Abstract:
An electromagnetic logging tool is disclosed that includes a support; and at least one four-coil array disposed on the support, wherein the at least one four-coil array comprises: a transmitter, a bucking coil, a receiver, and a trim coil. A method for balancing an induction array is disclosed that includes applying an alternating current to a transmitter of the induction array that comprises the transmitter, a bucking coil and a receiver; measuring a mutual coupling between the transmitter and the receiver; and adding an extra bucking coil, if the mutual coupling exceeds a selected criterion.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS  
       [0001]     The present application claims priority of U.S. Provisional Patent Application Ser. No. 60/657,174 filed on Feb. 28, 2005. The Provisional Application is incorporated by reference in its entirety. This application is related to U.S. Application, titled “Selectable Tap Induction Coil”, filed concurrently with this document and assigned to the present assignee. 
     
    
     BACKGROUND OF INVENTION  
       [0002]     1. Field of the Invention  
         [0003]     The invention relates generally to electromagnetic (EM) well logging. In particular, embodiments of the present invention relate to methods and apparatus for improving EM well logging sensitivities.  
         [0004]     2. Background Art  
         [0005]     During the exploration and production of oil and gas, many well logging techniques are deployed to log data of the geological formations. The data contain information that can be used to locate subsurface hydrocarbon reservoirs and to determine types and quantities of subsurface hydrocarbons. In such logging processes, a tool may be lowered into a borehole traversing a subsurface formation, either after the well has been drilled or during the drilling process. A typical logging tool includes a “sonde”, that emits, for example, acoustic or EM waves to interact with the surrounding formation. The signals produced from such interactions are then detected and measured by one or more sensors on the instrument. By processing the detected signals, a profile or log of the formation properties can be obtained.  
         [0006]     Logging techniques known in the art include “wireline” logging, logging-while-drilling (LWD), measurement-while-drilling (MWD), and logging-while-tripping (LWT). Wireline logging involves lowering an instrument into an already-drilled borehole at the end of an electrical cable to obtain measurements as the instrument is moved along the borehole. LWD and MWD involve disposing an instrument in a drilling assembly for use while a borehole is being drilled through earth formations. LWT involves disposing sources or sensors within the drill string to obtain measurements while the string is being withdrawn from the borehole.  
         [0007]      FIG. 1  shows a general illustration of a typical drilling rig with a drill string carrying a downhole logging tool in a borehole. The rotary drilling rig shown in  FIG. 1  comprises a mast  1  rising above the ground  2  and is fitted with a lifting gear  3 . The lifting gear  3  has a crown block  7  fixed to the top of the mast  1 , a vertically traveling block  8  with a hook  9  attached, a cable  10  passing around blocks  7  and  8  to form on one side a dead line  10   a  anchored to a fixed point  11  and on the other side an active line  10   b  that winds round the drum of a winch  12 . A drill string  4  formed of several segments of hollow drilling pipes connected end-to-end is suspended from the hook  9  by means of a swivel  13 , which is linked by a hose  14  to a mud pump  15 . The mud pump  15  pumps drilling mud into the well  6 , via the hollow pipes of the drill string  4  and out of the bit  5  to float the rock cuttings out of the well  6 . The drilling mud may be drawn from a mud pit  16 , which may also be fed with surplus mud from the well  6 . The drill string  4  may be elevated by turning the lifting gear  3  with the winch  12 . When raising or lowering drill pipes, the drill string  4  needs to be temporarily unhooked from the lifting gear  3 , during which the weight of the string  4  is supported by wedges  17 . The wedges  17  are anchored in a conical recess  18  in a rotating table  19  that is mounted on a platform  20 . The lower portion of the drill string  4  may include one or more instruments  30  for investigating downhole drilling conditions or for investigating the properties of the geological formations. In the case of sonic logging, the instrument  30  may include at least one transmitter and a plurality of receivers.  
         [0008]     Variations in the height h of the traveling block  8  during the raising cycle of the drill string operations are measured by means of a sensor  23  which may be an angle-of-rotation sensor coupled to the faster pulley of the crown block  7 . The weight applied to the hook  9  may also be measured by means of a strain gauge  24  inserted into the dead line  10   a  of the cable  10  to measure its tension. Sensors  23  and  24  are connected by lines  25  and  26  to a processing unit  27  having a clock incorporated therein. A recorder  28  is connected to the processing unit  27 , which is preferably a computer. In addition, the downhole tool  30  may include a processing unit  30   a.  The downhole processing unit  30   a  and/or the surface processing unit  27 , which may include a memory, may be used to perform the data analysis and determination of formation properties.  
         [0009]     For downhole tools, EM logging tools are among the widely used. EM logging tools are implemented with antennas that are operable as transmitters and/or receivers. The antennas are typically solenoid coils. Referring to  FIG. 2 , a coil  211  is shown comprising of insulated conducting wires having one or more turns wound around a support  214 . During operation, the coil  211  may function as a transmitter antenna when it is energized with an alternating current or an oscillating electrical signal  212 . The transmitter antenna emits EM waves through the borehole mud and into the surrounding earth formation. The coil  211  may also function as a receiver antenna that collects EM signals carrying information about the interactions between the EM waves and the mud/formation.  
         [0010]     The coil  211  carrying a varying current  212  will produce a magnetic dipole having a magnetic moment. The strength of the magnetic moment is proportional to the electric current in the wire, the number of turns of the wire, and the area encompassed by the coil. The direction and strength of the magnetic moment can be represented by a vector  213  in a direction parallel to the longitudinal axis of the coil. In conventional induction logging instruments, the transmitter and receiver antennas are mounted with their axes aligned with the longitudinal axis of the instrument. Thus, these tools are implemented with antennas having longitudinal magnetic dipoles (LMD). When an LMD antenna is placed in a borehole and energized to transmit EM energy, the induced electric currents flow around the antenna in the borehole and in the surrounding earth formations, and no net current flows up or down the borehole.  
         [0011]     Recent EM well logging tools have tilted or transverse coils, i.e., the coil&#39;s axis is not parallel with the longitudinal axis of the support. Consequently, the antenna has a transverse or tilted magnetic dipole (TMD). The TMD configuration permits a tool to have a three-dimensional evaluation capability, such as information about resistivity anisotropy or locations and orientations of dips and faults. In addition, directional sensitivity of the data can be used for directional drilling. Logging instruments equipped with TMD-antennas have been described in U.S. Pat. Nos. 6,147,496, 4,319,191, 5,757,191, and 5,508,616. Under certain conditions, a TMD-antenna may cause a net current to flow up or down the borehole. Some TMD-antennas are configured with multiple coils. For example, a particular TMD-antenna design includes a set of three coils, and such an antenna is known as a triaxial antenna.  
         [0012]     In wireline applications, the antennas are typically enclosed in a housing made of tough non-conductive materials such as a laminated fiberglass material. In LWD applications, the antennas are generally encased into a metallic support so that it can withstand the hostile environment and conditions encountered during drilling. Alternatively, logging instruments may be made of composite materials, thus, providing a non-conductive structure for mounting the antennas. U.S. Pat. Nos. 6,084,052, 6,300,762, 5,988,300, 5,944,124, and UK Patent GB 2337546 disclose examples of composite-material-based instruments and tubulars for oilfield applications.  
         [0013]     Induction logging is a well-known form of EM logging. In this type of logging, induction tools are used to produce a conductivity or resistivity profile of earth formations surrounding a borehole. U.S. Pat. Nos. 3,340,464, 3,147,429, 3,179,879, 3,056,917, and 4,472,684 disclose typical well logging tools based on induction logging.  
         [0014]     A conventional induction logging tool or “sonde” may include a transmitter antenna and a receiver antenna. Note that the designation of a transmitter and a receiver is for clarity of illustration. One skilled in the art would appreciate that a transmitter may be used as a receiver and a receiver may also be used as a transmitter depending on the application. Each antenna may include one or more coils, and may be mounted on the same support member or on different support members, i.e., the transmitter antenna and the receiver antenna may be on different tool sections. The antennas are axially spaced from each other in the longitudinal direction of the tool.  
         [0015]     In use, the transmitter antenna is energized with an alternating current. This generates an EM field that induces eddy currents in the earth formation surrounding the borehole. The intensity of the eddy currents is proportional to the conductivity of the formation. The EM field generated by the eddy currents, in turn, induces an electromotive force in one or more receiving coils. Phase-locked detection, amplification, and digitization of this electromotive force signal determines the amplitude and the phase of the voltage on the receiver coil. By recording and processing the receiver voltages, an earth formation conductivity profile can be obtained. U.S. Pat. No. 5,157,605 discloses an induction array well logging tool used to collect the voltage data.  
         [0016]     In principle, a conductivity profile may be obtained by simply measuring the voltages on the receiver. In practice, the receiver voltages are not only affected by “true” signals traveling through the formation, but are also affected by a direct coupling between the transmitter and the receiver. It is well known that the sensitivity of measurements obtained from induction-type loggings are adversely affected by the direct transmitter-to-receiver couplings (mutual couplings).  
         [0017]     Mathematically, the amplitude and phase of the received signal voltage may be expressed as a complex number (i.e., a phasor voltage). Accordingly, the apparent conductivity σ a  (as measured by a receiver induction array) is expressed in terms of its real and imaginary parts, 
 
σ a =σ R +σ x . 
 
         [0018]     The real part σ R  represents the true signal from the earth formation, while the imaginary part σ x  includes direct coupling that may be several orders of magnitude larger than the value of σ R , when the array is unbalanced. This can be seen from a well-known formula describing the conductivity measured by a two-coil (one transmitter and one receiver) array, when the transmitter is simplified as a point dipole,  
                 σ   α     =         σ   R     +     σ   X       =       -       2   ⁢   i       ω   ⁢           ⁢   μ         ⁢         (     1   -   ikL     )     ⁢     ⅇ     ik   ⁢           ⁢   L           L   2             ,           (   1   )             
 
 where ω is the frequency, μ is the magnetic permeability of a (homogeneous) medium, k 2 =iωμσ, σ is the conductivity of the medium, and L is the transmitter-receiver spacing. Defining a skin depth as 
 
δ=√{square root over (2/(ωμσ)))}(so that  k =(1 +i )/δ) 
 
 and expanding σ a  using the powers of L/δ, one obtains:  
                 σ   R     +     ⅈ   ⁢           ⁢     σ   X         =     σ   -       2   ⁢   i       ω   ⁢           ⁢   μ   ⁢           ⁢     L   2         -         2   ⁢   L   ⁢           ⁢   σ       3   ⁢           ⁢   δ       ⁢     (     1   -   i     )       +     O   ⁡     (       L   2     ⁢     /     ⁢     δ   2       )                 (   2   )             
 
 The first term on the right-hand side of Eq. ( 2 ) is the formation conductivity σ of interest. The second term, −2i/(ωμL 2 ), contributes to π x  only. It does not depend on σ and corresponds to the direct mutual transmitter-receiver coupling that exists in the air. An out-of-balance induction array can have a very large value of σ x , especially when L is small. Therefore, in order for an induction tool to achieve a high sensitivity, the induction array must be balanced to reduce the value of σ x . 
 
         [0019]     As illustrated in  FIG.3 , an induction-type logging instrument  300  typically includes a “bucking” coil  311  in order to eliminate or reduce direct coupling between the transmitter  312  and the main receiver  313 . The purpose of having two receiver coils, the main coil  313  and the bucking coil  311 , in a balanced configuration is to cancel the transmitter-main-coil coupling using the transmitter-bucking-coil coupling. The bucking coil  311  may be placed between the transmitter  312  and the main coil  313 . Practical induction arrays have always been mutually-balanced using a bucking coil. This is necessitated by the fact that the direct coupling between a transmitter and a receiver is usually several orders of magnitude stronger than the true signals, the latter being strongly attenuated when traveling through the earth formation. The instrument&#39;s  300  longitudinal axis is represented as a dashed line in  FIG. 3 .  
         [0020]     The minimum configuration for a mutually-balanced array is a three-coil array as illustrated in  FIG. 3 , including a transmitter  312  (T), a main receiver  313  (R 1 ) and a bucking coil  311  (R 2 ). In order to balance the array, the locations of the main receiver and bucking coils (Z main  and Z buck ) and the numbers of turns in these coils (N main  and N buck ) are chosen such that the sum of their responses is close to zero in the air. That is, the voltages on the two receiver coils satisfy the relation: V R1 +V R2 =0, in the air. Consequently, the responses from a logging operation will be the sum of the T−R 1  coupling responses and the T−R 2  coupling responses.  
         [0021]     In theory, the required positions of the receiver coils can be calculated with high precision, even when the transmitter is a finite-size solenoid. In practice, all geometrical parameters, such as the positions (z buck  and z main ) and the radii (r trans , r buck , and r main ) of the coils, have finite ranges of variations. The direct mutual couplings T−R 1  and T−R 2  are very sensitive to even minute changes in some of the geometrical parameters, and, therefore, large variations in measured signals may result from small errors or variations in, for example, the radii of the coils. Therefore, when an EM tool is manufactured, the configuration of the coils may need to be further adjusted from the calculated configuration. In a practical configuration, both positions of the bucking coil and the main receiver coil (z buck  and z main ) relative to the position of the transmitter (z=0) would need to be carefully adjusted with a high degree of precision in order to minimize direct mutual couplings. Such balancing or adjusting could be very difficult and demanding.  
         [0022]     One prior art method of fine-tuning the antenna is to use moveable coils so that the locations of the coils (e.g., the main receiver or bucking coils) may be altered to minimize the direct coupling. For example, if the direct coupling (reflected as residual σ x ) for a particular array is substantial, it can be minimized (or reduced to zero) by altering the location of an antenna, such as the bucking coil z buck . However, as a practical matter, it is preferred that the tool or antenna has no moving parts.  
         [0023]     An alternative method for balancing the array is to add a conductive loop near one of the coils (e.g., the receiver coil) to permit fine adjustment. Another approach is to adjust with the number of turns in the bucking coil. However, this approach is often impractical because removing or adding a single turn in a coil may produce large changes in σ x . This is especially true when the distance between the bucking coil and the transmitter coil is short. Therefore, there still exists a need for new approaches to balancing induction arrays.  
       SUMMARY OF THE INVENTION  
       [0024]     One aspect of the invention relates to electromagnetic logging tools. An electromagnetic logging tool in accordance with one embodiment of the invention includes a support; and at least one four-coil array disposed on the support, wherein the at least one four-coil array comprises: a transmitter, a bucking coil, a receiver, and a trim coil.  
         [0025]     Another aspect of the invention relates to methods for balancing an induction array. A method for balancing an induction array in accordance with one embodiment of the invention includes applying an alternating current to a transmitter of the induction array that comprises the transmitter, a bucking coil and a receiver; measuring a mutual coupling between the transmitter and the receiver; and adding an extra bucking coil, if the mutual coupling exceeds a selected criterion  
         [0026]     Other aspects and advantages of the invention will become apparent from the following description and the attached claims. 
     
    
     BRIEF SUMMARY OF THE DRAWINGS  
       [0027]      FIG. 1  illustrates a conventional drilling rig and a drill string with a downhole logging tool in a borehole.  
         [0028]      FIG. 2  shows schematic illustrating a structure of a conventional magnetic coil.  
         [0029]      FIG. 3  illustrates a prior art three-coil antenna.  
         [0030]      FIG. 4  shows a four-coil antenna array in accordance with one embodiment of the present invention.  
         [0031]      FIG. 5A-5C  illustrate sensitivities of σ x  and σ R  to one extra turn of bucking coil in different positions of an SA (shallow array), an MA (medium array), and a DA (deep array), respectively, in accordance with one embodiment of the invention.  
         [0032]      FIGS. 6A-6C  illustrate raw data of the 3-coil and the 4-coil responses for the SA, the MA, and the DA in accordance with one embodiment of the invention.  
         [0033]      FIGS. 6D-6F  illustrate de-convolved data of the 3-coil and the 4-coil responses for the SA, the MA, and the DA in accordance with one embodiment of the invention.  
         [0034]      FIG. 7  illustrates positions of the extra coil for each of the induction arrays in accordance with one embodiment of the invention.  
         [0035]      FIG. 8  shows a method for balancing an induction array in accordance with one embodiment of the present invention. 
     
    
     DETAILED DESCRIPTION  
       [0036]     Embodiments of the invention relate to a new approach to balancing induction arrays such that the mutual couplings between the transmitter and receivers can be effectively removed. As noted above, mutual couplings (as reflected in σ x ) between the transmitter and receiver can be several orders of magnitude larger than the signals that return from the formation.  
         [0037]     Embodiments of the invention use extra bucking coils (or trim coils) to provide further balancing such that the receivers in the induction arrays will have significantly lower σ x  signals. Such tool will be able to provide more accurate and/or more sensitive measurements under a wide range of conditions. The extra bucking coils should be designed to be more controllable to provide fine balancing of the arrays. Note that for clarity of description, the following will use transmitters, bucking coils, receivers (or main receivers), and extra bucking coils (or trim coils) to describe four-coil arrays in accordance with embodiments of the invention. One of ordinary skill in the art would appreciate that these different terms are intended to describe their different functions and these antennas/coils may have same or similar physical structures. Further, extra bucking coils and trim coils are considered synonymous and will be used interchangeably in the following description.  
         [0038]     When a transmitter and a receiver are treated as point dipoles, the mutual couplings between them vary with the spacing between them according to a function of 1/L 3  (an extra power of 1/L coming from the 1/L-dependence of the tool factor K). Therefore, the balancing condition of an induction array is met when,  
             M   main       L   main   3       +       M   buck       L   buck   3         =   0       
 
 where M main  and M buck  are magnetic moments of the main and bucking coils. If all coil turns have the same geometry, then M main  and M buck  equal M 0 N main  and M 0 N buck , respectively, where M 0  is the magnetic moment of a single turn. Therefore, the balance condition is met when  
                   N   main       L   main   3       +       N   buck       L   buck   3         =   0.           (   3   )             
 
 To satisfy this condition, the two terms on the left side of Eq. (3) are of the opposite signs. This condition can be achieved by winding the wires of the bucking and main receiver coils in the opposite directions. The 1/L 3 -dependence of the mutual couplings suggest that shorter arrays will be significantly more sensitive to spacing (L) variations. In other words, minor spacing changes will have a much larger impact on the accuracy of the shorter arrays. Likewise, any external factors, such as temperature and pressure, that may affect the accuracy of the array would also have more impact on the accuracy of the shorter arrays. As a result, shorter arrays typically require larger error specifications. 
 
         [0039]     For example, Table 1 below shows the impact of small changes of various parameters on a three array tool, which includes a Shallow Array (SA), a Medium Array (MA), or a Deep Array (DA). Results in Table 1 show changes in σ R  and σ x  (in units of mS/m) for these arrays with respect to changes in the locations and radii of the transmitter, the bucking receiver and the main receiver, as well as changes in coil diameters (or radii), mandrel diameters (or radii), and numbers of turns of the bucking or main receiver coil.  
                                                                                   TABLE 1                           Sensitivity of σ R  and σ X  (in mS/m) to changes of geometrical       parameters and numbers of turns.                    σ R     σ X     σ R     σ X     σ R     σ X         Parameter   Change   (SA)   (SA)   (MA)   (MA)   (DA)   (DA)                    Z main     0.001″   −0.08   36.68   −0.005   5.48   0.0003   0.65       Z buck     0.001″   0.20   −59.76   0.018   −9.86   −0.0004   −0.91       γ max     0.001″   0.07   −1.34   0.005   −0.17   −0.007   −0.05       γ min     0.001″   −41.91   −895.44   −11.71   −244.06   −3.35   −66.08       γ buck     0.001″   41.09   888.35   11.51   242.59   3.26   65.48       all coil   0.001″   0.14   −2.68   0.009   −0.35   −0.015   −0.097       radii       γ maximal     0.001″   −0.006   −16.37   −0.006   −3.33   0.012   −0.08       N buck     1 turn   −7.6   2635   −1.62   1128   0.035   57.3       N main     1 turn   1.9   −669   0.29   −199   −0.013   −20.4                  
 
         [0040]     As noted above, mutual couplings vary with 1/L 3  (L is the spacing between the transmitter and the receiver), which includes the K-factor of the tool that varies as 1/L. Therefore, it is expected that the shallow array (SA) is the most sensitive to changes in positions (z main  and z buck ), while the deep array (DA) is the least sensitive. Results in Table 1 also show that minor changes in the radii of the main receiver or bucking coils can result in huge changes in σ x . These dramatic changes due to radius variations may arise from radius mismatch between the main and bucking coils. In addition, there may also be effects related to slight eccentricity of the coils, etc. Therefore, it is desirable to have some way to correct for these effects when manufacturing the instrument.  
         [0041]     Table 1 also shows that a single turn change in the number of turns in the bucking or main receivers have a significant impact on the sensitivity of the σ x  signals. The effect is more significant with the shallow array (SA) than with the deep array (DA). These results indicate that it would be difficult (if not impossible) to balance an array by changing the number of turns of a bucking coil, especially the bucking coil of an SA.  
         [0042]     In view of the above, embodiments of the invention use an extra bucking coil to provide more controllable balancing of an induction array. The extra bucking coils (or trim coils) are preferably disposed farther away (compared to the main receiver or bucking coil) from the transmitter so that the additional bucking coil would be more controllable than the conventional bucking coil.  
         [0043]     The farther the trim coil is from the transmitter, the less is its sensitivity to various factors that influence mutual couplings, such as the number of turns of coils, locations, and radii. Thus, a trim coil with a longer spacing from the transmitter can provide better controllability. An ideal solution is to find a location where the trim coil would contribute a desirable amount of effect to the mutual couplings between the transmitter and the receiver, so that σ x  of the main receiver can be brought reasonably close to zero.  
         [0044]     In accordance with embodiments of the invention, after the induction antenna coils (the transmitter, the receiver main coil and the first bucking coil) are wound, the mutual couplings are measured. Theoretically, for integer values of N main  and N buck , it is possible to find L main  and L buck  to satisfy the condition in Equation (3) with any precision. However, in practice, the sum of the two terms on the left-hand side of Equation (3) will be non-zero due to finite tolerance. Let&#39;s assume the experimentally determined mutual coupling is Δ exp :  
           [         N   math       L   main   3       +       N   buck       L   buck   3         ]     exp     =       Δ   exp     .         
 
 The residual mutual couplings may be reduced close to zero by adding trim coils in such a way that 
 
 N   trim   /L   trim   3 =Δ exp  
 
 (or as close to −Δ exp  as possible). Therefore, we get:  
           [         N   main       L   math   3       +       N   buck       L   buck   3       +       N   trim       L   trim   3         ]     exp     =   0.       
 
         [0045]     The above analysis can be extended to more than one trim coils, which could be located at increasing spacings to provide fine tuning.  
         [0046]     By adding a trim coil, the system becomes a four-coil array. In preferred embodiments, the trim coil is further away from the transmitter than is the main coil. A configuration of a four-coil array, including a transmitter  412 , a bucking coil  411 , a main receiver coil  413 , and an extra bucking coil (trim coil)  414 , is illustrated in  FIG. 4 .  
         [0047]     In accordance with one embodiment of the invention, when making such an antenna array, the main and the bucking coils are first wound, then mutual coupling between the transmitter and the receiver (as reflected in σ x ) in the air is measured. Based on this measurement, the location and the number of turns of the extra bucking coil can be determined such that it can bring the value of σ x  as close to zero as possible (or within a tolerance range). After the number of turns is determined, the extra bucking coil is wound and the tool is then over-wrapped to protect the coils.  
         [0048]     The sensitivity of one turn of a trim coil at different spacings from the main receiver is shown in  FIGS. 5A-5C , for SA, MA and DA. Both the σ R  and σ x  changes are shown. For example, as shown in  FIG. 5A , for SA, one turn of the trim coil at 6″ spacing from the main receiver introduces about 200 mS/m change in σ x . If the array without this extra bucking coil has a value of, for example, σ x =±1250 mS/m in the air, then one would need ±6 turns (the sign depends on the direction of windings) to reduce it to 50 mS/m. In this way, the resulting σ x  can be brought to within ±100 mS/m, which is half of the one turn contribution. Similar considerations are also applicable to the longer arrays (MA and DA). For example, for MA, one turn of a trim coil at 11″ spacing from the main receiver can alter the σ x  by about 60 mS/m, and for DA, one turn of the trim coil at 4″ spacing from the main receiver would contribute about 12 mS/m to the σ x .  
         [0049]     The above theoretical calculation has been verified by several tools, which have been used to experimentally measure array characteristics and performances. Two of these sondes (designated as A and B), each containing a shallow array (SA), have been used to test the extra bucking coil in accordance embodiments of the present invention. Both sondes have high direct-coupling errors that would benefit from using the extra coil. The sonde errors are measured and are determined if an extra bucking coil is needed. In the case of these sondes, an extra bucking coil was added a few inches from the main receiver coil and the appropriate number of turns are wound. Measured sonde errors, before and after adding the trim coil, are tabulated in Table 2.  
                                                   TABLE 2                           Sonde errors before and after addition of an extra bucking coil.                Initial Sonde Error   Sonde Error With Trim           (mS/m)   Coil (mS/m)            Sonde   σ R     σ X     σ R     σ X                 A   −161   1308   −149   −184       B   −106   2213   −116    274                  
 
 It is apparent from Table 2 that the extra bucking coil is very effective in balancing the arrays. The actual amount σ x  contributed by the extra bucking coil is very close to the theoretical modeling results. 
 
         [0050]     The above description shows that it is possible to reduce the undesired σ x  of a main receiver by an extra bucking coil (trim coil). However, for this approach to be useful, the extra bucking coil should not degrade (or at least only minimally affect) the performance (such as vertical resolution) of the original array. Understanding how an extra bucking coil may affect the response of a main receiver would be helpful when designing an antenna array.  
         [0051]     The 2D axial Born response (T. Habashy and B. Anderson, “ Reconciling Differences in Depth of Investigation between  2- Mhz Phase Shift and Attenuation Resistivity Measurements ,” SPWLA 32 nd  Annual Logging Symposium, Midland, Tex., 1991) for a two-coil sonde to a point located at p, z in a cylindrical coordinate system is given by:  
                   g   cc     ⁡     (     ρ   ,   z   ,   σ     )       =       L   2     ⁢       ρ   3         r   T   3     ⁢     r   R   3         ⁢     (     1   -     ⅈ   ⁢           ⁢     kr   T         )     ⁢     (     1   -     ⅈ   ⁢           ⁢   k   ⁢           ⁢     r   R         )     ⁢     ⅇ     ik   ⁡     (       r   r     +     r   n       )             ,           (   4   )             
 
         [0052]     where r T  and r R  are, respectively, the distances from the transmitter and the receiver to the spatial point where the function is defined, L is the spacing between the two coils, k 2 =iωμσ, ω is the frequency of the transmitter current, μ is the magnetic permeability, and σ is the formation conductivity. The conductivity measured at a depth z can be expressed (in the low-contrast limit) by the convolution operation:  
                 σ   α     ⁡     (   z   )       =       ∫   0   ∞     ⁢       ⅆ   ρ     ⁢       ∫     -   ∞     ∞     ⁢       ⅆ     z   ′       ⁢     g   ⁡     (     ρ   ,     z   -     z   ′       ,   σ     )       ⁢       σ   ⁡     (     ρ   ,     z   ′       )       .                     (   5   )             
 
         [0053]     If the function above is integrated over the radius p, the result is called the vertical response function, given by the expression:  
                   g   ,     ⁡     (     z   ,   σ     )       =       ∫   0   ∞     ⁢       ⅆ   ρ     ⁢           ⁢     g   ⁡     (     ρ   ,   z   ,   σ     )             ,           (   6   )             
 
 where g(p,z,σ) is given by Eq. (4). The vertical response function provides an indication of how the tool will perform in resolving layers with different conductivities in the formations. 
 
         [0054]     For a multi-coil array with a single transmitter, one can sum and weight the individual coil pairs so that  
                 g   M     =         ∑   l     ⁢     (         TR   l       L   l       ⁢     g   l       )           ∑   l     ⁢     (       TR   l       L   l       )           ,           (   7   )             
 
         [0055]     where g i  is either the 2D or the integrated response function (Eq. 4 or Eq. 6) for the i-th receiver, and g M  is the multi-coil function.  
         [0056]     Vertical response functions for a conventional 3-coil array and a 4-coil array in accordance with one embodiment of the invention are compared in  FIGS. 6A-6F .  FIGS. 6A, 6B , and  6 C show the raw data of the 3-coil responses and the 4-coil responses for the shallow (SA 3  and SA 4 ), medium (MA 3  and MA 4 ), and deep (DA 3  and DA 4 ) arrays, respectively. In this example, the trim-to-main ratios of the numbers of turns, N trim /N main , were take as 0.25 for the SA (this represents the worst-case scenario), 0.05 for the MA, and 0.02 for the DA. It is clear from  FIGS. 6A-6C  that the differences in vertical response for the 3-coil and the 4-coil arrays are very small, suggesting that the extra bucking coil does not adversely impact the vertical resolution of the array.  
         [0057]      FIGS. 6D-6F  show the deconvolved responses for the same arrays (the 4-ft resolution response). Again, the differences between the 3-coil and 4-coil responses are very small, suggesting that the 4-coil design in accordance with embodiments of the invention would produce acceptable measurements, while providing better controllability in reducing σ x , as compared with the 3-coil configuration.  
         [0058]     In addition to preserving vertical resolution of the tools, the extra bucking coil should have no or little impact on the accuracy of the tool readings and should have similar tolerance for wellbore irregularities. This has been found to be the case. Mutual coupling balancing using an extra bucking coil, in accordance with embodiments of the invention, has been found to produce acceptable results for a reasonable range of the number of turns of coils in the extra bucking coil with respect to vertical resolution, log accuracy, shoulder bed response, and well irregularities.  
         [0059]     As noted above, it is preferred that the extra bucking coils be located with a larger spacing from the transmitter than the main receiver is. A typical induction tools has multiple arrays. Therefore, the locations for designing the extra bucking coils are not without limitation. Positioning an extra bucking coil in each array becomes a matter of finding locations along the tool axis that would not interfere with other components such as coils, pressure bulkheads, sensor electrodes, and yet can meet the requirements discussed above.  
         [0060]     The induction array and resistivity sensor designs of existing tools (such as the Array Induction Tool provided under the trade name of AIT® and Dual Induction Tool sold under the trade name of DIT® by Schlumberger) permit several locations for the placement of the extra bucking coils. Examples of where an extra bucking coil may be located in a 3-array tool are shown in Table 3 and illustrated graphically in  FIG. 7 . In Table 3, the position ranges are given from the corresponding main receiver coils, and the sensitivity is measured by the amount of σ x  contributed by one turn of the extra bucking coil.  
                                     TABLE 3                           Locations of the extra coil for each array; the position       is from center of the corresponding main coil.                    Position   Coil   Sensitivity           Array   Range (in)   Position (in)   (mS/m/turn)                       SA   3.6-5.1   4.8   250           (shallow)           MA   7.9-9.4   8.5    75           (medium)           DA (deep)   4.9-8.1   6.2    15                      
 
         [0061]      FIG. 7  shows the positions of the extra bucking coils in the SA, the MA, and the DA in accordance an embodiment of the present invention. For the SA, the bucking coil  111  is located between the transmitter  110  and the main receiver coil  112 . The extra bucking coil  113  is located further away from the transmitter  110  than is the main receiver coil  112 . For the MA, the bucking coil  114  is optionally located near the main receiver coil  112  of the SA, and is located between the transmitter  110  and the main receiver coil  115  for the MA. The extra bucking coil  116  for the MA is further away than the main receiver coil  115  for the MA. For the DA, the main receiver coil  118  is located between the bucking coil  117  and the extra bucking coil  119 .  
         [0062]      FIG. 8  shows a method of balancing an induction array in accordance with one embodiment of the invention. As shown, a transmitter in an induction array is energized with an alternating current (step  121 ) and a mutual coupling between the transmitter and the receiver is measured (step  122 ). Based on the measured mutual coupling, one can determine a good location and a number of turns for an extra bucking coil (a trim coil) that can be used to minimize the mutual coupling (step  123 ), as reflected in the imaginary signal portion of the phasor voltage signal detected by the receiver. Then, an extra bucking coil is added to the array (step  124 ) to further balance the array. If necessary, the mutual coupling can be re-determined and any further adjustment of the extra bucking coil may be performed (step  125 ). In accordance with embodiments of the invention, it is desirable that the mutual coupling (as reflected in σ x ) in an array is as small as possible or below a tolerance (or criterion). The selected criterion may be based on the magnitudes of the imaginary signals (σ x ) as compared with the real signals in phasor voltage signals detected by a receiver, taking into account the array tolerance (Δσ R ), which are different for different arrays. If the wrong-phase rejection factor is W (in the contemporary electronics, it can be as high as 100 or 200), the criterion would read  
         [0063]     The above description illustrates embodiments of the invention using an extra bucking coil for balancing an induction array. One of ordinary skill in the art would appreciate that embodiments of the invention may be used with a wide range of tools, including wireline tools, LWD, MWD, and LWT tools. In addition, such tools may also include one or more conventional three-coil arrays and/or one or more electrodes, such as those used in conventional conductivity/resistivity tools.  
         [0064]     Advantages of the present invention include one or more of the following. Extra bucking coils can provide better controlled balancing of the arrays. Induction tools of the invention are better balanced to have much smaller σ x , signals, which will be more tolerant of variations in environmental factors that may cause mismatches of the bucking coils. Induction tools of the invention have similar performance characteristics in terms of vertical resolutions, accuracy of resistivity measurements, and responses various features in the boreholes (shoulder bed effects, cave effects, etc.).