  through an input/output device  2515  and stores it in a memory  2520  where it is prepared for execution through compiling and linking, if necessary, and then executed. In one embodiment, the system accepts inputs through an input/output device  2515 , such as a keyboard or keypad, mouse, touchpad, touch screen, etc., and provides outputs through an input/output device  2515 , such as a monitor or printer. In one embodiment, the system stores the results of calculations in memory  2520  or modifies such calculations that already exist in memory  2520 . 
         [0067]    In one embodiment, the results of calculations that reside in memory  2520  are made available through a network  2525  to a remote real time operating center  2530 . In one embodiment, the remote real time operating center  2530  makes the results of calculations available through a network  2535  to help in the planning of oil wells  2540  or in the drilling of oil wells  2540 . 
         [0068]    The word “coupled” herein means a direct connection or an indirect connection. 
         [0069]    The text above describes one or more specific embodiments of a broader invention. The invention also is carried out in a variety of alternate embodiments and thus is not limited to those described here. The foregoing description of an embodiment of the invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Many modifications and variations are possible in light of the above teaching. It is intended that the scope of the invention be limited not by this detailed description, but rather by the claims appended hereto.