Patent Publication Number: US-9405034-B2

Title: Electromagnetic logging in time domain with use of specific current pulses

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to estimating a property of an earth formation. More specifically, the invention relates to apparatus and methods for performing transient electromagnetic sounding in a borehole to determine the property. 
     2. Description of the Related Art 
     Exploration and production of hydrocarbons requires knowledge of a subsurface earth formation that may contain reservoirs of the hydrocarbons. Transient electromagnetic (TEM) sounding at the surface of the earth is one way of gathering this knowledge. 
     Traditional TEM sounding is based on applying a series of electromagnetic energy pulses to the earth formation using a switch-off current mode. The series of pulses requires identical sharp pulses of source current applied to a transmitter coil. The pulses induce circulations of current in the earth formation. The circulations of current in turn induce electromagnetic signals in a receiver. A characteristic of the electromagnetic signals can be related to a property of the formation such as conductivity. 
     Traditional TEM sounding at the surface of the earth can have some drawbacks. One drawback is that earth formations containing reservoirs of hydrocarbons may be located deep inside the earth. The reservoirs may be beyond the reach of the electromagnetic energy pulses. Another drawback is that traditional TEM sounding may not be able to discriminate between different distances into the formation. 
     Therefore, what are needed are techniques to measure a property of an earth formation deep inside the earth using TEM sounding. Preferably, the techniques can discriminate between different distances into the earth formation. 
     BRIEF SUMMARY OF THE INVENTION 
     Disclosed is a method for estimating a property of an earth formation in a zone of investigation, the earth formation having the zone of investigation and another zone, the method including: applying a compound pulse of electromagnetic energy to the earth formation, the compound pulse having a shape to induce a predominately greater amount of current in the zone of investigation than in the another zone, wherein the compound pulse includes a first pulse and a second pulse, the first pulse having an amplitude different from the amplitude of the second pulse; receiving a response of electromagnetic energy from the current induced by the compound pulse; and estimating the property from the response of electromagnetic energy. 
     Also disclosed is an apparatus for estimating a property of an earth formation in a zone of investigation, the earth formation having the zone of investigation and another zone, the apparatus including: a logging instrument; a transmitter; a receiver; and a processing system configured to implement the following instructions: applying a compound pulse of electromagnetic energy to the earth formation, the compound pulse having a shape to induce a predominately greater amount of current in the zone of investigation than in the another zone, wherein the compound pulse includes a first pulse and a second pulse, the first pulse having an amplitude different from the amplitude of the second pulse; receiving a response of electromagnetic energy from the current induced by the compound pulse; and estimating the property from the response of electromagnetic energy. 
     Further disclosed is a computer program product stored on machine-readable media having machine-executable instructions for estimating a property of an earth formation in a zone of investigation, the earth formation including the zone of investigation and another zone, by implementing a method including: applying a compound pulse of electromagnetic energy to the earth formation, the compound pulse having a shape to induce a predominately greater amount of current in the zone of investigation than in the another zone, wherein the compound pulse includes a first pulse and a second pulse, the first pulse having an amplitude different from the amplitude of the second pulse; receiving a response of electromagnetic energy from the current induced by the compound pulse; and estimating the property from the response of electromagnetic energy. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The subject matter, which is regarded as the invention, is particularly pointed out and distinctly claimed in the claims at the conclusion of the specification. The foregoing and other features and advantages of the invention are apparent from the following detailed description taken in conjunction with the accompanying drawings, wherein like elements are numbered alike, in which: 
         FIG. 1  illustrates an exemplary embodiment of a logging instrument disposed in a borehole penetrating the earth; 
         FIG. 2  depicts aspects of the logging instrument configured for transient electromagnetic logging in one dimensional cylindrical geometry; 
         FIG. 3  depicts aspects of responses to different types of impulses; 
         FIG. 4  depicts aspects of responses to different types of impulses in a multilayer cylindrical medium; 
         FIG. 5  depicts aspects of responses to different types of impulses with water-based mud in the borehole and a water-bearing formation; 
         FIG. 6  depicts aspects of responses to different types of impulses for a three-layer environment; 
         FIG. 7  illustrates a graph showing a relationship between resistivity and radius; 
         FIG. 8  illustrates a magnetic dipole in a homogeneous environment; 
         FIG. 9  depicts aspects of transient electromagnetic field excitation by a current impulse with an arbitrary shape; and 
         FIG. 10  presents one example of a method for estimating a property of an earth formation in a zone of investigation. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Disclosed are embodiments of techniques for performing transient electromagnetic (TEM) sounding from within a borehole penetrating an earth formation. The techniques, which include apparatus and method, call for emitting an electromagnetic energy pulse with a certain shape. The shape is optimized to induce circulating currents in a zone of investigation in the formation that is a specific distance from the borehole. In addition, the shape is optimized to minimize induced circulating currents in other zones of the formation. Thus, electromagnetic energy induced in a receiver in the borehole will result mainly from the circulating currents in the target zone. The electromagnetic energy is related a property of the formation, and, therefore, the property can be estimated by measuring the induced electromagnetic energy. 
     Certain definitions are presented here for convenience. The terms “pulse” and “impulse” relate to electromagnetic energy of a certain shape and finite duration transmitted into the earth formation. The terms pulse and impulse may be used interchangeably herein. The terms “compound” or “complex” relate to a pulse of electromagnetic energy that has two or more components. Each of the components has a distinguishing feature. Non-limiting examples of the distinguishing feature include amplitude, duration, shape, and polarity. In one embodiment, a compound pulse may include two components also referred to as pulses such that the compound pulse is made up of the two pulses. The terms compound and complex may be used interchangeably herein. The term “predominately” relates to an amount of current induced in a zone of investigation in the earth formation relative to an amount of current induced in another zone in the earth formation. A predominately greater amount of current induced in the zone of interest will provide a response of electromagnetic energy that can be related to a property of the earth formation in the zone of investigation. As used herein, the term predominately relates at least to a minimum amount of increase in currents induced in the zone of investigation with respect to other zones, the minimum amount being necessary to be able to estimate a property of the earth formation in the zone of investigation from the response. The term “q” relates to an effective cross-section of a receiver coil. In one embodiment, the effective cross-section is determined by multiplying the cross-sectional area of one turn in the coil by the number of turns in the coil. 
     Referring to  FIG. 1 , an exemplary embodiment of a well logging instrument  10  is shown disposed in a borehole  2 . The borehole  2  is drilled through earth  3  and penetrates a formation  4 , which include various formation layers  4 A- 4 E. In the embodiment of  FIG. 1 , the logging instrument  10  is lowered into and withdrawn from the borehole  2  by use of an armored electrical cable  5  or similar conveyance as is known in the art. In other embodiments, the logging instrument  10  may perform measurements, referred to as logging-while-drilling (LWD), during drilling operations or during a temporary halt. 
     The logging instrument  10  as shown in  FIG. 1  is configured to estimate a property of the formation  4  using TEM sounding. Referring to  FIG. 1 , the logging instrument  10  includes a transmitter  6  and a receiver  7 . The transmitter  6  transmits an electromagnetic energy pulse  8  into the borehole  2  and the formation  4 . The electromagnetic energy pulse  8  induces currents  9  in the formation  4  mainly at a zone of investigation  14  related to the shape of the pulse  8 . The currents  9  in turn induce electromagnetic energy  11  in the receiver  7 . In the embodiment of  FIG. 1 , the logging instrument  10  includes an electronic unit  12 . The electronic unit  12  can be configured to operate the logging instrument  10 . Examples of operations performed by the electronic unit  12  include transmitting the electromagnetic energy pulse  8  with a specific shape and receiving data related to the electromagnetic energy  11  induced in the receiver  7 . In addition, the electronic unit  12  can be configured to transmit the data to a processing system  13  at the surface of the earth  3  using the electrical cable  5  or a telemetry system for LWD applications. In addition, for LWD applications, the data can be stored in the electronic unit  12  for later retrieval when the logging instrument  10  is removed from the borehole  2 . 
     In the embodiment of  FIG. 1 , the processing system  13  is configured to determine the specific shape needed to induce the circulating currents  9  at the distance D from the borehole  2 . In addition, the processing system  13  is configured to estimate a property of the formation  4  at distance D using the data received from the logging instrument  10 . 
     Examples of the techniques disclosed herein are presented next. We perform resolving five radial zones within 2 feet of the borehole  2  using time domain electromagnetic fields. One of the basic ideas behind the disclosed approach is to evaluate certain sequences of specific current pulses (to the transmitter  6 ) that may be very sensitive to specific areas of the formation  4  (up to five zones within two feet). As tools for investigation, we use a representation of the response as a sum of the input from isolated zones and we use the results of inversion for the estimate. Studying this option, we developed mathematical modeling, initial guess, and software tools for solving forward and inverse problems. Details of the mathematical analysis and calculations are disclosed later on in this document. 
     Referring to  FIG. 2 , the logging instrument  10  with two coils configured for TEM logging is considered. The logging instrument  10  includes the following parameters: the instrument  10  includes the transmitter  6  of moment M z (t) and the receiver coil  7 . Different current shapes are considered. The configuration is positioned on longitudinal axis  20  of the borehole  2  with resistivity R m  and radius r m . The resistivity of the formation is R t . The radial resistivity profile in the invasion zone of radius r xo  is described by a stepwise function R xo (r). This profile was represented by five subzones. 
     Next, a compound impulse  21  (referring to  FIG. 2 ) is used to create the induced current  9  with a specific spatial distribution. An arbitrary pulse P(τ) can be presented as a sum of short rectangular pulses. If N rectangular pulses of durations Δτ are used, a total response  22  (referring to  FIG. 2 ) can be presented as follows: 
     
       
         
           
             
               
                 
                   
                     E 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       N 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         P 
                         ⁡ 
                         
                           ( 
                           
                             τ 
                             i 
                           
                           ) 
                         
                       
                       · 
                       
                         [ 
                         
                           
                             ⅇ 
                             ⁡ 
                             
                               ( 
                               
                                 t 
                                 - 
                                 
                                   τ 
                                   i 
                                 
                                 - 
                                 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     τ 
                                   
                                   2 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             ⅇ 
                             ⁡ 
                             
                               ( 
                               
                                 t 
                                 - 
                                 
                                   τ 
                                   i 
                                 
                                 + 
                                 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     τ 
                                   
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     The compound pulse  21  shown in  FIG. 2  is made up of two components, a first pulse  23  and a second pulse  24 . Other embodiments of the compound pulse  21  can include more. 
     The contribution to the measured signal from cylindrical area D (r 1 ≦r≦r 2 ;  z   1 ≦ z ≦ z   2 ) can be expressed (for a rectangular pulse) as the following integral: 
     
       
         
           
             
               
                 
                   
                     Emf 
                     D 
                   
                   = 
                   
                     
                       
                         
                           
                             M 
                             z 
                           
                           ⁢ 
                           q 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           μ 
                         
                         
                           32 
                           ⁢ 
                           π 
                           ⁢ 
                           
                             π 
                           
                         
                       
                       · 
                       
                         
                           ∂ 
                           
                               
                           
                         
                         
                           ∂ 
                           t 
                         
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           
                             ( 
                             
                               
                                 μ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 σ 
                               
                               t 
                             
                             ) 
                           
                           
                             5 
                             / 
                             2 
                           
                         
                         · 
                         
                           
                             ∫ 
                             
                               r 
                               1 
                             
                             
                               r 
                               2 
                             
                           
                           ⁢ 
                           
                             
                               ∫ 
                               
                                 
                                   z 
                                   _ 
                                 
                                 1 
                               
                               
                                 
                                   z 
                                   _ 
                                 
                                 2 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   r 
                                   3 
                                 
                                 ⁢ 
                                 
                                   ⅇ 
                                   
                                     
                                       - 
                                       
                                         
                                           μ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           σ 
                                         
                                         
                                           4 
                                           ⁢ 
                                           t 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           r 
                                           2 
                                         
                                         + 
                                         
                                           
                                             z 
                                             _ 
                                           
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   r 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   
                                     z 
                                     _ 
                                   
                                 
                               
                               
                                 
                                   [ 
                                   
                                     
                                       r 
                                       2 
                                     
                                     + 
                                     
                                       
                                         ( 
                                         
                                           z 
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                                             _ 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                   ] 
                                 
                                 
                                   3 
                                   / 
                                   2 
                                 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     We use a uniformly conductive medium of conductivity σ or resistivity ρ=1/σ. The source of the field is a vertical magnetic dipole of a moment M z  (see  FIG. 2 ). 
     In  FIG. 3 , the results of calculations based on equation (1) and equation (2) are given. Our goal is to create the induced current  9  with a specific spatial distribution, most of which is located in a region of interest (i.e., the zone of investigation  14 ) during the entire transient process. The parameters used in the creation of  FIG. 3  are as follows: the D area has the following boundaries: r 1 =0.3 m, r 2  0.7  ,  z   1  0.5 m,  z   2 =1 m; the moment of transmitter coil  6  is M z =1 A*m*m; for receiver coil  7  q=1 m*m; amplitude of the basic pulse A 1 =1 and duration is 1000 nsec.; amplitude of the additional (negative) pulse, A 2 =0.7 and duration is 25 nsec.; the resistivity ρ=1 Ohm-m; length of tool (from transmitter  6  to receiver  7 ) is 0.5 m. 
     First, let us consider the case of step function excitation. The response is shown in  FIG. 3  by the curves marked “step”. The total response (curve  31 ) differs greatly from the contribution from the D area (curve  32 ). This is an indication that a significant part of the current flows outside the D area. 
     Therefore, to increase the amount of current flowing inside the D area we propose to use a compound impulse  21  shown in  FIG. 2 . The impulse consists of a basic rectangular pulse followed by an additional shorter pulse with negative amplitude. The physics behind the idea is that the small negative pulse will eliminate the currents from a near-borehole zone, which is not part of the D area. 
     The curves marked “impulse” show the total response (curve  33 ) and the response from the D area (curve  34 ). They are very close to each other indicating that we succeeded in creating an electric field with the prescribed spatial distribution. Using the compound impulse, we kept a significant part of the current in the D area during the total transient process. 
     A complex impulse is used to create a transient electromagnetic field in a multilayer cylindrical environment. An algorithm named CYLTEM was created using equation (1) to compute the electric fields in the time domain caused by the compound pulse  21  in a cylindrically-layered environment. In  FIG. 4 , an example of such a computation is given. The environment described in Table 1 is the near-borehole zone (at distances of up to 0.6 m from the borehole axis), subdivided into five subzones based on their resistivities. 
                     TABLE 1                  Near Borehole Zone                         N   ρ (Ohm-m)   r bound  (m)                                 1   10   0.1       2   10   0.2       3   30   0.3       4   10   0.4       5   5   0.5       6   2   0.6       7   10   ∞                    
The parameters used in the creation of  FIG. 4  are as follows: transient signal from 2-coil logging instrument  10  in a multilayer cylindrical medium; the moment of the transmitter coil  6  is M z =1 A*m*m; for receiver coil  7  q=1 m*m; the amplitudes (relative) of the complex impulse  20  are: 100, 10, 1, 0.1, 0.01; duration is 5×100 nsec.; tool  10  length is 0.3 m (from transmitter  6  to receiver  7 ).
 
     In  FIG. 5 , results of computing electrical fields in another example are given. The environment is again the near-borehole zone (at distances of up to 0.6 m from the borehole axis). However, water-based mud is used and the formation  4  is water-bearing (R m =2 Ohm-m; R t =2 Ohm-m). The invasion zone is represented by five radial zones (referred to as “model  3   b ”). The parameters used in the creation of  FIG. 5  are as follows: transient signal from 2-coil logging instrument  10  in a multilayer cylindrical medium (model  3   b ); the moment of transmitter coil  6  is M z =1 A*m*m; for receiver coil  7  q=1 m*m; the amplitudes of the complex impulse  21  are: 100, 10, 1, 0.1, 0.01; duration is 5×100 nsec.; tool  10  length is 0.3 m (from transmitter  6  to receiver  7 ). 
     In  FIG. 4  and  FIG. 5 , curves of the transient signal are given for the turn-off mode (i.e., step) and for excitation using the complex pulse  21 . Applying the complex pulse  21  enables us to greatly decrease the dynamic range of the response. 
     In  FIG. 6 , results are shown for a three-layer environment created by placing the cylindrical layer into a homogeneous environment. The curves represent a relatively anomalous signal for a step-like turn-off of the current at the source (curve  61 ) and for the complex pulse  21  (curve  62 ). As shown in  FIG. 6 , using the outbalancing negative rectangular pulse lets us increase the input from 27% to 43%. The parameters used in the creation of  FIG. 6  are as follows: model is cylindrical layer (Resistance=2 Ohm-m) into homogeneous (Resistance=2 Ohm-m) environment; the amplitude of the basic pulse A 1 =1 and duration is 1000 nsec.; the amplitude of the additional (negative) pulse, A 2 =0.7, duration is 100 nsec.; length of tool is 0.5 m. (from the transmitter  6  to the receiver  7 ). 
     Next, the transient mode is estimated using inversion. To investigate the possibility of using transient fields to study the near-borehole zone, we constructed an inversion algorithm based on the Born Approximation of the forward problem. The main advantage of this approach is that it establishes the linear dependence of a signal on disturbances in the conductivity. 
     Thus, the complete field created from N circular disturbances in a linear (Born) approximation may be expressed as follows: 
                     Emf   ⁡     (   t   )       =         Emf   0     ⁡     (   t   )       +       ∑     i   =   1     N     ⁢           ⁢           G   i     ⁡     (   t   )       ·   Δ     ⁢           ⁢     σ   i                   (   3   )               
where Emf 0 (t) is the normal signal from a homogeneous environment and G i  are some geometrical factors.
 
     Equation (4) yields a linear set for defining disturbances if we substitute the measured full signal into the left part and consider the known conductivity of the reference homogeneous environment. However, we may also use the values of the apparent resistivity. 
     Some examples of inversion using equation (3) are presented. We will compute synthetic data using CYLTEM and the response at the axis for the model presented in Table 1, for the step-like pulse. This model is also presented in  FIG. 7  as a graph showing the relationship between resistivity and radius (see curve  71 ). We use a simplified model because we are using a linear approximation based on a homogeneous reference model. The length of the instrument  10  used to create  FIG. 7  is 0.5 meter from the transmitter  6  to the receiver  7 . Inversion results are indicated by curve  72 , and show that logging using Transient Electromagnetic Sounding lets us recover resistivities for five subzones. Values closest to true values were received after the first inversion step using the simplest reference environment. Further corrections are possible through iterations based on a more complex referent model. 
     It has been shown that using the proper pulse shape may control creation of measurement anomalies in the zone of interest. Inversion examples have shown that, within the zone from the borehole to the distance of 0.6 m away from the borehole axis, five subzones could be identified based on their resistivities, even when logging with a regular rectangular pulse. In addition, it has been shown that complex pulse usage can be used for blocking the impact from a conductive borehole. 
     A mathematical description of transient electromagnetic measurements using the compound current impulse  21  is presented next. First, a transient electromagnetic field in a homogenous environment is considered. Let us consider the behavior of a transient electromagnetic field in a uniformly conductive medium of a conductivity σ or resistivity ρ=1/σ. We choose the cylindrical system of coordinates (r, φ, z). The source of the field is a vertical magnetic dipole of moment M z (t) placed at the origin of this system  FIG. 8 . 
     Suppose that a moment M z  is switched off instantaneously so it can be described as a step function of time: 
     
       
         
           
             
               
                 
                   
                     
                       M 
                       z 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               M 
                               z 
                             
                           
                           
                             
                               t 
                               &lt; 
                               0 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               t 
                               &gt; 
                               0 
                             
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     The expression for the electric field in a frequency domain can be written as: 
     
       
         
           
             
               
                 
                   
                     E 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       φ 
                       ⁡ 
                       
                         ( 
                         
                           r 
                           , 
                           
                             z 
                             _ 
                           
                           , 
                           ω 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       - 
                       ⅈ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ω 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       μ 
                       · 
                       
                         
                           M 
                           z 
                         
                         
                           4 
                           ⁢ 
                           π 
                         
                       
                       · 
                       
                         
                           ∂ 
                           
                               
                           
                         
                         
                           ∂ 
                           r 
                         
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             ⅇ 
                             
                               - 
                               kR 
                             
                           
                           R 
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     Applying the Fourier transform for the frequency domain, we can also obtain an expression for E φ  in the time domain for step-function excitation: 
                         E   φ     ⁡     (     r   ,     z   _     ,   t     )       =           μ   ⁢           ⁢     M   z         4   ⁢   μ       ·       ∂               ∂   r         ⁢       ∫     -   ∞     ∞     ⁢         ⅇ     -   kR       R     ⁢     ⅇ       -   ⅈω     ⁢           ⁢   t               ⁢                   (   6   )               
where k 2 =−iωμσ and R=√{square root over (r 2 + z   2 )} (quasi-stationary approximation).
 
     Let us substitute s=−iω and represent Equation (8) as an inverse Laplace transform: 
                       E   φ     ⁡     (     r   ,     z   _     ,   t     )       =           μ   ⁢           ⁢     M   z         4   ⁢   μ       ·       ∂               ∂   r         ⁢     {       1   R     ⁢       1     2   ⁢   πⅈ       ·       ∫     γ   -     i   ⁢           ⁢   ∞         γ   +     i   ⁢           ⁢   ∞         ⁢       ⅇ       -   α     ⁢     s         ⁢           ⁢     ⅇ   st     ⁢     ⅆ   s             }               (   7   )               
where α=R√{square root over (μσ)}, and γ is a constant. This is a reference integral, and we obtain:
 
     
       
         
           
             
               
                 
                   
                     
                       E 
                       φ 
                     
                     ⁡ 
                     
                       ( 
                       
                         r 
                         , 
                         
                           z 
                           _ 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           M 
                           z 
                         
                         
                           8 
                           ⁢ 
                           
                             π 
                             
                               3 
                               / 
                               2 
                             
                           
                           ⁢ 
                           σ 
                         
                       
                       · 
                       
                         
                           ( 
                           
                             
                               μ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               σ 
                             
                             t 
                           
                           ) 
                         
                         
                           3 
                           / 
                           2 
                         
                       
                       · 
                       
                         
                           ∂ 
                           
                               
                           
                         
                         
                           ∂ 
                           r 
                         
                       
                     
                     ⁢ 
                     
                       
                         ⅇ 
                         
                           - 
                           
                             
                               μ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               σ 
                               ⁢ 
                               
                                 
                                   
                                     r 
                                     2 
                                   
                                   + 
                                   
                                     
                                       z 
                                       _ 
                                     
                                     2 
                                   
                                 
                               
                             
                             
                               4 
                               ⁢ 
                               t 
                             
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     Suppose that a vertical measuring coil (i.e., receiver coil  7 ) of effective cross-section q is at the z-axis. Based on equation (8) we can write the expression for the electromotive force Emf induced in the receiver coil  7 : 
                   Emf   =           M   z     ⁢   q       8   ⁢   π   ⁢     π     ⁢   σ       ·       (       μ   ⁢           ⁢   σ     t     )       5   /   2       ·       exp   ⁡     (       -       μ   ⁢           ⁢   σ       4   ⁢   t         ·     z   2       )       .               (   9   )               
It can be concluded that the time attenuation of Emf is governed by the function t −5/2  if the time range t&gt;&gt;0 is considered.
 
     Proceeding from the results obtained for the step function excitation, we will consider the electromagnetic field created by an impulse with an arbitrary shape. Let us assume that the impulse shape is described by the function P(τ). The electric field related to the impulse P(τ) can be calculated through the convolution P(τ) with the function e(t). The function e(t) is the electromagnetic field arising from the excitation with a step function. 
     We will use the numerical analog of the convolution integral. As it is shown in  FIG. 9 , an arbitrary pulse P(τ) can be presented as a sum of short rectangular pulses. If N rectangular pulses of durations Δτ are used, the total response can be presented as follows: 
                     E   ⁡     (   t   )       =       ∑     i   =   1     N     ⁢           ⁢       P   ⁡     (     τ   i     )       ·       [       ⅇ   ⁡     (     t   -       τ   i     ⁢       Δ   ⁢           ⁢   τ     2         )       -     ⅇ   ⁡     (     t   -     τ   i     +       Δ   ⁢           ⁢   τ     2       )         ]     .                 (   10   )               
Note that the representation of equation (10) lets us reduce computational time. In fact, we can calculate the response from the step function excitation only once. Then, the values of this function obtained for a broad time range can be used in equation (10) many times.
 
     Next, the measured signal as a sum of partial responses from different parts of the medium is presented. Special attention is paid to evaluating methods for creating a desirable distribution of currents in space. We would like to be able to generate currents with a maximum intensity in the region of interest (i.e., the target zone or the zone of investigation  14 ). Now let us study the role of the impulse shape in creating the necessary distribution of the current  9 . With this purpose in mind, we present the measured response as a sum of partial responses from different parts of the medium. The current flow in the medium can be subdivided into currents  9  flowing within rings that have the same axis as the dipole inducing the currents  9 . 
     Suppose that the cross-section of the ring of radius r is dr·d z . According to the Ohm&#39;s law, the current ΔJ that flows in the ring can be described in a following manner:
 
Δ J=E   φ ( r, z ,t )·σ· dr·d z     (11)
 
where E φ (r, z ,t) is an electric field in a uniform media:
 
     
       
         
           
             
               
                 
                   
                     
                       E 
                       φ 
                     
                     ⁡ 
                     
                       ( 
                       
                         r 
                         , 
                         
                           z 
                           _ 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           M 
                           z 
                         
                         · 
                         r 
                       
                       
                         16 
                         ⁢ 
                         π 
                         ⁢ 
                         
                           π 
                         
                         ⁢ 
                         σ 
                       
                     
                     · 
                     
                       
                         ( 
                         
                           
                             μ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             σ 
                           
                           t 
                         
                         ) 
                       
                       
                         5 
                         / 
                         2 
                       
                     
                     · 
                     
                       
                         exp 
                         ⁡ 
                         
                           [ 
                           
                             
                               - 
                               
                                 
                                   μ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   σ 
                                 
                                 
                                   4 
                                   ⁢ 
                                   t 
                                 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   r 
                                   2 
                                 
                                 + 
                                 
                                   
                                     z 
                                     _ 
                                   
                                   2 
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     We can use the Biot-Savart law and calculate the magnetic field related to the ring with current ΔJ. In our case, the measuring coil is located at the axis ( FIG. 8 ), then an expression for the electromagnetic field of a circular current can be used. Thus, we obtain the expression for the electromotive force. The signal is generated by elementary current ring: 
     
       
         
           
             
               
                 
                   dEmf 
                   = 
                   
                     q 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     μ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       σ 
                       · 
                       
                         
                           ∂ 
                           
                             E 
                             φ 
                           
                         
                         
                           ∂ 
                           t 
                         
                       
                       · 
                       
                         
                           
                             
                               r 
                               2 
                             
                             ⁢ 
                             
                               ⅆ 
                               r 
                             
                             ⁢ 
                             
                               ⅆ 
                               
                                 z 
                                 _ 
                               
                             
                           
                           
                             
                               2 
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     r 
                                     2 
                                   
                                   + 
                                   
                                     ( 
                                     
                                       z 
                                       - 
                                       
                                         
                                           z 
                                           _ 
                                         
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 ] 
                               
                             
                             
                               3 
                               / 
                               2 
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     The contribution to the measured signal arising from the cylindrical area D (r 1 ≦r≦r 2 ;  z   1 ≦ z ≦ z   2 ) can be expressed as the following integral: 
     
       
         
           
             
               
                 
                   
                     Emf 
                     D 
                   
                   = 
                   
                     
                       
                         
                           M 
                           z 
                         
                         ⁢ 
                         q 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         μ 
                       
                       
                         32 
                         ⁢ 
                         π 
                         ⁢ 
                         
                           π 
                         
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                             
                         
                       
                       
                         ∂ 
                         t 
                       
                     
                     ⁢ 
                     
                       
                         { 
                         
                           
                             
                               ( 
                               
                                 
                                   μ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   σ 
                                 
                                 t 
                               
                               ) 
                             
                             
                               5 
                               / 
                               2 
                             
                           
                           · 
                           
                             
                               ∫ 
                               
                                 r 
                                 1 
                               
                               
                                 r 
                                 2 
                               
                             
                             ⁢ 
                             
                               
                                 ∫ 
                                 
                                   
                                     z 
                                     _ 
                                   
                                   1 
                                 
                                 
                                   
                                     z 
                                     _ 
                                   
                                   2 
                                 
                               
                               ⁢ 
                               
                                 
                                   
                                     r 
                                     3 
                                   
                                   ⁢ 
                                   
                                     ⅇ 
                                     
                                       
                                         - 
                                         
                                           
                                             μ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             σ 
                                           
                                           
                                             4 
                                             ⁢ 
                                             t 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             r 
                                             2 
                                           
                                           + 
                                           
                                             
                                               z 
                                               _ 
                                             
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     r 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ⅆ 
                                     
                                       z 
                                       _ 
                                     
                                   
                                 
                                 
                                   
                                     [ 
                                     
                                       
                                         r 
                                         2 
                                       
                                       + 
                                       
                                         ( 
                                         
                                           z 
                                           - 
                                           
                                             
                                               z 
                                               _ 
                                             
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                     ] 
                                   
                                   
                                     3 
                                     / 
                                     2 
                                   
                                 
                               
                             
                           
                         
                         } 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     The inversion of transient logging data is presented next. To investigate the possibility of using transient fields to study the near-borehole zone, we constructed an inversion algorithm based on the Born Approximation of the forward problem. The main advantage of this approach is that it establishes the linear dependence of a signal on disturbances in the conductivity. 
     Let us consider the electric field in a homogeneous environment: 
                     E   φ     =     ⅈ   ⁢           ⁢   ω   ⁢           ⁢     μ   ·         M   z     ⁢   r       4   ⁢       π   ⁢   R     3         ·     (     1   +   kR     )     ·       ⅇ     -   kR       .                 (   15   )               
If a circular disturbance of conductivity Δσ with a cross-section drd z  appears at the first approximation, it may be interpreted as an extraneous current:
 
Δ J=Δσ·E   φ ( r, z   )· drd z ,   (16)
 
which creates an additional (anomalous) field. The signal from this source at the axis can be defined by means of equation (17) based on the reciprocity law:
 
                     Δ   ⁢           ⁢     Emf   ⁡     (   z   )         =     ⅈ   ⁢           ⁢   ω   ⁢           ⁢     μ   ·       Δ   ⁢           ⁢     J   ⁡     (     r   ,     z   _       )       ⁢     qr   2         2   ⁢     R   3         ·     (     1   +   kR     )     ·       ⅇ     -   kR       .                 (   17   )               
Substituting the current equation (16) into equation (17), we obtain the following results for an anomalous electromotive force:
 
                     Δ   ⁢           ⁢     Emf   ⁡     (   z   )         =       -     ω   2       ⁢     μ   2     ⁢   Δ   ⁢           ⁢     σ   ·         M   z     ⁢     qr   3         8   ⁢       π   ⁢   R     3     ⁢       R   _     3         ·     (     1   +   kR     )     ·     ⅇ     -   kR       ·     (     1   +     k   ⁢     R   _         )     ·     ⅇ       -   k     ⁢     R   _           ⁢     ⅆ   r     ⁢     ⅆ     z   _                 (   18   )               
where R=√{square root over (r 2 +(z− z ) 2 )},  R =√{square root over (r 2 + z   2 )},  z − is a point in the excited environment containing the secondary current; and z is an observation point at the axis.
 
     Using Fourier transformation we proceed to the time domain and arrive at this representation of the anomalous field of a thin circular disturbance of conductivity: 
                     Δ   ⁢           ⁢     Emf   ⁡     (   t   )         =             M   z     ⁢     qr   3         8   ⁢   π   ⁢     π     ⁢     R   3     ⁢       R   _     3         ·         μ   2     ⁢   Δ   ⁢           ⁢   σ       t   2       ·     ⅇ     -     x   2         ·   x   ·     [       2   ⁢     x   4       -     5   ⁢     x   2       +         R   ⁢     R   _     ⁢   μ   ⁢           ⁢   σ       4   ⁢   t       ·     (       4   ⁢     x   4       -     20   ⁢     x   2       +   15     )         ]       ⁢     ⅆ   r     ⁢     ⅆ     z   _                 (   19   )               
where
 
             x   =           μ   ⁢           ⁢   σ       4   ⁢   t         ·       (     R   +     R   _       )     .             
This should then be integrated in respect to r and  z . Thus, the complete field from N circular disturbances in a linear (Born) approximation may be expressed as follows:
 
                     Emf   ⁡     (   t   )       =         Emf   0     ⁡     (   t   )       +       ∑     i   =   1     N     ⁢           ⁢           G   i     ⁡     (   t   )       ·   Δ     ⁢           ⁢     σ   i                   (   20   )               
where Emf 0 (t) is the normal signal from a homogeneous environment (e.g. equation (12)), and G i  is found from equation (19).
 
     If we substitute the measured full signal into the left part of equation (20) and consider the conductivity of the reference homogeneous environment known, equation (20) yields a linear set for defining disturbances. However, we may use the values of the apparent resistivity. At much later times, equation (12) enables us to define the apparent resistivity as follows: 
     
       
         
           
             
               
                 
                   
                     ρ 
                     τ 
                   
                   = 
                   
                     
                       μ 
                       
                         4 
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         t 
                       
                     
                     · 
                     
                       
                         
                           ( 
                           
                             
                               μ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 M 
                                 z 
                               
                               ⁢ 
                               q 
                             
                             
                               Emf 
                               · 
                               t 
                             
                           
                           ) 
                         
                         
                           2 
                           / 
                           3 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
       FIG. 10  presents one example of a method  100  for estimating a property of the earth formation  4  in the zone of investigation  14  where the earth formation  4  includes the zone of investigation  14  and another zone. The method  100  calls for (step  101 ) applying the compound pulse  21  to the earth formation  4 . The compound pulse  21  has a shape that induces a predominately greater amount of the current  9  in the zone of investigation  14  than in the another zone. The compound pulse  21  includes the first pulse  23  and the second pulse  24 , the first pulse  23  having an amplitude different from the amplitude of the second pulse  24 . Further, the method  100  calls for (step  102 ) receiving the response  22  from the current  9  induced in the formation  4  by the compound pulse  21 . Further, the method  100  includes (step  103 ) estimating the property from the response  22 . 
     In support of the teachings herein, various analysis components may be used, including a digital and/or an analog system. For example, the electronic unit  12  and the processing unit  13  may include the digital and/or analog system. The system may have components such as a processor, storage media, memory, input, output, communications link (wired, wireless, pulsed mud, optical or other), user interfaces, software programs, signal processors (digital or analog) and other such components (such as resistors, capacitors, inductors and others) to provide for operation and analyses of the apparatus and methods disclosed herein in any of several manners well-appreciated in the art. It is considered that these teachings may be, but need not be, implemented in conjunction with a set of computer executable instructions stored on a computer readable medium, including memory (ROMs, RAMs), optical (CD-ROMs), or magnetic (disks, hard drives), or any other type that when executed causes a computer to implement the method of the present invention. These instructions may provide for equipment operation, control, data collection and analysis and other functions deemed relevant by a system designer, owner, user or other such personnel, in addition to the functions described in this disclosure. 
     Further, various other components may be included and called upon for providing for aspects of the teachings herein. For example, a power supply (e.g., at least one of a generator, a remote supply and a battery), cooling component, heating component, magnet, electromagnet, sensor, electrode, transmitter, receiver, transceiver, antenna, controller, optical unit, electrical unit, electromechanical unit, connectors or supports may be included in support of the various aspects discussed herein or in support of other functions beyond this disclosure. 
     Elements of the embodiments have been introduced with either the articles “a” or “an.” The articles are intended to mean that there are one or more of the elements. The terms “including” and “having” and their derivatives are intended to be inclusive such that there may be additional elements other than the elements listed. The conjunction “or” when used with a list of at least two terms is intended to mean any term or combination of terms. The terms “first,” “second” and “third” are used to distinguish elements and do not denote a particular order. 
     It will be recognized that the various components or technologies may provide certain necessary or beneficial functionality or features. Accordingly, these functions and features as may be needed in support of the appended claims and variations thereof, are recognized as being inherently included as a part of the teachings herein and a part of the invention disclosed. 
     While the invention has been described with reference to exemplary embodiments, it will be understood that various changes may be made and equivalents may be substituted for elements thereof without departing from the scope of the invention. In addition, many modifications will be appreciated by those skilled in the art to adapt a particular instrument, situation or material to the teachings of the invention without departing from the essential scope thereof. Therefore, it is intended that the invention not be limited to the particular embodiment disclosed as the best mode contemplated for carrying out this invention, but that the invention will include all embodiments falling within the scope of the appended claims.