Abstract:
A method and a system are provided for allowing determination of a direction and distance from a tool to anomaly in a formation. The apparatus for performing the method includes at least one transmitter and at least one receiver. An embodiment of the method includes transmitting electromagnetic signals from the at least one transmitter through the formation near the wellbore and detecting responses at the at least one receiver induced by the electromagnetic signals. The method may further include determining the direction from the device to the anomaly based on the detected responses. The method may also include calculating at least one of an apparent azimuth angle and an apparent dip angle based on the responses, monitoring the at least one calculated apparent angle over time, and determining the direction to the anomaly after the at least one monitored apparent angle deviates from a zero value.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
   This is a divisional application of application Ser. No. 10/701,735, filed 5 Nov. 2003, which is hereby incorporated by reference 

   FIELD OF THE INVENTION 
   The present invention relates to a method and system for locating an anomaly and in particular to finding the direction and distance to a resistive or conductive anomaly in a formation surrounding a borehole in drilling applications. 
   BACKGROUND OF THE INVENTION 
   In logging while drilling (LWD) geo-steering applications, it is advantageous to detect the presence of a formation anomaly ahead of or around a bit or bottom hole assembly. While currently available techniques are capable of detecting the presence of an anomaly, they are not capable of determining the location of the anomaly with sufficient depth or speed. 
   In formation evaluation, the depth of investigation of most logging tools, wire line or LWD has been limited to a few feet from the borehole. One such tool is disclosed in U.S. Pat. No. 5,678,643 to Robbins, et al. U.S. Pat. No. 5,678,643 to Robbins, et al. discloses an LWD tool for locating an anomaly. The tool transmits acoustic signals into a wellbore and receives returning acoustic signals including reflections and refractions. Receivers detect the returning acoustic signals and the time between transmission and receipt can be measured. Distances and directions to detected anomalies are determined by a microprocessor that processes the time delay information from the receivers. As set forth above, the depth of investigation facilitated by the tool is limited. 
   Another technique that provides limited depth of investigation is disclosed in U.S. Pat. No. 6,181,138 to Hagiwara. This technique for locating an anomaly utilizes tilted coil induction tools and frequency domain excitation techniques. In order to achieve a depth of investigation with such a tool, a longer tool size would be required. However, longer tools generally result in poor spatial resolution. 
   In order to increase depth capabilities, transient electromagnetic (EM) methods have been proposed. One such method for increasing the depth of investigation is proposed in U.S. Pat. No. 5,955,884 to Payton, et al. The tool disclosed in this patent utilizes electric and electromagnetic transmitters to apply electromagnetic energy to a formation at selected frequencies and waveforms that maximize radial depth of penetration into the target formation. In this transient EM method, the current is generally terminated at a transmitter antenna and temporal change of voltage induced in a receiver antenna is monitored. This technique has allowed detection of an anomaly at distances as deep as ten to one hundred meters. However, while Payton discloses a transient EM method enabling detection of an anomaly, it does not provide a technique for determining the direction of the anomaly. 
   Other references, such as PCT application WO/03/019237 also disclose the use of directional resistivity measurements in logging applications. This reference uses the measurements for generating an image of an earth formation after measuring the acoustic velocity of the formation and combining the results. This reference does not disclose a specific method for determining distance and direction to an anomaly. 
   When logging measurements are used for well placement, detection or identification of anomalies can be critical. Such anomalies may include for example, a fault, a bypassed reservoir, a salt dome, or an adjacent bed or oil-water contact. It would be beneficial to determine both the distance and the direction of the anomaly from the drilling site. 
   Tri-axial induction logging devices, including wire-line and LWD devices are capable of providing directional resistivity measurements. However, no method has been proposed for utilizing these directional resistivity measurements to identify the direction to an anomaly. 
   Accordingly, a new solution is needed for determining the direction and distance from a tool to an anomaly. Furthermore, a real time solution having an increased depth of analysis is needed so that the measurements can be immediately useful to equipment operators. 
   SUMMARY OF THE INVENTION 
   In one aspect, an embodiment of the present invention is directed to a method for determining a direction to an anomaly in a formation near a wellbore. The method is implemented using a device including at least one transmitter and at least one receiver. The method includes transmitting electromagnetic signals from the transmitter through the formation near the wellbore and detecting responses at the receiver induced by the electromagnetic signals. The method additionally includes determining the direction and distance from the device to the anomaly based on the detected responses. 
   In a further aspect, a method for determining a direction and distance to an anomaly in a formation near a wellbore is provided. The method is accomplished using a device with at least one transmitter for transmitting electromagnetic signals and at least one receiver for detecting responses. The method includes calculating at least one of an apparent azimuth angle and an apparent dip angle based on the responses and monitoring the at least one calculated apparent angle over time. The method additionally includes determining the direction to the anomaly after the at least one monitored apparent angle deviates from a zero value. 
   In yet another aspect, an embodiment of the invention provides a method for determining a direction and distance to an anomaly in a formation near a wellbore. The method is implemented using a device including at least one transmitter for transmitting electromagnetic signals and a receiver for detecting responses. The method includes calculating at least one of an apparent azimuth angle and an apparent dip angle based on the responses and monitoring the at least one calculated apparent angle over time. The method additionally includes measuring the distance to the anomaly when the at least one monitored apparent angle reaches an asymptotic value. 
   In another embodiment, the invention provides a computer readable medium for storing computer executable instructions for performing a method for determining a direction to an anomaly in a system comprising a device with at least one transmitter for transmitting electromagnetic signals and a receiver for detecting responses. The method comprises calculating at least one of an apparent azimuth angle and an apparent dip angle based on the responses, monitoring the at least one calculated apparent angle over time, and determining the direction to the anomaly when the at least one monitored apparent angle reaches an asymptotic value. 
   In another embodiment, the invention provides a system for determining a direction to an anomaly in a formation near a wellbore. The system comprises at least one transmitter for transmitting electromagnetic signals through the formation near the wellbore, at least one receiver for detecting responses induced by the electromagnetic signal, and a computer readable medium storing instructions for determining the direction from the device to the anomaly based on the detected responses. 
   In another embodiment, the invention provides a computer readable medium for storing computer executable instructions for performing a method for determining a direction to an anomaly in a system comprising a device with at least one transmitter for transmitting electromagnetic signals and a receiver for detecting responses. The method comprises calculating at least one of an apparent azimuth angle and an apparent dip angle based on the responses, monitoring the at least one calculated apparent angle over time, and determining the direction to the anomaly after the at least one monitored angle deviates from a zero value. 
   In another embodiment, the invention provides a method for locating an anomaly in relation to a logging tool including at least one transmitter and at least one receiver in a formation near a wellbore. The method comprises measuring a voltage response over time at the at least one receiver, calculating an apparent conductivity over time based on the voltage response, and locating the anomaly based on variation of the apparent conductivity over time. 
   In another embodiment, the invention provides a method for locating an anomaly in relation to a logging tool including at least one transmitter and at least one receiver in a formation near a wellbore. The method comprises calculating an apparent conductivity over a selected time span based on the voltage response, determining a time at which the apparent conductivity deviates from a constant value, and ascertaining the anomaly in a region specified by the determined time. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The present invention is described in detail below with reference to the attached drawing figures, wherein: 
       FIG. 1  is a block diagram showing a system in accordance with embodiment of the invention; 
       FIG. 2  is a flow chart illustrating a method in accordance with an embodiment of the invention; 
       FIG. 3  is a graph illustrating directional angles between tool coordinates and anomaly coordinates; 
       FIG. 4A  is a graph showing a resistivity anomaly in a tool coordinate system; 
       FIG. 4B  is a graph showing a resistivity anomaly in an anomaly coordinate system; 
       FIG. 5  is a graph illustrating tool rotation within a borehole; 
       FIG. 6  is a graph showing directional components; 
       FIG. 7  is a schematic showing apparent conductivity with a coaxial tool; 
       FIG. 8  is a schematic showing apparent conductivity wit a coplanar tool. 
       FIG. 9  is a graph of a modeled voltage coaxial, component and cross-component responses; 
       FIG. 10  is a graph of a modeled voltage coaxial, component and cross-component responses; 
       FIG. 11  is a graph of apparent dip angle; 
       FIG. 12  is a graph of modeled apparent conductivity; 
       FIG. 13  is a graph of the modeled ratio of apparent coplanar and apparent coaxial responses without a cross-component measurement; 
       FIG. 14  is a graph of modeled apparent dip angles at varying approach angles; 
       FIG. 15  is a graph of the modeled apparent dip at varying distances; 
       FIG. 16  is a graph of the modeled voltage response of a coaxial tool; 
       FIG. 17  is a graph of modeled voltage responses at varying times; 
       FIG. 18  is a graph of the modeled voltage responses for a differing tool transmitter receiver spacing; 
       FIG. 19  is a graph of modeled voltage responses at varying times for a specific tool configuration; 
       FIG. 20  is a graph of modeled voltage responses at varying times for a specific tool configuration; 
       FIG. 21  is a graph of modeled voltage response of a coaxial tool at varying distances; 
       FIG. 22  is a graph of modeled apparent resistivity for the voltage responses depicted in  FIG. 21 ; 
       FIG. 23  is a graph of modeled apparent conductivity in a two-layer model; 
       FIG. 24  is a graph of a ration of modeled apparent resistivities for the responses depicted in  FIGS. 22 and 23 ; 
       FIG. 25  is a graph of modeled apparent resistivities for varying tool configurations; 
       FIG. 26  is a graph of modeled apparent resistivities for varying model layer resistivity ratios; 
       FIG. 27  is a graph of modeled apparent resistivities for varying model layer resistivity ratios; 
       FIG. 28  is a graph of modeled apparent resistivities for large values of time; 
       FIG. 29  is a graph of modeled apparent resistivity for varying model layer resistivity ratios; 
       FIG. 30  is a graph of modeled transitions of apparent resistivity as a function of time; 
       FIG. 31  is a graph of modeled distance to a layer as a function of transition time; and 
       FIG. 32  is a graph of modeled apparent resistivities at varying distances. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   Embodiments of the invention relate to a system and method for determining distance and direction to an anomaly in a formation within a wellbore. Both frequency domain excitation and time domain excitation have been used to excite electromagnetic fields for use in anomaly detection. In frequency domain excitation, a device transmits a continuous wave of a fixed or mixed frequency and measures responses at the same band of frequencies. In time domain excitation, a device transmits a square wave signal, triangular wave signal, pulsed signal or pseudo-random binary sequence as a source and measures the broadband earth response. Sudden changes in transmitter current cause signals to appear at a receiver caused by induction currents in the formation. The signals that appear at the receiver are called transient responses because the receiver signals start at a first value and then decay or increase with time to a constant level. The technique disclosed herein implements the time domain excitation technique. 
   As set forth below, embodiment of the invention propose a general method to determine a direction to a resistive or conductive anomaly using transient EM responses. As will be explained in detail, the direction to the anomaly is specified by a dip angle and an azimuth angle. Embodiments of the invention propose to define an apparent dip (θ app (t)) and an apparent azimuth (ψ app (t)) by combinations of tri-axial transient measurements. An apparent direction ({θ app (t), ψ app (t)}) approaches a true direction ({θ, ψ}) as a time (t) increases. The θ app (t) and ψ app (t) both initially read zero when an apparent conductivity σ coaxial (t) and σ coplanar (t) from coaxial and coplanar measurements both read the conductivity around the tool. The apparent conductivity will be further explained below and can also be used to determine the location of an anomaly in a wellbore. 
     FIG. 1  illustrates a system that may be used to implement the embodiments of the method of the invention. A surface computing unit  10  may be connected with an electromagnetic measurement tool  2  disposed in a wellbore  4  and supported by a cable  12 . The cable  12  may be constructed of any known type of cable for transmitting electrical signals between the tool  2  and the surface computing unit  10 . One or more transmitters  16  and one are more receivers  18  may be provided for transmitting and receiving signals. A data acquisition unit  14  may be provided to transmit data to and from the transmitters  16  and receivers  18  to the surface computing unit  10 . 
   Each transmitter  16  and each receiver  18  may be tri-axial and thereby contain components for sending and receiving signals along each of three axes. Accordingly, each transmitter module may contain at least one single or multi-axis antenna and may be a 3-orthogonal component transmitter. Each receiver may include at least one single or multi-axis electromagnetic receiving component and may be a 3-orthogonal component receiver. 
   The data acquisition unit  14  may include a controller for controlling the operation of the tool  2 . The data acquisition unit  14  preferably collects data from each transmitter  16  and receiver  18  and provides the data to the surface computing unit  10 . 
   The surface computing unit  10  may include computer components including a processing unit  30 , an operator interface  32 , and a tool interface  34 . The surface computing unit  10  may also include a memory  40  including relevant coordinate system transformation data and assumptions  42 , a direction calculation module  44 , an apparent direction calculation module  46 , and a distance calculation module  48 . The surface computing unit  10  may further include a bus  50  that couples various system components including the system memory  40  to the processing unit  30 . The computing system environment  10  is only one example of a suitable computing environment and is not intended to suggest any limitation as to the scope of use or functionality of the invention. Furthermore, although the computing system  10  is described as a computing unit located on a surface, it may optionally be located below the surface, incorporated in the tool, positioned at a remote location, or positioned at any other convenient location. 
   The memory  40  preferably stores the modules  44 ,  46 , and  48 , which may be described as program modules containing computer-executable instructions, executed by the surface computing unit  10 . The program module  44  contains the computer executable instruction necessary to calculate a direction to an anomaly within a wellbore. The program module  46  includes the computer executable instructions necessary to calculate an apparent direction as will be further explained below. The program module  48  contains the computer executable instructions necessary to calculate a distance to an anomaly. The stored data  46  includes data pertaining to the tool coordinate system and the anomaly coordinate system and other data required for use by the program modules  44 ,  46 , and  48 . These program modules  44 ,  46 , and  48 , as well as the stored data  42 , will be further described below in conjunction with embodiments of the method of the invention. 
   Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Moreover, those skilled in the art will appreciate that the invention may be practiced with other computer system configurations, including hand-held devices, multiprocessor systems, microprocessor-based or programmable consumer electronics, minicomputers, mainframe computers, and the like. The invention may also be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network. In a distributed computing environment, program modules may be located in both local and remote computer storage media including memory storage devices. 
   Although the computing system  10  is shown as having a generalized memory  40 , the computing system  10  would typically includes a variety of computer readable media. By way of example, and not limitation, computer readable media may comprise computer storage media and communication media. The computing system memory  40  may include computer storage media in the form of volatile and/or nonvolatile memory such as a read only memory (ROM) and random access memory (RAM). A basic input/output system (BIOS), containing the basic routines that help to transfer information between elements within computer  10 , such as during start-up, is typically stored in ROM. The RAM typically contains data and/or program modules that are immediately accessible to and/or presently being operated on by processing unit  30 . By way of example, and not limitation, the computing system  10  includes an operating system, application programs, other program modules, and program data. 
   The components shown in the memory  40  may also be included in other removable/nonremovable, volatile/nonvolatile computer-storage media. For example only, a hard disk drive may read from or write to nonremovable, nonvolatile magnetic media, a magnetic disk drive may read from or write to a removable, non-volatile magnetic disk, and an optical disk drive may read from or write to a removable, nonvolatile optical disk such as a CD ROM or other optical media. Other removable/non-removable, volatile/non-volatile computer storage media that can be used in the exemplary operating environment include, but are not limited to, magnetic tape cassettes, flash memory cards, digital versatile disks, digital video tape, solid state RAM, solid state ROM, and the like. The drives and their associated computer storage media discussed above and illustrated in  FIG. 1 , provide storage of computer readable instructions, data structures, program modules and other data for the computing system  10 . 
   A user may enter commands and information into the computing system  10  through input devices such as a keyboard and pointing device, commonly referred to as a mouse, trackball or touch pad. Input devices may include a microphone, joystick, satellite dish, scanner, or the like. These and other input devices are often connected to the processing unit  30  through the operator interface  32  that is coupled to the system bus  50 , but may be connected by other interface and bus structures, such as a parallel port or a universal serial bus (USB). A monitor or other type of display device may be connected to the system bus  50  via an interface, such as a video interface. In addition to the monitor, computers may also include other peripheral output devices such as speakers and printer, which may be connected through an output peripheral interface. 
   Although many other internal components of the computing system  10  are not shown, those of ordinary skill in the art will appreciate that such components and the interconnection are well known. Accordingly, additional details concerning the internal construction of the computer  10  need not be disclosed in connection with the present invention. 
     FIG. 2  is a flow chart illustrating the procedures involved in a method of the invention. Generally, in procedure A, the transmitters  16  transmit electromagnetic signals. In procedure B, the receivers  18  receive transient responses. In procedure C, the system processes the transient responses to determine a distance and direction to the anomaly. 
     FIGS. 3-6  illustrate the technique for implementing procedure C for determining distance and direction to the anomaly. 
   Tri-Axial Transient EM Responses 
     FIG. 3  illustrates directional angles between tool coordinates and anomaly coordinates. A transmitter coil T is located at an origin that serves as the origin for each coordinate system. A receiver R is placed at a distance L from the transmitter. An earth coordinate system, includes a Z-axis in a vertical direction and an X-axis and a Y-axis in the East and the North directions, respectively. The deviated borehole is specified in the earth coordinates by a deviation angle θ b  and its azimuth angle ψ b . A resistivity anomaly A is located at a distance D from the transmitter in the direction specified by a dip angle (θ a ) and its azimuth (ψ a ). 
   In order to practice embodiments of the method,  FIG. 4A  shows the definition of a tool/borehole coordinate system having x, y, and z axes. The z-axis defines the direction from the transmitter T to the receiver R. The tool coordinates in  FIG. 4A  are specified by rotating the earth coordinates (X, Y, Z) in  FIG. 3  by the azimuth angle (ψ b ) around the Z-axis and then rotating by θ b  around the y-axis to arrive at the tool coordinates (x, y, z). The direction of the anomaly is specified by the dip angle (∂) and the azimuth angle (φ) where: 
   
     
       
         
           
             
               
                 
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   Similarly,  FIG. 4B  shows the definition of an anomaly coordinate system having a, b, and c axes. The c-axis defines the direction from the transmitter T to the center of the anomaly A. The anomaly coordinates in  FIG. 4B  are specified by rotating the earth coordinates (X, Y, Z) in  FIG. 3  by the azimuth angle (ψ a ) around the Z-axis and subsequently rotating by θ a  around the b-axis to arrive at the anomaly coordinates (a, b, c). In this coordinate system, the direction of the borehole is specified in a reverse order by the azimuth angle (φ) and the dip angle (∂). 
   Transient Responses in Two Coordinate Systems 
   The method is additionally based on the relationship between the transient responses in two coordinate systems. The magnetic field transient responses at the receivers [R x , R y , R z ] which are oriented in the [x, y, z] axis direction of the tool coordinates, respectively, are noted as 
                   [           V   xx           V   xy           V   xz               V   yx           V   yy           V   yz               V   zx           V   zy           V   zz           ]     =       [           R   x               R   y               R   z           ]     ⁡     [       M   x     ⁢           ⁢     M   y     ⁢           ⁢     M   z       ]               (   3   )               
from a magnetic dipole source in each axis direction, [M x , M y , M z ].
 
   When the resistivity anomaly is distant from the tool, the formation near the tool is seen as a homogeneous formation. For simplicity, the method may assume that the formation is isotropic. Only three non-zero transient responses exist in a homogeneous isotropic formation. These include the coaxial response and two coplanar responses. Coaxial response V zz (t) is the response when both the transmitter and the receiver are oriented in the common tool axis direction. Coplanar responses, V xx (t) and V yy (t), are the responses when both the transmitter T and the receiver R are aligned parallel to each other but their orientation is perpendicular to the tool axis. All of the cross-component responses are identically zero in a homogeneous isotropic formation. Cross-component responses are either from a longitudinally oriented receiver with a transverse transmitter, or vise versa. Another cross-component response is also zero between a mutually orthogonal transverse receiver and transverse transmitter. 
   The effect of the resistivity anomaly is seen in the transient responses as time increases. In addition to the coaxial and the coplanar responses, the cross-component responses V ij (t) (i≠j; i,j=x, y, z) become non-zero. 
   The magnetic field transient responses may also be examined in the anomaly coordinate system. The magnetic field transient responses at the receivers [R a , R b , R c ] that are oriented in the [a, b, c] axis direction of the anomaly coordinates, respectively, may be noted as 
                   [           V   aa           V   ab           V   ac               V   ba           V   bb           V   bc               V   ca           V   cb           V   cc           ]     =       [           R   a               R   b               R   c           ]     ⁡     [       M   a     ⁢           ⁢     M   b     ⁢           ⁢     M   c       ]               (   4   )               
from a magnetic dipole source in each axis direction, [M a , M b , M c ].
 
   When the anomaly is large and distance compared to the transmitter-receiver spacing, the effect of spacing can be ignored and the transient responses can be approximated with those of the receivers near the transmitter. Then, the method assumes that axial symmetry exists with respect to the c-axis that is the direction from the transmitter to the center of the anomaly. In such an axially symmetric configuration, the cross-component responses in the anomaly coordinates are identically zero in time-domain measurements. 
   
     
       
         
           
             
               
                 
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   The magnetic field transient responses in the tool coordinates are related to those in the anomaly coordinates by a simple coordinate transformation P(∂, φ) specified by the dip angle (∂) and azimuth angle (φ). 
   
     
       
         
           
             
               
                 
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                 ) 
               
             
           
         
       
     
   
   Determination of Target Direction 
   The assumptions set forth above contribute to determination of target direction, which is defined as the direction of the anomaly from the origin. When axial symmetry in the anomaly coordinates is assumed, the transient response measurements in the tool coordinates are constrained and the two directional angles may be determined by combinations of tri-axial responses. 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           V 
                           xx 
                         
                       
                       
                         
                           V 
                           xy 
                         
                       
                       
                         
                           V 
                           xz 
                         
                       
                     
                     
                       
                         
                           V 
                           yx 
                         
                       
                       
                         
                           V 
                           yy 
                         
                       
                       
                         
                           V 
                           yz 
                         
                       
                     
                     
                       
                         
                           V 
                           zx 
                         
                       
                       
                         
                           V 
                           zy 
                         
                       
                       
                         
                           V 
                           zz 
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       
                         P 
                         ⁡ 
                         
                           ( 
                           
                             ϑ 
                             , 
                             ϕ 
                           
                           ) 
                         
                       
                       tr 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               V 
                               aa 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               V 
                               aa 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               V 
                               cc 
                             
                           
                         
                       
                       ] 
                     
                   
                   ⁢ 
                   
                     P 
                     ⁡ 
                     
                       ( 
                       
                         ϑ 
                         , 
                         ϕ 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 8 
                 ) 
               
             
           
         
       
     
   
   In terms of each tri-axial response 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         V 
                         xy 
                       
                       = 
                         
                       ⁢ 
                       
                         
                           
                             ( 
                             
                               
                                 
                                   V 
                                   aa 
                                 
                                 ⁢ 
                                 
                                   cos 
                                   2 
                                 
                                 ⁢ 
                                 ϑ 
                               
                               + 
                               
                                 
                                   V 
                                   cc 
                                 
                                 ⁢ 
                                 
                                   sin 
                                   2 
                                 
                                 ⁢ 
                                 ϑ 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             cos 
                             2 
                           
                           ⁢ 
                           ϕ 
                         
                         + 
                         
                           
                             V 
                             aa 
                           
                           ⁢ 
                           
                             sin 
                             2 
                           
                           ⁢ 
                           ϕ 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         V 
                         yy 
                       
                       = 
                         
                       ⁢ 
                       
                         
                           
                             ( 
                             
                               
                                 
                                   V 
                                   aa 
                                 
                                 ⁢ 
                                 
                                   cos 
                                   2 
                                 
                                 ⁢ 
                                 ϑ 
                               
                               + 
                               
                                 
                                   V 
                                   cc 
                                 
                                 ⁢ 
                                 
                                   sin 
                                   2 
                                 
                                 ⁢ 
                                 ϑ 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             sin 
                             2 
                           
                           ⁢ 
                           ϕ 
                         
                         + 
                         
                           
                             V 
                             aa 
                           
                           ⁢ 
                           
                             cos 
                             2 
                           
                           ⁢ 
                           ϕ 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         V 
                         zz 
                       
                       = 
                         
                       ⁢ 
                       
                         
                           
                             V 
                             aa 
                           
                           ⁢ 
                           
                             sin 
                             2 
                           
                           ⁢ 
                           ϑ 
                         
                         + 
                         
                           
                             V 
                             cc 
                           
                           ⁢ 
                           
                             cos 
                             2 
                           
                           ⁢ 
                           ϑ 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 9 
                 ) 
               
             
           
           
             
               
                 
                   
                     
                       
                         V 
                         xy 
                       
                       = 
                         
                       ⁢ 
                       
                         
                           V 
                           yx 
                         
                         = 
                         
                           
                             - 
                             
                               ( 
                               
                                 
                                   V 
                                   aa 
                                 
                                 - 
                                 
                                   V 
                                   cc 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             sin 
                             2 
                           
                           ⁢ 
                           ϑ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ϕ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ϕ 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         V 
                         zx 
                       
                       = 
                         
                       ⁢ 
                       
                         
                           V 
                           xz 
                         
                         = 
                         
                           
                             - 
                             
                               ( 
                               
                                 
                                   V 
                                   aa 
                                 
                                 - 
                                 
                                   V 
                                   cc 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ϑ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ϑ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ϕ 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         V 
                         yz 
                       
                       = 
                         
                       ⁢ 
                       
                         
                           V 
                           zy 
                         
                         = 
                         
                           
                             - 
                             
                               ( 
                               
                                 
                                   V 
                                   aa 
                                 
                                 - 
                                 
                                   V 
                                   cc 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ϑ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ϑsinϕ 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 10 
                 ⁢ 
                   
                 ) 
               
             
           
         
       
     
   
   The following relations can be noted: 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           V 
                           xx 
                         
                         + 
                         
                           V 
                           yy 
                         
                         + 
                         
                           V 
                           zz 
                         
                       
                       = 
                       
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             V 
                             aa 
                           
                         
                         + 
                         
                           V 
                           cc 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           V 
                           xx 
                         
                         - 
                         
                           V 
                           yy 
                         
                       
                       = 
                       
                         
                           ( 
                           
                             
                               V 
                               cc 
                             
                             - 
                             
                               V 
                               aa 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           sin 
                           2 
                         
                         ⁢ 
                         
                           ϑ 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   cos 
                                   2 
                                 
                                 ⁢ 
                                 ϕ 
                               
                               - 
                               
                                 
                                   sin 
                                   2 
                                 
                                 ⁢ 
                                 ϕ 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           V 
                           yy 
                         
                         - 
                         
                           V 
                           zz 
                         
                       
                       = 
                       
                         
                           - 
                           
                             ( 
                             
                               
                                 V 
                                 cc 
                               
                               - 
                               
                                 V 
                                 aa 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 cos 
                                 2 
                               
                               ⁢ 
                               ϑ 
                             
                             - 
                             
                               
                                 sin 
                                 2 
                               
                               ⁢ 
                               ϑ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 sin 
                                 2 
                               
                               ⁢ 
                               ϕ 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           V 
                           zz 
                         
                         - 
                         
                           V 
                           xx 
                         
                       
                       = 
                       
                         
                           ( 
                           
                             
                               V 
                               cc 
                             
                             - 
                             
                               V 
                               aa 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 cos 
                                 2 
                               
                               ⁢ 
                               ϑ 
                             
                             - 
                             
                               
                                 sin 
                                 2 
                               
                               ⁢ 
                               ϑ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 cos 
                                 2 
                               
                               ⁢ 
                               ϕ 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 11 
                 ) 
               
             
           
         
       
     
   
   Several distinct cases can be noted. In the first of these cases, when none of the cross-components is zero, V xy ≠0 nor V yz ≠0 nor V zx ≠0, then the azimuth angle φ is not zero nor π/2 (90°), and can be determined by, 
   
     
       
         
           
             
               
                 
                   
                     
                       ϕ 
                       = 
                       
                         
                           1 
                           2 
                         
                         ⁢ 
                         
                           tan 
                           
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             
                               V 
                               xy 
                             
                             + 
                             
                               V 
                               yx 
                             
                           
                           
                             
                               V 
                               xx 
                             
                             - 
                             
                               V 
                               yy 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       ϕ 
                       = 
                       
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               V 
                               yz 
                             
                             
                               V 
                               xz 
                             
                           
                         
                         = 
                         
                           
                             tan 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               V 
                               zy 
                             
                             
                               V 
                               zx 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 12 
                 ) 
               
             
           
         
       
     
   
   By noting the relation, 
                     V   xy       V   xz       =       tan   ⁢           ⁢   ϑsin   ⁢           ⁢   ϕ   ⁢           ⁢   and   ⁢           ⁢       V   xy       V   yz         =     tan   ⁢           ⁢   ϑcos   ⁢           ⁢   ϕ               (   13   )               
the dip (deviation) angle ∂ is determined by,
 
   
     
       
         
           
             
               
                 
                   tan 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   ϑ 
                 
                 = 
                 
                   
                     
                       
                         ( 
                         
                           
                             V 
                             xy 
                           
                           
                             V 
                             xz 
                           
                         
                         ) 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             V 
                             xy 
                           
                           
                             V 
                             yz 
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
             
             
               
                 ( 
                 14 
                 ) 
               
             
           
         
       
     
   
   In the second case, when V xy =0 and V yz =0, then ∂=0 φ=0 or π (180°) or φ=±π/2 (90°) and ∂=±π/2 (90°), as the coaxial and the coplanar responses should differ from each other (V aa ≠V cc ). 
   If φ=0, then the dip angle ∂ is determined by, 
   
     
       
         
           
             
               
                 ϑ 
                 = 
                 
                   
                     - 
                     
                       1 
                       2 
                     
                   
                   ⁢ 
                   
                     tan 
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       
                         V 
                         xz 
                       
                       + 
                       
                         V 
                         zx 
                       
                     
                     
                       
                         V 
                         xx 
                       
                       - 
                       
                         V 
                         zz 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 15 
                 ) 
               
             
           
         
       
     
   
   If φ=π(180°), then the dip angle ∂ is determined by, 
   
     
       
         
           
             
               
                 ϑ 
                 = 
                 
                   
                     + 
                     
                       1 
                       2 
                     
                   
                   ⁢ 
                   
                     tan 
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       
                         V 
                         xz 
                       
                       + 
                       
                         V 
                         zx 
                       
                     
                     
                       
                         V 
                         xx 
                       
                       - 
                       
                         V 
                         zz 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 16 
                 ) 
               
             
           
         
       
     
   
   Also, with regard to the second case, If ∂=0, then V xx =V yy  and V zz =0. If φ=±π/2 (90°) and ∂=±π/2 (90°), then V zz =V xx  and V zz =0. These instances are further discussed below with relation to the fifth case. 
   In the third case, when V xy =0 and V xx =0, then φ=±π/2 (90°) or ∂=0 or φ=0 and ∂=±π/2 (90°). 
   If φ=π/2, then the dip angle ∂ is determined by, 
   
     
       
         
           
             
               
                 ϑ 
                 = 
                 
                   
                     - 
                     
                       1 
                       2 
                     
                   
                   ⁢ 
                   
                     tan 
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       
                         V 
                         yz 
                       
                       + 
                       
                         V 
                         zy 
                       
                     
                     
                       
                         V 
                         yy 
                       
                       - 
                       
                         V 
                         zz 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 17 
                 ) 
               
             
           
         
       
     
   
   If φ=−π/2, then the dip angle ∂ is determined by, 
   
     
       
         
           
             
               
                 ϑ 
                 = 
                 
                   
                     + 
                     
                       1 
                       2 
                     
                   
                   ⁢ 
                   
                     tan 
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       
                         V 
                         yz 
                       
                       + 
                       
                         V 
                         yx 
                       
                     
                     
                       
                         V 
                         yy 
                       
                       - 
                       
                         V 
                         zz 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 18 
                 ) 
               
             
           
         
       
     
   
   Also with regard to the third case, If ∂=0, then V zz =V yy  and V yz =0. If φ=0 and ∂=±π/2 (90°), V yy =V xx  and V yz =0. These situations are further discussed below with relation to the fifth case. 
   In the fourth case, V xx =0 and V yz =0, then ∂32 0 or π(180°) or ±π/2 (90°). 
   If ∂=±π/2, then the azimuth angle φ is determined by, 
   
     
       
         
           
             
               
                 ϕ 
                 = 
                 
                   
                     - 
                     
                       1 
                       2 
                     
                   
                   ⁢ 
                   
                     tan 
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       
                         V 
                         xy 
                       
                       + 
                       
                         V 
                         yx 
                       
                     
                     
                       
                         V 
                         xx 
                       
                       - 
                       
                         V 
                         zz 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 19 
                 ) 
               
             
           
         
       
     
   
   Also with regard to the fourth case, if ∂=0 or π(180°), then V xx =V yy  and V yz =0. This situation is also shown below with relation to the fifth case. 
   In the fifth case, all cross components vanish, V xx =V yx =V xy =0, then ∂=0, or ∂=±π/2 (90°) and φ=0 or ±π/2 (90°). 
   If V xx =V yy  then ∂=0 or π(180°). 
   If V yy =V xx  then ∂=±π/2 (90°) and φ=0. 
   If V zz =V xx  then ∂=±π/2 (90°) and φ=±π/2 (90°). 
   Tool Rotation Around the Tool/Borehole Axis 
   In the above analysis, all transient responses V ij (t) (i, j=x, y, z) are specified by the x-, y-, and z-axis directions of the tool coordinates. However, the tool rotates inside the borehole and the azimuth orientation of the transmitter and the receiver no longer coincides with the x- or y-axis direction as shown in  FIG. 5 . If the measured responses are {tilde over (V)}{tilde over (V i,j )}(ĩ,{tilde over (j)}={tilde over (x)},{tilde over (y)},z) where {tilde over (x)} and {tilde over (y)} axis are the direction of antennas fixed to the rotating tool, and ψ is the tool&#39;s rotation angle, then 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           V 
                           
                             
                               x 
                               ~ 
                             
                             ⁢ 
                             
                               x 
                               ~ 
                             
                           
                         
                       
                       
                         
                           V 
                           
                             
                               x 
                               ~ 
                             
                             ⁢ 
                             
                               y 
                               ~ 
                             
                           
                         
                       
                       
                         
                           V 
                           
                             
                               x 
                               ~ 
                             
                             ⁢ 
                             z 
                           
                         
                       
                     
                     
                       
                         
                           V 
                           
                             
                               y 
                               ~ 
                             
                             ⁢ 
                             
                               x 
                               ~ 
                             
                           
                         
                       
                       
                         
                           V 
                           
                             
                               y 
                               ~ 
                             
                             ⁢ 
                             
                               y 
                               ~ 
                             
                           
                         
                       
                       
                         
                           V 
                           
                             
                               y 
                               ~ 
                             
                             ⁢ 
                             z 
                           
                         
                       
                     
                     
                       
                         
                           V 
                           
                             z 
                             ⁢ 
                             
                               x 
                               ~ 
                             
                           
                         
                       
                       
                         
                           V 
                           
                             z 
                             ⁢ 
                             
                               y 
                               ~ 
                             
                           
                         
                       
                       
                         
                           V 
                           zz 
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       
                         R 
                         ⁡ 
                         
                           ( 
                           ψ 
                           ) 
                         
                       
                       tr 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               V 
                               xx 
                             
                           
                           
                             
                               V 
                               xy 
                             
                           
                           
                             
                               V 
                               xz 
                             
                           
                         
                         
                           
                             
                               V 
                               yx 
                             
                           
                           
                             
                               V 
                               yy 
                             
                           
                           
                             
                               V 
                               yz 
                             
                           
                         
                         
                           
                             
                               V 
                               zx 
                             
                           
                           
                             
                               V 
                               zy 
                             
                           
                           
                             
                               V 
                               zz 
                             
                           
                         
                       
                       ] 
                     
                   
                   ⁢ 
                   
                     R 
                     ⁡ 
                     
                       ( 
                       ψ 
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 20 
                 ) 
               
             
           
           
             
               
                 
                   R 
                   ⁡ 
                   
                     ( 
                     ψ 
                     ) 
                   
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ψ 
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ψ 
                         
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ψ 
                         
                       
                       
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ψ 
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
             
             
               
                 ( 
                 21 
                 ) 
               
             
           
         
       
     
   
   Then, 
   
     
       
         
           
             
               
                 
                   
                     V 
                     
                       
                         x 
                         ~ 
                       
                       ⁢ 
                       
                         x 
                         ~ 
                       
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               V 
                               aa 
                             
                             ⁢ 
                             
                               cos 
                               2 
                             
                             ⁢ 
                             ϑ 
                           
                           + 
                           
                             
                               V 
                               cc 
                             
                             ⁢ 
                             
                               sin 
                               2 
                             
                             ⁢ 
                             ϑ 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           cos 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             ϕ 
                             - 
                             ψ 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         V 
                         aa 
                       
                       ⁢ 
                       
                         
                           sin 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             ϕ 
                             - 
                             ψ 
                           
                           ) 
                         
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     V 
                     
                       
                         y 
                         ~ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         y 
                         ~ 
                       
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               V 
                               aa 
                             
                             ⁢ 
                             
                               cos 
                               2 
                             
                             ⁢ 
                             ϑ 
                           
                           + 
                           
                             
                               V 
                               cc 
                             
                             ⁢ 
                             
                               sin 
                               2 
                             
                             ⁢ 
                             ϑ 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           sin 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             ϕ 
                             - 
                             ψ 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         V 
                         aa 
                       
                       ⁢ 
                       
                         
                           cos 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             ϕ 
                             - 
                             ψ 
                           
                           ) 
                         
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     V 
                     zz 
                   
                   = 
                   
                     
                       
                         V 
                         aa 
                       
                       ⁢ 
                       
                         sin 
                         2 
                       
                       ⁢ 
                       ϑ 
                     
                     + 
                     
                       
                         V 
                         cc 
                       
                       ⁢ 
                       
                         cos 
                         2 
                       
                       ⁢ 
                       ϑ 
                     
                   
                 
               
             
             
               
                 ( 
                 22 
                 ) 
               
             
           
           
             
               
                 
                   
                     V 
                     
                       
                         x 
                         ~ 
                       
                       ⁢ 
                       
                         y 
                         ~ 
                       
                     
                   
                   = 
                   
                     
                       V 
                       
                         
                           y 
                           ~ 
                         
                         ⁢ 
                         
                           x 
                           ~ 
                         
                       
                     
                     = 
                     
                       
                         - 
                         
                           ( 
                           
                             
                               V 
                               aa 
                             
                             - 
                             
                               V 
                               cc 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         sin 
                         2 
                       
                       ⁢ 
                       
                         ϑcos 
                         ⁡ 
                         
                           ( 
                           
                             ϕ 
                             - 
                             ψ 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             ϕ 
                             - 
                             ψ 
                           
                           ) 
                         
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     V 
                     
                       z 
                       ⁢ 
                       
                         x 
                         ~ 
                       
                     
                   
                   = 
                   
                     
                       V 
                       
                         
                           x 
                           ~ 
                         
                         ⁢ 
                         z 
                       
                     
                     = 
                     
                       
                         - 
                         
                           ( 
                           
                             
                               V 
                               aa 
                             
                             - 
                             
                               V 
                               cc 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ϑsinϑcos 
                         ⁡ 
                         
                           ( 
                           
                             ϕ 
                             - 
                             ψ 
                           
                           ) 
                         
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     V 
                     
                       
                         y 
                         ~ 
                       
                       ⁢ 
                       z 
                     
                   
                   = 
                   
                     
                       V 
                       
                         z 
                         ⁢ 
                         
                           y 
                           ~ 
                         
                       
                     
                     = 
                     
                       
                         - 
                         
                           ( 
                           
                             
                               V 
                               aa 
                             
                             - 
                             
                               V 
                               cc 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ϑsinϑsin 
                         ⁡ 
                         
                           ( 
                           
                             ϕ 
                             - 
                             ψ 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 23 
                 ) 
               
             
           
         
       
     
   
   The following relations apply: 
   
     
       
         
           
             
               
                 
                   
                     
                       V 
                       
                         
                           x 
                           ~ 
                         
                         ⁢ 
                         
                           x 
                           ~ 
                         
                       
                     
                     + 
                     
                       V 
                       
                         
                           y 
                           ~ 
                         
                         ⁢ 
                         
                           y 
                           ~ 
                         
                       
                     
                     + 
                     
                       V 
                       zz 
                     
                   
                   = 
                   
                     
                       2 
                       ⁢ 
                       
                         V 
                         aa 
                       
                     
                     + 
                     
                       V 
                       cc 
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     
                       V 
                       
                         
                           x 
                           ~ 
                         
                         ⁢ 
                         
                           x 
                           ~ 
                         
                       
                     
                     - 
                     
                       V 
                       
                         
                           y 
                           ~ 
                         
                         ⁢ 
                         
                           y 
                           ~ 
                         
                       
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           V 
                           cc 
                         
                         - 
                         
                           V 
                           aa 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       sin 
                       2 
                     
                     ⁢ 
                     ϑ 
                     ⁢ 
                     
                       { 
                       
                         
                           
                             cos 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               ϕ 
                               - 
                               ψ 
                             
                             ) 
                           
                         
                         - 
                         
                           
                             sin 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               ϕ 
                               - 
                               ψ 
                             
                             ) 
                           
                         
                       
                       } 
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     
                       V 
                       
                         
                           y 
                           ~ 
                         
                         ⁢ 
                         
                           y 
                           ~ 
                         
                       
                     
                     - 
                     
                       V 
                       zz 
                     
                   
                   = 
                   
                     
                       - 
                       
                         ( 
                         
                           
                             V 
                             cc 
                           
                           - 
                           
                             V 
                             aa 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           
                             cos 
                             2 
                           
                           ⁢ 
                           ϑ 
                         
                         - 
                         
                           
                             sin 
                             2 
                           
                           ⁢ 
                           
                             
                               ϑsin 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 ϕ 
                                 - 
                                 ψ 
                               
                               ) 
                             
                           
                         
                       
                       } 
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     
                       V 
                       zz 
                     
                     - 
                     
                       V 
                       
                         
                           x 
                           ~ 
                         
                         ⁢ 
                         
                           x 
                           ~ 
                         
                       
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           V 
                           cc 
                         
                         - 
                         
                           V 
                           aa 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       { 
                       
                         
                           
                             cos 
                             2 
                           
                           ⁢ 
                           ϑ 
                         
                         - 
                         
                           
                             sin 
                             2 
                           
                           ⁢ 
                           
                             
                               ϑcos 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 ϕ 
                                 - 
                                 ψ 
                               
                               ) 
                             
                           
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 ( 
                 24 
                 ) 
               
             
           
         
       
     
   
   Consequently, 
   
     
       
         
           
             
               
                 
                   
                     ϕ 
                     - 
                       
                     ⁢ 
                     ψ 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
                       tan 
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         
                           V 
                           
                             
                               x 
                               ~ 
                             
                             ⁢ 
                             
                               y 
                               ~ 
                             
                           
                         
                         + 
                         
                           V 
                           
                             
                               y 
                               ~ 
                             
                             ⁢ 
                             
                               x 
                               ~ 
                             
                           
                         
                       
                       
                         
                           V 
                           
                             
                               x 
                               ~ 
                             
                             ⁢ 
                             
                               x 
                               ~ 
                             
                           
                         
                         - 
                         
                           V 
                           
                             
                               y 
                               ~ 
                             
                             ⁢ 
                             
                               y 
                               ~ 
                             
                           
                         
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     ϕ 
                     - 
                     ψ 
                   
                   = 
                   
                     
                       
                         tan 
                         
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           V 
                           
                             
                               y 
                               ~ 
                             
                             ⁢ 
                             z 
                           
                         
                         
                           V 
                           
                             
                               x 
                               ~ 
                             
                             ⁢ 
                             z 
                           
                         
                       
                     
                     = 
                     
                       
                         tan 
                         
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           V 
                           
                             z 
                             ⁢ 
                             
                               y 
                               ~ 
                             
                           
                         
                         
                           V 
                           
                             z 
                             ⁢ 
                             
                               x 
                               ~ 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 25 
                 ) 
               
             
           
         
       
     
   
   The azimuth angle φ is measured from the tri-axial responses if the tool rotation angle ψ is known. To the contrary, the dip (deviation) angle ∂ is determined by 
                   tan   ⁢           ⁢   ϑ     =           (       V       x   ~     ⁢     y   ~           V       x   ~     ⁢   z         )     2     +       (       V       x   ~     ⁢     y   ~           V       y   ~     ⁢   z         )     2                 (   26   )               
without knowing the tool orientation ψ.
 
   Apparent Dip Angle And Azimuth Angle And The Distance To The Anomaly 
   The dip and the azimuth angle described above indicate the direction of a resistivity anomaly determined by a combination of tri-axial transient responses at a time (t) when the angles have deviated from a zero value. When t is small or close to zero, the effect of such anomaly is not apparent in the transient responses as all the cross-component responses are vanishing. To identify the anomaly and estimate not only its direction but also the distance, it is useful to define the apparent azimuth angle φ app (t) by, 
                       ϕ   app     ⁡     (   t   )       =       1   2     ⁢     tan     -   1       ⁢           V   xy     ⁡     (   t   )       +       V   yx     ⁡     (   t   )               V   xx     ⁡     (   t   )       -       V   yy     ⁡     (   t   )               ⁢     
     ⁢         ϕ   app     ⁡     (   t   )       =         tan     -   1       ⁢         V   yz     ⁡     (   t   )           V   xz     ⁡     (   t   )           =       tan     -   1       ⁢         V   zy     ⁡     (   t   )           V   zx     ⁡     (   t   )                       (   27   )               
and the effective dip angle ∂ app (t) by
 
                   tan   ⁢           ⁢       ϑ   app     ⁡     (   t   )         =           (         V   xy     ⁡     (   t   )           V   xz     ⁡     (   t   )         )     2     +       (         V   xy     ⁡     (   t   )           V   yz     ⁡     (   t   )         )     2                 (   28   )               
for the time interval when φ app (t)≠0 nor π/2 (90°). For simplicity, the case examined below is one in which none of the cross-component measurements is identically zero:
   V   xy ( t )≠0 , V   yz ( t )≠0, and  V   zx ( t )≠0. 
   For the time interval when φ app (t)=0, ∂ app (t) is defined by, 
   
     
       
         
           
             
               
                 
                   
                     ϑ 
                     app 
                   
                   ⁡ 
                   
                     ( 
                     t 
                     ) 
                   
                 
                 = 
                 
                   
                     - 
                     
                       1 
                       2 
                     
                   
                   ⁢ 
                   
                     tan 
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       
                         
                           V 
                           xz 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       + 
                       
                         
                           V 
                           zx 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     
                       
                         
                           V 
                           xx 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       - 
                       
                         
                           V 
                           zz 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 29 
                 ) 
               
             
           
         
       
     
   
   For the time interval when φ app (t)=π/2 (90°), ∂ app (t) is defined by, 
   
     
       
         
           
             
               
                 
                   
                     ϑ 
                     app 
                   
                   ⁡ 
                   
                     ( 
                     t 
                     ) 
                   
                 
                 = 
                 
                   
                     - 
                     
                       1 
                       2 
                     
                   
                   ⁢ 
                   
                     tan 
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       
                         
                           V 
                           yz 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       + 
                       
                         
                           V 
                           zy 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     
                       
                         
                           V 
                           yy 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       - 
                       
                         
                           V 
                           zz 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 30 
                 ) 
               
             
           
         
       
     
   
   When t is small and the transient responses do not see the effect of a resistivity anomaly at distance, the effective angles are identically zero, φ app (t)=∂ app (t)=0. As t increases, when the transient responses see the effect of the anomaly, φ app (t) and ∂ app (t) begin to show the true azimuth and the true dip angles. The distance to the anomaly may be indicated at the time when φ app (t) and ∂ app (t) start deviating from the initial zero values. As shown below in a modeling example, the presence of an anomaly is detected much earlier in time in the effective angles than in the apparent conductivity (σ app (t)). Even if the resistivity of the anomaly may not be known until σ app (t) is affected by the anomaly, its presence and the direction can be measured by the apparent angles. With limitation in time measurement, the distant anomaly may not be seen in the change of σ app (t) but is visible in φ app (t) and ∂ app (t). 
   MODELING EXAMPLE 
   A simplified modeling example exists when a resistivity anomaly is a massive salt dome, and the salt interface may be regarded as a plane interface. For further simplification, it can be assumed that the azimuth of the salt face is known. Accordingly, the remaining unknowns are the distance D to the salt face from the tool, the isotropic or anisotropic formation resistivity, and the approach angle (or dip angle) θ as shown in  FIG. 6 . 
     FIGS. 9 and 10  show the voltage from the coaxial V zz (t), coplanar V xx (t), and the cross-component V xx (t) measurements for L=1 m, for θ=30°, and at salt distance D=10 m and D=100 m respectively. The apparent dip θ app (t) is defined by, 
   
     
       
         
           
             
               
                 
                   
                     θ 
                     app 
                   
                   ⁡ 
                   
                     ( 
                     t 
                     ) 
                   
                 
                 = 
                 
                   
                     - 
                     
                       1 
                       2 
                     
                   
                   ⁢ 
                   
                     tan 
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             V 
                             zx 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         + 
                         
                           
                             V 
                             xz 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                       
                         
                           
                             V 
                             zz 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         - 
                         
                           
                             V 
                             xx 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                     
                     . 
                   
                 
               
             
             
               
                 ( 
                 31 
                 ) 
               
             
           
         
       
     
   
     FIG. 11  shows the apparent dip (θ app (t)) for the L=1 m tool assembly when the salt face is D=10 m away and at the approach angle of θ= 30 °. 
   In addition, the apparent conductivity (σ app (t)), from both the coaxial (V zz (t)) and the coplanar (V xx (t)) responses is shown in  FIG. 12 , wherein the approach angle (θ) and salt face distance (D) are the same as in  FIG. 11 . 
   Also plotted is the ratio, σ app-coplanar (t)/σ app-coaxial (t), that is available without cross-component V xx (t) measurements as shown in  FIG. 13 , wherein the approach angle (θ) and salt face distance (D) are the same as in  FIG. 3 . 
   Note that the direction to the salt face is immediately identified in the apparent dip θ app (t) plot of  FIG. 11  as early as 10 −4  second when the presence of the resistivity anomaly is barely detected in the apparent conductivity (σ app (t)) plot of  FIG. 12 . It takes almost 10 −3  second for the apparent conductivity to approach an asymptotic σ app (later t) value and for the apparent conductivity ratio to read θ=30°. 
     FIG. 14  shows the apparent dip θ app (t) for the L=1 m tool assembly when the salt face is D=10 m away, but at different angles between the tool axis and the target. The approach angle (θ) may be identified at any angle. 
     FIG. 15  compares the apparent dip θ app (t) for different salt face distances (D) and different angles between the tool axis and the target. 
   The distance to the salt face can be also determined by the transition time at which θ app (t) takes an asymptotic value. Even if the salt face distance (D) is 100 m, it can be identified and its direction can be measured by the apparent dip θ app (t). 
   In summary, the method considers the coordinate transformation of transient EM responses between tool-fixed coordinates and anomaly-fixed coordinates. When the anomaly is large and far away compared to the transmitter-receiver spacing, one may ignore the effect of spacing and approximate the transient EM responses with those of the receivers near the transmitter. Then, one may assume axial symmetry exists with respect to the c-axis that defines the direction from the transmitter to the anomaly. In such an axially symmetric configuration, the cross-component responses in the anomaly-fixed coordinates are identically zero. With this assumption, a general method is provided for determining the direction to the resistivity anomaly using tri-axial transient EM responses. 
   The method defines the apparent dip θ app (t) and the apparent azimuth ψ app (t) by combinations of tri-axial transient measurements. The apparent direction {θ app (t), ψ app (t)} reads the true direction {θ,ψ} at later time. The θ app (t) and ψ app (t) both read zero when t is small and the effect of the anomaly is not sensed in the transient responses or the apparent conductivity. The conductivities (σ coaxial (t) and σ coplanar (t)) from the coaxial and coplanar measurements both indicate the conductivity of the near formation around the tool. 
   Deviation of the apparent direction ({θ app (t), ψ app (t)}) from zero identifies the anomaly. The distance to the anomaly is measured by the time when the apparent direction ({θ app (t), ψ app (t)}) approaches the true direction ({θ, ψ}). The distance can be also measured from the change in the apparent conductivity. However, the anomaly is identified and measured much earlier in time in the apparent direction than in the apparent conductivity. 
   Apparent Conductivity 
   As set forth above, apparent conductivity can be used as an alternative technique to apparent angles in order to determine the location of an anomaly in a wellbore. The time-dependent apparent conductivity can be defined at each point of a time series at each logging depth. The apparent conductivity at a logging depth z is defined as the conductivity of a homogeneous formation that would generate the same tool response measured at the selected position. 
   In transient EM logging, transient data are collected as a logging depth or tool location z as a time series of induced voltages in a receiver loop. Accordingly, time dependent apparent conductivity (σ(z, t)) may be defined at each point of the time series at each logging depth, for a proper range of time intervals depending on the formation conductivity and the tool specifications. 
   Apparent Conductivity For a Coaxial Tool 
   The induced voltage of a coaxial tool with transmitter-receiver spacing L in the homogeneous formation of conductivity (σ) is given by, 
   
     
       
         
           
             
               
                 
                   
                     V 
                     zZ 
                   
                   ⁡ 
                   
                     ( 
                     t 
                     ) 
                   
                 
                 = 
                 
                   C 
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           
                             μ 
                             o 
                           
                           ⁢ 
                           σ 
                         
                         ) 
                       
                       
                         3 
                         2 
                       
                     
                     
                       8 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         t 
                         
                           5 
                           2 
                         
                       
                     
                   
                   ⁢ 
                   
                     ⅇ 
                     
                       - 
                       
                         u 
                         2 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 32 
                 ) 
               
             
           
           
             
               
                 
                   
                     where   
                   
                   ⁢ 
                   
                     u 
                     2 
                   
                 
                 = 
                 
                   
                     
                       
                         μ 
                         o 
                       
                       ⁢ 
                       σ 
                     
                     4 
                   
                   ⁢ 
                   
                     
                       L 
                       2 
                     
                     t 
                   
                   ⁢ 
                   
                       and   
                   
                   ⁢ 
                   C 
                   ⁢ 
                   
                       is  a  constant. 
                   
                 
               
             
             
               
                   
               
             
           
         
       
     
   
     FIG. 7  illustrates a coaxial tool in which both a transmitter coil (T) and a receiver coil (R) are wound around the common tool axis. The symbols σ 1  and σ 2  may represent the conductivities of two formation layers. This tool is used to illustrate the voltage response for different values of t and L in  FIGS. 16-18 , where σ 1 =σ 2 . 
     FIG. 16  shows the voltage response of the coaxial tool with L=01 m in a homogeneous formation for various formation resistivities (R) from 1000 ohm-m to 0.1 ohm-m. The voltage is positive at all times t for t&gt;0. The slope of the voltage is nearly constant 
                 ∂   ln     ⁢           ⁢       V   zZ     ⁡     (   t   )             ∂   ln     ⁢           ⁢   t       ≈     -     5   2             
in the time interval between 10 −8  second and 1 second (and later) for any formation resistivity larger than 10 ohm-m. The slope changes sign at an earlier time around 10 −6  second when the resistivity is low as 0.1 ohm-m.
 
     FIG. 17  shows the voltage response as a function of formation resistivity at different times (t) for the same coaxial tool spacing (L=01 m). For the resistivity range from 0.1 ohm-m to 100 ohm-m, the voltage response is single valued as a function of formation resistivity for the measurement time (t) later than 10 −6  second. At smaller times (t), for instance at 10 −7  second, the voltage is no longer single valued. The same voltage response is realized at two different formation resistivity values. 
     FIG. 18  shows the voltage response as a function of formation resistivity for a larger transmitter-receiver spacing of L=10 m on a coaxial tool. The time interval when the voltage response is single valued is shifted toward larger times (t). The voltage response is single valued for resistivity from 0.1 ohm-m to 100 ohm-m, for the measurement time (t) later than 10 −4  second. At smaller values of t, for instance at t=10 −5  second, the voltage is no longer single valued. The apparent conductivity from a single measurement (coaxial, single spacing) alone is not well defined. 
   For relatively compact transmitter-receiver spacing (L=1 m to 10 m), and for the time measurement interval where t is greater than 10 −6  second, the transient EM voltage response is mostly single valued as a function of formation resistivity between 0.2-ohm-m and 100 ohm-m (and higher). This enables definition of the time-changing apparent conductivity from the voltage response (V zZ (t)) at each time measurement as: 
                   C   ⁢         (       μ   o     ⁢       σ   app     ⁡     (   t   )         )       3   2         8   ⁢           ⁢     t     5   2           ⁢     ⅇ     -         u   app     ⁡     (   t   )       2           =       V   zZ     ⁡     (   t   )               (   33   )               
where
 
                 u   app     ⁡     (   t   )       2     =           μ   o     ⁢       σ   app     ⁡     (   t   )         4     ⁢       L   2     t             
and V zZ (t) on the right hand side is the measured voltage response of the coaxial tool. From a single type of measurement (coaxial, single spacing), the greater the spacing L, the larger the measurement time (t) should be to apply the apparent conductivity concept. The σ app (t) should be constant and equal to the formation conductivity in a homogeneous formation: σ app (t)=σ. The deviation from a constant (σ) at time (t) suggests a conductivity anomaly in the region specified by time (t).
 
   Apparent Conductivity For a Coplanar Tool 
   The induced voltage of the coplanar tool with transmitter-receiver spacing L in the homogeneous formation of conductivity (σ) is given by, 
                     V   xX     ⁡     (   t   )       =     C   ⁢         (       μ   0     ⁢       σ   app     ⁡     (   t   )         )       3   2         8   ⁢           ⁢     t     5   2           ⁢     (     1   -     u   2       )     ⁢     ⅇ     -     u   2                   (   34   )               
where
 
             u   2     =           μ   0     ⁢       σ   app     ⁡     (   t   )           4   ⁢           ⁢   t       ⁢     L   2             
and C is a constant. At small values of t, the coplanar voltage changes polarity depending on the spacing L and the formation conductivity.
 
     FIG. 8  illustrates a coplanar tool in which the transmitter (T) and the receiver (R) are parallel to each other and oriented perpendicularly to the tool axis. The symbols σ 1  and σ 2  may represent the conductivities of two formation layers. This tool is used to illustrate the voltage response for different values of t and L in  FIGS. 19 and 20  below, where σ 1 =σ 2 . 
     FIG. 19  shows the voltage response of a coplanar tool with a length L=01 m as a function of formation resistivity at different times (t). For the resistivity range from 0.1 ohm-m to 100 ohm-m, the voltage response is single valued as a function of formation resistivity for values of t larger than 10 −6  second. At smaller values of t, for instance at t=10 7  second, the voltage changes polarity and is no longer single valued. 
     FIG. 20  shows the voltage response as a function of formation resistivity at different times (t) for a longer coplanar tool with a length L=05 m. The time interval when the voltage response is single valued is shifted towards larger values of t. 
   Similarly to the coaxial tool response, the time-changing apparent conductivity is defined from the coplanar tool response V xX (t) at each time of measurement as, 
                   C   ⁢         (       μ   0     ⁢       σ   app     ⁡     (   t   )         )       3   2         8   ⁢           ⁢     t     5   2           ⁢     (     1   -         u   app     ⁡     (   t   )       2       )     ⁢     ⅇ     -         u   app     ⁡     (   t   )       2           =       V   xX     ⁡     (   t   )               (   35   )               
where
 
                 u   app     ⁡     (   t   )       2     =           μ   o     ⁢       σ   app     ⁡     (   t   )         4     ⁢       L   2     t             
and V xX (t) on the right-hand side is the measured voltage response of the coplanar tool. The longer the spacing, the larger the value t should be to apply the apparent conductivity concept from a single type of measurement (coplanar, single spacing). The σ app (t) should be constant and equal to the formation conductivity in a homogeneous formation: σ app (t)=σ.
 
   Apparent Conductivity For A Pair Of Coaxial Tools 
   When there are two coaxial receivers, the ratio between the pair of voltage measurements is given by, 
                       V   zZ     ⁡     (       L   1     ;   t     )           V   zZ     ⁡     (       L   2     ;   t     )         =     ⅇ       -         μ   o     ⁢   σ       4   ⁢           ⁢   t         ⁢     (       L   1   2     -     L   2   2       )                 (   36   )               
where L 1  and L 2  are transmitter-receiver spacing of two coaxial tools.
 
   Conversely, the time-changing apparent conductivity is defined for a pair of coaxial tools by, 
                     σ   app     ⁡     (   t   )       =         -     ln   ⁡     (         V   zZ     ⁡     (       L   1     ;   t     )           V   zZ     ⁡     (       L   2     ;   t     )         )           (       L   1   2     -     L   2   2       )       ⁢       4   ⁢           ⁢   t       μ   o                 (   37   )               
at each time of measurement. The σ app (t) should be constant and equal to the formation conductivity in a homogeneous formation: σ app (t)=σ.
 
   The apparent conductivity is similarly defined for a pair of coplanar tools or for a pair of coaxial and coplanar tools. The σ app (t) should be constant and equal to the formation conductivity in a homogeneous formation: σ app (t)=σ. The deviation from a constant (σ) at time (t) suggests a conductivity anomaly in the region specified by time (t). 
   Analysis Of Coaxial Transient Response In Two-Layer Models 
   To illustrate usefulness of the concept of apparent conductivity, the transient response of a tool in a two-layer earth model, as in  FIG. 7  for example, can be examined. A coaxial tool with a transmitter-receiver spacing L may be placed in a horizontal well. Apparent conductivity (σ app (t)) reveals three parameters including: (1) the conductivity (σ 1 =0.1 S/m) of a first layer in which the tool is placed; (2) the conductivity (σ 2 =1 S/m) of an adjacent bed; and (3) the distance of the tool (horizontal borehole) to the layer boundary, d=1, 5, 10, 25, and 50 m. 
   Under a more general circumstance, the relative direction of a borehole and tool to the bed interface is not known. In the case of horizontal well logging, it&#39;s trivial to infer that the tool is parallel to the interface as the response does not change when the tool moves. 
   The voltage response of the L=01 m transmitter-receiver offset coaxial tool at different distances is shown in  FIG. 21 . Information can be derived from these responses using apparent conductivity as further explained with regard to  FIG. 22 .  FIG. 22  shows the voltage data of  FIG. 21  plotted in terms of apparent conductivity. The apparent conductivity plot shows conductivity at small t, conductivity at large t, and the transition time that moves as the distance (d) changes 
   As will be further explained below, in a two-layer resistivity profile, the apparent conductivity as t approaches zero can identify the layer conductivity around the tool, while the apparent conductivity as t approaches infinity can be used to determine the conductivity of the adjacent layer at a distance. The distance to a bed boundary from the tool can also be measured from the transition time observed in the apparent conductivity plot. The apparent conductivity plot for both time and tool location may be used as an image presentation of the transient data. Similarly,  FIG. 23  illustrates the apparent conductivity in a two-layer model where σ 1 =1 S/m (R 1 =1 ohm-m) and σ 2 =0.1 S/m (R 2 =1 ohm-m). 
   Conductivity At Small Values Of t 
   At small values of t, the tool reads the apparent conductivity of the first layer around the tool. At large values of t, the tools reads 0.4 S/m for a two-layer model where σ 1 =0.1 S/m (R 1 =10 ohm-m) and σ 2 =1 S/m (R 2 =1 ohm-m), which is an average between the conductivities of the two layers. The change of distance (d) is reflected in the transition time. 
   Conductivity at small values of t is the conductivity of the local layer where the tool is located. At small values of t, the signal reaches the receiver directly from the transmitter without interfering with the bed boundary. Namely, the signal is affected only by the conductivity around the tool. Conversely, the layer conductivity can be measured easily by examining the apparent conductivity at small values of t. 
   Conductivity At Large Values Of t 
   Conductivity at large values of t is some average of conductivities of both layers. At large values of t, nearly half of the signals come from the formation below the tool and the remaining signals come from above, if the time for the signal to travel the distance between the tool and the bed boundary is small. 
     FIG. 24  compares the σ app (t) plot of  FIGS. 22 and 23  for L=1 m and d=1 m where the resistivity ratio R 1 /R 2  is 10:1 in  FIG. 22  and 1:10 in  FIG. 23 . Though not shown, the conductivity at large values of t has a slight dependence on d. When the dependence is ignored, the conductivity at large values of t is determined solely by the conductivities of the two layers and is not affected by the location of the tool in layer  1  or layer  2 . 
     FIG. 25  compares the σ app (t) plots for d=1 m but with different spacings L. The σ app (t) reaches the nearly constant conductivity at large values of t as L increases. However, the conductivity at large values of t is almost independent of the spacing L for the range of d and the conductivities considered. 
     FIG. 26  compares the σ app (t) plots for d=1 m and L=1 m but for different resistivity ratios. The apparent conductivity at large t is proportional to σ 1  for the same ratio (σ 1 /σ 2 ). For instance:
 σ app ( t→∞; R   1   /R   2 10 ohm m, R   1 =10 ohm- m )=10*σ app ( t→∞; R   1   /R   2 =10 ohm- m, R   1 =100 ohm- m ) 
     FIG. 27  shows example s of the σ app (t) plots for d=1 m and L=1 m but for different sensitivity ratios of the target layer  2  while the local conductivity (σ 1 ) is fixed at 1 S/m (R 1 =1 ohm-m). The apparent conductivity at large values of t is determined by the target layer  2  conductivity, as shown in  FIG. 28  when σ 1  is fixed at 1 S/m. 
   Numerically, the late time conductivity may be approximated by the square root average of two-layer conductivities as: 
   
     
       
         
           
             
               
                 
                   
                     
                       σ 
                       app 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             t 
                             → 
                             ∞ 
                           
                           ; 
                           
                             σ 
                             1 
                           
                         
                         , 
                         
                           σ 
                           2 
                         
                       
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       
                         σ 
                         1 
                       
                     
                     + 
                     
                       
                         σ 
                         2 
                       
                     
                   
                   2 
                 
               
             
             
               
                 ( 
                 39 
                 ) 
               
             
           
         
       
     
   
   To summarize, the conductivity at large values of t (as t approaches infinity) can be used to estimate the conductivity (σ 2 ) of the adjacent layer when the local conductivity (σ 1 ) near the tool is known, for instance from the conductivity as t approaches 0 as illustrated in  FIG. 29 . 
   Estimation of d, The Distance To The Adjacent Bed 
   The transition time at which the apparent conductivity (σ app (t) starts deviating from the local conductivity (σ 1 ) towards the conductivity at large values of t depends on d and L, as shown in  FIG. 30 . 
   For convenience, the transition time (t c ) can be defined as the time at which the σ app (t) takes the cutoff conductivity (σ c ). In this case, the cutoff conductivity is represented by the arithmetic average between the conductivity as t approaches zero and the conductivity as t approaches infinity. The transition time (t c ) is dictated by the ray path: 
                         (     L   2     )     2     +     d   2       ,             (   40   )               
that is the shortest distance for the EM signal traveling from the transmitter to the bed boundary, to the receiver, independently of the resistivity of the two layers. Conversely, the distance (d) can be estimated from the transition time (t c ), as shown in  FIG. 31 .
 
   Other Uses Of Apparent Conductivity 
   Similarly to conventional induction tools, the apparent conductivity (σ app (t) is useful for analysis of the error in transient signal processing. The effect of the noise in transient response data may be examined as the error in the conductivity determination. 
   A plot of the apparent conductivity (σ app (z; t)) for different distances (d) in both the z and t coordinates may serve as an image presentation of the transient data as shown in  FIG. 32  for a L=01 m tool. The z coordinate references the tool depth along the borehole. The σ app (z; t) plot shows the approaching bed boundary as the tool moves along the borehole. 
   The apparent conductivity should be constant and equal to the formation conductivity in a homogeneous formation. The deviation from a constant conductivity value at time (t) suggests the presence of a conductivity anomaly in the region specified by time (t). 
   In summary, the method allows real-time location of an anomaly in a borehole. The location of the anomaly is defined by its distance and direction from an origin. As demonstrated, the distance and direction can be determined based on magnetic field transient responses. 
   The present invention has been described in relation to particular embodiments, which are intended in all respects to be illustrative rather than restrictive. Alternative embodiments will become apparent to those skilled in the art to which the present invention pertains without departing from its scope. 
   From the foregoing, it will be seen that this invention is one well adapted to attain all the ends and objects set forth above, together with other advantages, which are obvious and inherent to the system and method. It will be understood that certain features and sub-combinations are of utility and may be employed without reference to other features and sub-combinations. This is contemplated and within the scope of the claims.