Patent Publication Number: US-2016245072-A1

Title: Magnetic Gradient and Curvature Based Ranging Method

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims priority to and the benefit of U.S. Provisional Patent Application No. 61/894,460, filed 24 Oct. 2013, which is incorporated by reference herein. 
    
    
     FIELD OF THE INVENTION 
     Disclosed embodiments relate generally to drilling and surveying subterranean boreholes such as for use in oil and natural gas exploration and more particularly to methods for determining a distance between a twin well and a magnetized target well using first spatial derivatives and second spatial derivatives of a measured magnetic field. 
     BACKGROUND INFORMATION 
     Magnetic ranging measurements may be used to obtain a distance and a direction to an adjacent well. For example, commonly assigned U.S. Pat. No. 7,656,161 discloses a technique in which a predetermined magnetic pattern is deliberately imparted to a plurality of casing tubulars. These tubulars, thus magnetized, are coupled together and lowered into the adjacent well (the target well) to form a magnetized section of casing string typically including a plurality of longitudinally spaced pairs of opposing magnetic poles. Measurements of the magnetic field may then be utilized to survey and guide drilling of a drilling well (e.g. a twin well) relative to the target well. The distance between the twin and target wells may be determined from various magnetic field measurements made in the twin well (as further disclosed in commonly assigned U.S. Pat. No. 7,617,049). These well twinning techniques may be advantageously utilized, for example, in steam assisted gravity drainage (SAGD) applications in which horizontal twin wells are drilled to recover heavy oil from tar sands. 
     While the above described methodology has been successfully utilized in well twinning applications, there is room for yet further improvement. For example, it can be difficult to accurately remove the earth&#39;s magnetic field from the measured magnetic field since the attitude of the drilling well is not generally known with precision. Moreover, since the distance between the two wells is obtained from the measured magnetic field strength (intensity), any changes in the strength of the casing magnetization may cause a corresponding error in the obtained distance (e.g., a decay in the casing magnetization may cause the distance to be underestimated). Therefore there is a need for improved ranging methodologies. 
     SUMMARY 
     Methods for determining a distance from a drilling well to a magnetized target well are disclosed. The methods include acquiring magnetic field measurements from the drilling well. A drill string is deployed in the drilling well and includes at least one magnetic field sensor in sensory range of magnetic flux emanating from the magnetized target well. The acquired magnetic field measurements are made at a plurality of spaced apart locations, e.g., at a plurality of spaced apart axial and/or radial locations in the drilling well. The acquired magnetic field measurements are processed to obtain a ratio including at least one of the following: (i) a ratio of a magnetic field intensity to a first spatial derivative of a magnetic field, (ii) a ratio of a magnetic field intensity to a second spatial derivative of a magnetic field, and (iii) a ratio of a first spatial derivative of a magnetic field to a second spatial derivative of the magnetic field. The ratio (or ratios) are then processed to obtain the distance from the drilling well to the magnetized target well. 
     The disclosed embodiments may provide various technical advantages. For example, the disclosed methods may improve the accuracy of the distances determined via magnetic ranging by reducing the dependence of the magnetic ranging measurements on the strength of the target magnetization. Moreover, certain of the disclosed embodiments may obviate the need to remove the earth&#39;s magnetic field from the measured magnetic field. 
     This summary is provided to introduce a selection of concepts that are further described below in the detailed description. This summary is not intended to identify key or essential features of the claimed subject matter, nor is it intended to be used as an aid in limiting the scope of the claimed subject matter. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       For a more complete understanding of the disclosed subject matter, and advantages thereof, reference is now made to the following descriptions taken in conjunction with the accompanying drawings, in which: 
         FIG. 1  depicts a prior art arrangement for a SAGD well twinning operation. 
         FIG. 2  depicts a prior art magnetization of a wellbore tubular. 
         FIG. 3  depicts a flow chart of one example of a disclosed method embodiment for determining a distance between a drilling well and a magnetized target well. 
         FIG. 4  depicts a plot of the magnetic field about a magnetized casing string. 
         FIGS. 5A and 5B  depict plots of the axial and radial components (B z  and B r ) of the magnetic field as a function of normalized axial position along the target well at various distances from the target well. 
         FIGS. 6A, 6B, and 6C  depict plots of the three independent first spatial derivatives of the magnetic field as a function of normalized axial position along the target well at various distances from the target well. 
         FIGS. 7A, 7B, 7C, and 7D  depict plots of the four independent second spatial derivatives of the magnetic field as a function of normalized axial position along the target well at various distances from the target well. 
         FIGS. 8A and 8B  depict plots of various ratios of a magnetic field intensity to a first spatial derivative of the magnetic field as a function of the actual distance to the magnetized target. 
         FIGS. 9A and 9B  depict plots of various ratios of a magnetic field intensity to a second spatial derivative of the magnetic field as a function of the actual distance to the magnetized target. 
         FIGS. 10A, 10B, 10C, and 10D  depict plots of various ratios of a first spatial derivative of a magnetic field to a second spatial derivative of the magnetic field as a function of the actual distance to the magnetized target. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  schematically depicts one example of a well twinning application such as a SAGD twinning operation. Common SAGD twinning operations require a horizontal twin well  20  to be drilled a substantially fixed distance substantially directly above a horizontal portion of the target well  30  (e.g., not deviating more than about 1-2 meters up or down or to the left or right of the lower well). In the exemplary embodiment shown, the lower (target) well  30  is drilled first, for example, using conventional directional drilling and MWD techniques. However, the disclosed embodiments are not limited in regard to which of the wells is drilled first. The target wellbore  30  is then cased using a plurality of premagnetized tubulars (such as those shown on  FIG. 2  described below) to form a magnetized casing string  35 . In the embodiment shown, drill string  24  includes at least one tri-axial magnetic field measurement sensor  28  deployed in close proximity to the drill bit  22 . Sensor  28  is used to passively measure the magnetic field about target well  30  as the twin well is drilled. Such passive magnetic field measurements are then utilized to guide continued drilling of the twin well  20  along a predetermined path relative to the target well  30  (e.g., as described in U.S. Pat. Nos. 7,617,049, 7,656,161, and 8,026,722, each of which is fully incorporated by reference herein). 
     With reference now to  FIG. 2 , one example tubular  60  magnetized as described in the &#39;722 patent is shown. The depicted tubular  60  embodiment includes a plurality of discrete magnetized zones  62  (typically three or more). Each magnetized zone  62  may be thought of as a discrete cylindrical magnet having a north N pole on one longitudinal end thereof and a south S pole on an opposing longitudinal end thereof such that a longitudinal magnetic flux  68  is imparted to the tubular  60 . Tubular  60  further includes a single pair of opposing north-north NN poles  65  at the midpoint thereof. The purpose of the opposing magnetic poles  65  is to focus magnetic flux outward from tubular  60  as shown at  70  (or inward for opposing south-south poles as shown at  72 ). The tubulars may be magnetized, for example, using the apparatus disclosed in U.S. Pat. No. 7,538,650, which is fully incorporated by reference herein. 
     With continued reference to  FIG. 1 , the casing string  35  is formed by joining (threadably connecting) premagnetized tubulars in the target well  30 . In one embodiment, the resultant string  35  has a single pair of opposing magnetic poles in the central region (the middle third) of each tubular. Thus the pairs of opposing magnetic poles (NN or SS) are spaced at intervals about equal to the length of tubulars, while the period of the magnetic field pattern (e.g., the distance from one a NN pair of opposing magnetic poles to the next NN pair) is about twice the length of a tubular. 
     As described above, drill string  20  may include a triaxial magnetic field sensor  28 . The depicted embodiment of the sensor  28  includes three mutually orthogonal magnetic field sensors, one of which is oriented substantially parallel with the borehole axis (M Z ). Sensor  28  may thus be considered as determining a plane (defined by M X  and M Y ) orthogonal to the borehole axis and a pole (M Z ) parallel to the borehole axis of the twin well, where M X , M Y , and M Z  represent measured magnetic field vectors in the x, y, and z directions. 
     The magnetic field about the magnetized casing string may be measured and represented, for example, as a vector whose orientation depends on the location of the measurement point within the magnetic field. In order to determine the magnetic field vector due to the target well (e.g., target well  30 ) at any point downhole, the magnetic field of the earth may be subtracted from the measured magnetic field vector using means known to those of ordinary skill in the art. The magnetic field of the earth (including both magnitude and direction components) may be known, for example, from previous geological survey data or a geomagnetic model. It will be understood that in certain embodiments such subtraction of the magnetic field of the earth is not required. 
     It will be appreciated that the disclosed embodiments are not limited to the depictions of  FIGS. 1 and 2 . For example, the disclosure is not limited to SAGD applications. Rather, exemplary methods in accordance with this disclosure may be utilized to drill twin wells having substantially any relative orientation for substantially any application. Moreover, the disclosure is not limited to any particular magnetization pattern or spacing of pairs of opposing magnetic poles on the target well. 
       FIG. 3  depicts a flow chart of one example of a disclosed method embodiment  100  for determining a distance between a drilling well and a magnetized target well (e.g., as depicted on  FIG. 1 ). The method includes acquiring a plurality of axially and or radially spaced magnetic field measurements at  110 . The magnetic field measurements may then be processed at  120  to compute first spatial derivatives and second spatial derivatives of the magnetic field. The first spatial derivatives and second spatial derivatives may be further processed at  130  to compute one or more of the following ratios: (i) a ratio of the magnetic field intensity to a first spatial derivative of the magnetic field, (ii) a ratio of the magnetic field intensity to a second spatial derivative of the magnetic field, and/or (iii) a ratio of a first spatial derivative of the magnetic field to a second spatial derivative of the magnetic field. The computed ratio or ratios may then be further processed to obtain the distance between the drilling well and the magnetized target well at  140 . 
     The plurality of axial and/or radially spaced magnetic field measurements may be acquired at  110  using magnetic field sensors deployed in a drill string in the drilling well (e.g., sensor  28  deployed in drill string  24  in drilling well  20  in  FIG. 1 ). In certain embodiments, the spaced magnetic field measurements may be made using a single triaxial magnetic field sensor. For example, axially spaced measurements may be obtained via moving the drill string axially in the wellbore (in the uphole or downhole direction) between measurements. Radially spaced measurements may be obtained by rotating an off-centered (eccentered) sensor to various toolface angles between measurements. In other embodiments, the drill string may include a plurality of axially and/or radially spaced magnetic field sensors. For example, two, three, or more axially spaced measurements may be acquired via corresponding magnetic field sensors deployed in the drill string (e.g., at half meter intervals along the length of the string). Radially spaced measurements may be acquired via corresponding magnetic field sensors deployed about the circumference of the drill string (e.g., first and second diametrically opposed sensors or three or more sensors deployed at suitable angular intervals about the circumference). Radially spaced measurements may also be acquired using corresponding sensors having different degrees of eccentricity (e.g., a central sensor and one or more eccentered sensors). The magnetic field sensors may also be offset both axially and radially (e.g., first and second axially spaced sensors having one or more eccentered sensors located axially between them). The disclosed method embodiments are not limited to any particular magnetic field sensor configuration and/or spacing. 
     The magnetic field measurements may be resolved into three orthogonal components which can in turn be defined, for example, as highside, lateral, and along-hole or axial directions (or x, y, and z directions as described above). The highside and lateral components may also be resolved into polar coordinates, designated, for example, by a radial intensity and a toolface-to-target direction. Four magnetic field gradients (first spatial derivatives of the magnetic field) may be defined based on the axial and radial components. However, since the magnetic field is magnetostatic and current-free, its curl is zero and only three of these gradients are independent as indicated below: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           B 
                           r 
                         
                       
                       
                         ∂ 
                         r 
                       
                     
                     ; 
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         ∂ 
                         
                           B 
                           z 
                         
                       
                       
                         ∂ 
                         z 
                       
                     
                     ; 
                   
                    
                   
                     
 
                   
                    
                   and 
                    
                   
                     
 
                   
                    
                   
                     
                       
                         ∂ 
                         
                           B 
                           r 
                         
                       
                       
                         ∂ 
                         z 
                       
                     
                     = 
                     
                       
                         ∂ 
                         
                           B 
                           z 
                         
                       
                       
                         ∂ 
                         r 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where B z  and B r  represent the intensity of the measured magnetic field in the axial (z) and radial (r) directions. Four independent second spatial derivatives of the magnetic field may also be obtained based on the axial and radial components of the magnetic field. They are as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ∂ 
                           2 
                         
                          
                         
                           B 
                           r 
                         
                       
                       
                         ∂ 
                         
                           r 
                           2 
                         
                       
                     
                     ; 
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         
                           ∂ 
                           2 
                         
                          
                         
                           B 
                           z 
                         
                       
                       
                         ∂ 
                         z 
                       
                     
                     ; 
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           
                             B 
                             r 
                           
                         
                         
                           ∂ 
                           
                             z 
                             2 
                           
                         
                       
                       = 
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           
                             B 
                             z 
                           
                         
                         
                           
                             ∂ 
                             r 
                           
                           · 
                           
                             ∂ 
                             z 
                           
                         
                       
                     
                     ; 
                   
                    
                   
                     
 
                   
                    
                   and 
                    
                   
                     
 
                   
                    
                   
                     
                       
                         
                           ∂ 
                           2 
                         
                          
                         
                           B 
                           z 
                         
                       
                       
                         ∂ 
                         
                           r 
                           2 
                         
                       
                     
                     = 
                     
                       
                         
                           ∂ 
                           2 
                         
                          
                         
                           B 
                           r 
                         
                       
                       
                         
                           ∂ 
                           r 
                         
                         · 
                         
                           ∂ 
                           z 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     It will be understood that at least two spaced apart magnetic field measurements are generally required to obtain a first spatial derivative of the magnetic field (a gradient of the magnetic field) and that at least three spaced apart magnetic field measurements are generally required to obtain a second spatial derivative of the magnetic field (a curvature of the magnetic field). 
     The magnetic field gradients may be computed at  120 , for example, from first and second spaced apart magnetic field measurements. For example, the gradient of the axial component of the magnetic field in the axial direction (∂B z /∂z) may be obtained as follows: 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         B 
                         z 
                       
                     
                     
                       ∂ 
                       z 
                     
                   
                   = 
                   
                     
                       Δ 
                        
                       
                           
                       
                        
                       
                         B 
                         z 
                       
                     
                     
                       Δ 
                        
                       
                           
                       
                        
                       z 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     where ΔB z  represents the difference in the axial component of the magnetic field between the first and second measurement positions (i.e., ΔB z =B z2 −B z1 ) and Δz represents the axial measurement spacing (the distance between the first and second measurement positions, i.e., Δz=z 2 −z 1 ). Gradients of the radial component of the magnetic field and/or in the radial direction may be similarly computed. 
     The second spatial derivatives may be computed at  120 , for example, from first, second, and third spaced apart magnetic field measurements. For example, the curvature of the axial component of the magnetic field in the axial direction (∂ 2  B z /∂z 2 ) may be obtained as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         2 
                       
                        
                       
                         B 
                         z 
                       
                     
                     
                       ∂ 
                       
                         z 
                         2 
                       
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               
                                 Δ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   B 
                                   z 
                                 
                               
                               
                                 Δ 
                                  
                                 
                                     
                                 
                                  
                                 z 
                               
                             
                              
                             
                               ( 
                               2 
                               ) 
                             
                           
                           - 
                           
                             
                               
                                 Δ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   B 
                                   z 
                                 
                               
                               
                                 Δ 
                                  
                                 
                                     
                                 
                                  
                                 z 
                               
                             
                              
                             
                               ( 
                               1 
                               ) 
                             
                           
                         
                         ) 
                       
                       
                         Δ 
                          
                         
                             
                         
                          
                         z 
                       
                     
                     = 
                     
                       
                         
                           B 
                           
                             z 
                              
                             
                                 
                             
                              
                             3 
                           
                         
                         - 
                         
                           2 
                            
                           
                             B 
                             
                               z 
                                
                               
                                   
                               
                                
                               2 
                             
                           
                         
                         + 
                         
                           B 
                           
                             z 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                       
                         
                           ( 
                           
                             Δ 
                              
                             
                                 
                             
                              
                             z 
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     where 
     
       
         
           
             
               
                 Δ 
                  
                 
                     
                 
                  
                 
                   B 
                   z 
                 
               
               
                 Δ 
                  
                 
                     
                 
                  
                 z 
               
             
              
             
               ( 
               1 
               ) 
             
           
         
       
     
     represents the magnetic field gradient between the first and second axial positions, 
     
       
         
           
             
               
                 Δ 
                  
                 
                     
                 
                  
                 
                   B 
                   z 
                 
               
               
                 Δ 
                  
                 
                     
                 
                  
                 z 
               
             
              
             
               ( 
               2 
               ) 
             
           
         
       
     
     represents the magnetic field gradient between the second and third axial positions, and Δz represents the axial measurement spacing. The second spatial derivatives may also be obtained, for example, by fitting three or more spaced measurements to a function such as a polynomial and then differentiating the function. Second spatial derivatives of the radial component of the magnetic field and/or in the radial direction may be similarly computed. 
     Owing to the dimensional constraints on downhole tools, the radial measurement spacing tends to be limited to about 0.1 meters or less. The spacing in the axial direction is not physically constrained in the same way; however, it may be advantageous for the axial measurement spacing to be less than about a few meters in order to maintain good resolution and to avoid complications caused by tool curvature. The short radial measurement spacing tends to increases sensitivity to noise such that in certain operations it may be advantageous to use the axially distributed measurements ∂B r /∂z, ∂B z /∂z, ∂ 2 B r /∂z 2 , and ∂ 2 B z /∂z 2  when possible. 
     Variations in the first spatial derivatives and the second spatial derivatives of the magnetic field with position relative to a magnetized target well may be evaluated using a magnetic model. For example, a magnetized casing string having a repeating magnetic pattern along the axis of the string (e.g., as described above with respect to  FIGS. 1 and 2 ) may be modelled as a repeating series of point sources (monopoles) and/or line sources distributed along the centerline of the string. For a monopole model, the field at any point (r, z) from a point source located at (0, zp) may expressed as follows: 
     
       
         
           
             
               
                 
                   
                     B 
                     z 
                   
                   = 
                   
                     
                       P 
                       
                         4 
                          
                         π 
                       
                     
                     · 
                     
                       
                         ( 
                         
                           z 
                           - 
                           zp 
                         
                         ) 
                       
                       
                         
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   z 
                                   - 
                                   zp 
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               r 
                               2 
                             
                           
                           ] 
                         
                         1.5 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                     B 
                     r 
                   
                   = 
                   
                     
                       P 
                       
                         4 
                          
                         π 
                       
                     
                     · 
                     
                       r 
                       
                         
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   z 
                                   - 
                                   zp 
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               r 
                               2 
                             
                           
                           ] 
                         
                         1.5 
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     where P represents the strength of each of the magnetic poles and 0≦p&lt;1 and represents the axial location along the repeating magnetic pattern (where the positions p=0, 1, . . . are adjacent NN opposing magnetic poles). For a line source model, the field at any point (r, z) from a line source of length L centered at (0, zp) may expressed as follows: 
     
       
         
           
             
               
                 
                   
                     B 
                     z 
                   
                   = 
                   
                     
                       P 
                       
                         4 
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         L 
                       
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               
                                 1 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           z 
                                           - 
                                           zp 
                                           - 
                                           
                                             L 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     + 
                                     
                                       r 
                                       2 
                                     
                                   
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               1 
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         z 
                                         - 
                                         zp 
                                         + 
                                         
                                           L 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                   + 
                                   
                                     r 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   
                     B 
                     r 
                   
                   = 
                   
                     
                       P 
                       
                         4 
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         Lr 
                       
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   z 
                                   - 
                                   zp 
                                   + 
                                   
                                     L 
                                     / 
                                     2 
                                   
                                 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           z 
                                           - 
                                           zp 
                                           + 
                                           
                                             L 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                     + 
                                     
                                       r 
                                       2 
                                     
                                   
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 z 
                                 - 
                                 zp 
                                 - 
                                 
                                   L 
                                   / 
                                   2 
                                 
                               
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         z 
                                         - 
                                         zp 
                                         - 
                                         
                                           L 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                   + 
                                   
                                     r 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
       FIG. 4  depicts a plot of the actual magnetic field about a magnetized casing string. The field is represented as a plot of the axial component of the magnetic field versus the radial component of the magnetic field. The magnetic field is further plotted at various radial distances from the string. The casing string was magnetized with a repeating pattern of opposing magnetic poles such that the pattern repeats with a period of twice the length of the tubulars that make up the string (as described above). It may be noted that the casing magnetization in this example is mildly asymmetric with the left side of the plot being larger than the right side, possibly indicating that joints magnetized with one polarity retained slightly more magnetization than the others (the disclosed embodiments are of course not limited in this regard). This fact will aid in determining the sensitivity of a ranging technique to the absolute magnetization of the target as ideally the calculated distance should be the same for both joints. 
     When ranging to a target well magnetized as described above, drilling may be stopped and magnetic surveys taken at locations corresponding to maximum radial flux from the target (i.e. adjacent the NN or SS opposing magnetic poles located at the approximate midpoint of each tubular). At these locations the axial field from the target tends to be small (near zero) while the radial field tends to be at a maximum. These locations correspond to the left and right sides of the plot depicted on  FIG. 4 . The gradients ∂B z /∂z and ∂B r /∂r are relatively large at these locations while ∂B r /∂z is small (near zero). Of the second spatial derivatives, ∂ 2 B r /∂r 2  and ∂ 2 B r /∂z 2  tend to be large, while ∂ 2 B z /∂r 2  and ∂ 2 B z /∂z 2  are small (near zero). Since measurements of small quantities tend to be susceptible to noise, it may be advantageous make use of the larger values ∂B r /∂r, ∂B z /∂z, ∂ 2 B r /∂r 2 , and ∂ 2 B r /∂z 2 , and particularly the long baseline measurements ∂B z /∂z and ∂ 2 B r /∂z 2 . 
       FIGS. 5A and 5B  depict plots of the axial and radial components (B z  and B r ) of the magnetic field as a function of normalized axial position along the target well at various distances from the target well. The joint ends are located at normalized axial positions of 1.0 and 2.0 while the opposing magnetic poles are located at normalized axial positions of 0.5, 1.5, and 2.5 (with SS opposing magnetic poles being located at 0.5 and 2.5 and a NN opposing magnetic pole being located at 1.5). Consistent with the plot depicted on  FIG. 4 , the radial component has maxima at axial positions of 0.5, 1.5, and 2.5 (adjacent to the opposing magnetic poles). 
       FIGS. 6A, 6B, and 6C  depict plots of the three independent magnetic field gradients (first spatial derivatives) as a function of normalized axial position along the target well at various distances from the target well.  FIG. 6A  depicts the gradient of the intensity of the radial magnetic field component in the radial direction ∂B r /∂r.  FIG. 6B  depicts the gradient of the intensity of the axial magnetic field component in the axial direction ∂B z /∂z. And  FIG. 6C  depicts the gradient of the intensity of the radial magnetic field component in the axial direction ∂B r /∂z (which is equal to the gradient of the intensity of the axial magnetic field component in the radial direction ∂B z /∂r).  FIGS. 6A and 6B  show that ∂B r /∂r and ∂B z /∂z have maxima at axial positions of 0.5, 1.5, and 2.5 (adjacent the opposing magnetic poles).  FIG. 6C  shows that ∂B r /∂z is approximately zero at the same axial positions. 
       FIGS. 7A, 7B, 7C, and 7D  depict plots of the four independent second spatial derivatives of the magnetic field as a function of normalized axial position along the target well at various distances from the target well.  FIG. 7A  depicts the second spatial derivative of the radial component of the magnetic field in the radial direction ∂ 2 B r /∂r 2 .  FIG. 7B  depicts the second spatial derivative of the radial component of the magnetic field in the axial direction ∂ 2 B r /∂z 2 .  FIG. 7C  depicts the second spatial derivative of the axial component of the magnetic field in the radial direction ∂ 2 B z /∂r 2 .  FIG. 7D  depicts the second spatial derivative of the axial component of the magnetic field in the axial direction ∂ 2 B z /∂z 2 .  FIGS. 7A and 7B  show that ∂ 2 B r /∂r 2  and ∂ 2 B r /∂z 2  have maxima at axial positions of 0.5, 1.5, and 2.5 (adjacent the opposing magnetic poles).  FIGS. 7C and 7D  show that ∂ 2 B z /∂r 2  and ∂ 2 B z /∂z 2  are approximately zero at the same axial positions. 
     When the magnetic field measurements are made at axial positions adjacent (or nearly adjacent) to the opposing magnetic poles, the magnetic field intensity, the first spatial derivatives, and the second spatial derivatives may be approximated, for example, from Equations 5 and 6 above (the monopole approximation). Thus, for example, when z=zp the magnetic field intensities may be expressed as follows: 
     
       
         
           
             
               
                 
                   
                     B 
                     z 
                   
                   ≈ 
                   0 
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
             
               
                 
                   
                     B 
                     r 
                   
                   ≈ 
                   
                     P 
                     
                       4 
                        
                       
                           
                       
                        
                       π 
                        
                       
                           
                       
                        
                       
                         r 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     The first spatial derivatives may be also be expressed, for example as follows: 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         B 
                         z 
                       
                     
                     
                       ∂ 
                       z 
                     
                   
                   ≈ 
                   
                     P 
                     
                       4 
                        
                       
                           
                       
                        
                       π 
                        
                       
                           
                       
                        
                       
                         r 
                         3 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         B 
                         r 
                       
                     
                     
                       ∂ 
                       r 
                     
                   
                   ≈ 
                   
                     P 
                     
                       4 
                        
                       
                           
                       
                        
                       π 
                        
                       
                           
                       
                        
                       
                         r 
                         3 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         B 
                         r 
                       
                     
                     
                       ∂ 
                       z 
                     
                   
                   ≈ 
                   
                     
                       ∂ 
                       
                         B 
                         z 
                       
                     
                     
                       ∂ 
                       r 
                     
                   
                   ≈ 
                   0 
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     The second spatial derivatives may also be expressed, for example, as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         2 
                       
                        
                       
                         B 
                         r 
                       
                     
                     
                       ∂ 
                       
                         r 
                         2 
                       
                     
                   
                   ≈ 
                   
                     
                       3 
                        
                       P 
                     
                     
                       2 
                        
                       
                           
                       
                        
                       π 
                        
                       
                           
                       
                        
                       
                         r 
                         4 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ∂ 
                         2 
                       
                        
                       
                         B 
                         r 
                       
                     
                     
                       ∂ 
                       
                         z 
                         2 
                       
                     
                   
                   ≈ 
                   
                     
                       3 
                        
                       P 
                     
                     
                       4 
                        
                       
                           
                       
                        
                       π 
                        
                       
                           
                       
                        
                       
                         r 
                         4 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ∂ 
                         2 
                       
                        
                       
                         B 
                         z 
                       
                     
                     
                       ∂ 
                       
                         r 
                         2 
                       
                     
                   
                   ≈ 
                   
                     
                       
                         ∂ 
                         2 
                       
                        
                       
                         B 
                         r 
                       
                     
                     
                       ∂ 
                       
                         z 
                         2 
                       
                     
                   
                   ≈ 
                   0 
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     As described above, the intent of the magnetic ranging measurements is to determine the relative position of the drilling well with respect to the magnetized target well, for example, via determining a distance and direction from the drilling well to the target well. The toolface direction (the direction in the plane normal to the tool axis) towards the target may be obtained from a ratio of the two components measured in that plane (e.g., a ratio of the x and y components of the measured magnetic field). The distance to the target may be found from a ratio of a magnetic field intensity to a first spatial derivative of the magnetic field, a ratio of a magnetic field intensity to a second spatial derivative of the magnetic field, and/or a ratio of a first spatial derivative of the magnetic field to a second spatial derivative of the magnetic field. The use of one or more of the following ratios may be advantageous in that the ratios are independent of the strength of the magnetic poles. The use of multiple ratios may further improve the accuracy of the obtained distance by giving corresponding multiple independent measurements. 
     When the magnetic field measurements are made at axial positions adjacent (or nearly adjacent) to the opposing magnetic poles, the ratios may be approximated from certain of Equations 9 through 16 above. The distance to the target may be expressed in terms of example ratios of a magnetic field intensity to a first spatial derivative of the magnetic field, for example, as follows: 
     
       
         
           
             
               
                 
                   r 
                   ≈ 
                   
                     
                       B 
                       r 
                     
                     
                       
                         ∂ 
                         
                           B 
                           z 
                         
                       
                       
                         ∂ 
                         z 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   r 
                   ≈ 
                   
                     
                       - 
                       2 
                     
                      
                     
                       
                         B 
                         r 
                       
                       
                         
                           ∂ 
                           
                             B 
                             r 
                           
                         
                         
                           ∂ 
                           r 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     The distance to the target may also be expressed in terms of example ratios of a magnetic field intensity to a second spatial derivative of the magnetic field, for example, as follows: 
     
       
         
           
             
               
                 
                   r 
                   ≈ 
                   
                     
                       [ 
                       
                         6 
                          
                         
                           
                             B 
                             r 
                           
                           
                             
                               
                                 ∂ 
                                 2 
                               
                                
                               
                                 B 
                                 r 
                               
                             
                             
                               ∂ 
                               
                                 r 
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                     
                       1 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   r 
                   ≈ 
                   
                     
                       [ 
                       
                         
                           - 
                           3 
                         
                          
                         
                           
                             B 
                             r 
                           
                           
                             
                               
                                 ∂ 
                                 2 
                               
                                
                               
                                 B 
                                 r 
                               
                             
                             
                               ∂ 
                               
                                 z 
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                     
                       1 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     The distance to the target may be further expressed in terms of example ratios of a first spatial derivative of the magnetic field to a second spatial derivative of the magnetic field, for example, as follows: 
     
       
         
           
             
               
                 
                   r 
                   ≈ 
                   
                     6 
                      
                     
                       
                         
                           ∂ 
                           
                             B 
                             z 
                           
                         
                         
                           ∂ 
                           z 
                         
                       
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           
                             B 
                             r 
                           
                         
                         
                           ∂ 
                           
                             r 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
             
               
                 
                   r 
                   ≈ 
                   
                     
                       - 
                       3 
                     
                      
                     
                       
                         
                           ∂ 
                           
                             B 
                             z 
                           
                         
                         
                           ∂ 
                           z 
                         
                       
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           
                             B 
                             r 
                           
                         
                         
                           ∂ 
                           
                             z 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   r 
                   ≈ 
                   
                     
                       - 
                       3 
                     
                      
                     
                       
                         
                           ∂ 
                           
                             B 
                             r 
                           
                         
                         
                           ∂ 
                           r 
                         
                       
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           
                             B 
                             r 
                           
                         
                         
                           ∂ 
                           
                             r 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
             
               
                 
                   r 
                   ≈ 
                   
                     1.5 
                      
                     
                       
                         
                           ∂ 
                           
                             B 
                             r 
                           
                         
                         
                           ∂ 
                           r 
                         
                       
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           
                             B 
                             r 
                           
                         
                         
                           ∂ 
                           
                             z 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     The performance of these functions (equations 17 through 24) may be estimated using the model of the magnetized target shown on  FIG. 4 . A transform may be developed to convert the ratio to its corresponding actual distance. The ratios between a magnetic field intensity and a first spatial derivative of the magnetic field (given in equations 17 and 18) are evaluated in the plots shown on  FIGS. 8A and 8B .  FIG. 8A  depicts a plot of the ratio 
     
       
         
           
             
               B 
               r 
             
             
               
                 ∂ 
                 
                   B 
                   z 
                 
               
               
                 ∂ 
                 z 
               
             
           
         
       
     
     versus actual distance at axial positions of 0.5, 1.5, and 2.5. In this example the ratio seems to be poorly suited to determining distance as it is substantially independent of distance.  FIG. 8B  depicts a plot of the ratio 
     
       
         
           
             
               - 
               2 
             
              
             
               
                 B 
                 r 
               
               
                 
                   ∂ 
                   
                     B 
                     r 
                   
                 
                 
                   ∂ 
                   r 
                 
               
             
           
         
       
     
     versus actual distance at axial positions of 0.5, 1.5, and 2.5. In this example, the ratio varies monotonically with distance. The separation between the two curves at larger distances indicates that the ratio may be somewhat sensitive to the absolute intensity of the magnetic poles. 
     The ratios between a magnetic field intensity and a second spatial derivative of the magnetic field (given in equations 19 and 20) are evaluated at normalized axial positions of 0.5, 1.5, and 2.5 in the plots shown on  FIGS. 9A and 9B .  FIG. 9A  depicts a plot of the ratio 
     
       
         
           
             
               ( 
               
                 6 
                  
                 
                   
                     B 
                     r 
                   
                   
                     
                       
                         ∂ 
                         2 
                       
                        
                       
                         B 
                         r 
                       
                     
                     
                       ∂ 
                       
                         r 
                         2 
                       
                     
                   
                 
               
               ) 
             
             
               1 
               2 
             
           
         
       
     
     versus actual distance while  FIG. 9B  depicts a plot of the ratio 
     
       
         
           
             
               ( 
               
                 
                   - 
                   3 
                 
                  
                 
                   
                     B 
                     r 
                   
                   
                     
                       
                         ∂ 
                         2 
                       
                        
                       
                         B 
                         r 
                       
                     
                     
                       ∂ 
                       
                         z 
                         2 
                       
                     
                   
                 
               
               ) 
             
             
               1 
               2 
             
           
         
       
     
     versus actual distance. In these examples, the ratios vary monotonically with distance and may therefore be suitable for use in distance determination. The separation between the two curves in each plot indicates that these ratios may be somewhat sensitive to the absolute intensity of the magnetic poles. 
     The ratios between a first spatial derivative of the magnetic field and a second spatial derivative of the magnetic field (given in equations 21 through 24) are evaluated at normalized axial positions of 0.5, 1.5, and 2.5 in the plots shown on  FIGS. 10A, 10B, 10C, and 10D .  FIG. 10A  depicts a plot of the ratio 
     
       
         
           
             6 
              
             
               
                 
                   ∂ 
                   
                     B 
                     z 
                   
                 
                 
                   ∂ 
                   z 
                 
               
               
                 
                   
                     ∂ 
                     2 
                   
                    
                   
                     B 
                     r 
                   
                 
                 
                   ∂ 
                   
                     r 
                     2 
                   
                 
               
             
           
         
       
     
     versus actual distance. In this example the ratio is a strong monotonic function of the distance making it a good candidate for distance determination.  FIG. 10B  depicts a plot of the ratio 
     
       
         
           
             
               - 
               3 
             
              
             
               
                 
                   ∂ 
                   
                     B 
                     z 
                   
                 
                 
                   ∂ 
                   z 
                 
               
               
                 
                   
                     ∂ 
                     2 
                   
                    
                   
                     B 
                     r 
                   
                 
                 
                   ∂ 
                   
                     z 
                     2 
                   
                 
               
             
           
         
       
     
     versus actual distance. The second spatial derivative in this ratio may also be determined by measuring ∂/∂z (∂B z /∂r) or ∂/∂r(∂B z /∂z).  FIG. 10C  depicts a plot of the ratio 
     
       
         
           
             
               - 
               3 
             
              
             
               
                 
                   ∂ 
                   
                     B 
                     r 
                   
                 
                 
                   ∂ 
                   r 
                 
               
               
                 
                   
                     ∂ 
                     2 
                   
                    
                   
                     B 
                     r 
                   
                 
                 
                   ∂ 
                   
                     r 
                     2 
                   
                 
               
             
           
         
       
     
     versus actual distance. In these examples, the ratios vary monotonically with distance and may therefore be suitable for use in distance determination. The separation between the two curves in  FIGS. 10A and 10B  indicates that these ratios may be somewhat sensitive to the absolute intensity of the magnetic poles. The ratio in  FIG. 10C  shows very little sensitivity to the absolute intensity of the magnetic poles.  FIG. 10D  depicts a plot of the ratio 
     
       
         
           
             1.5 
              
             
               
                 
                   ∂ 
                   
                     B 
                     r 
                   
                 
                 
                   ∂ 
                   r 
                 
               
               
                 
                   
                     ∂ 
                     2 
                   
                    
                   
                     B 
                     r 
                   
                 
                 
                   ∂ 
                   
                     z 
                     2 
                   
                 
               
             
           
         
       
     
     versus actual distance. The second spatial derivative in this ratio may also be determined by measuring ∂/∂r (∂B z /∂z) or ∂/∂z (∂B z /∂r). In this example, the ratio is not well correlated with distance. 
     It will be understood that method  100  may be performed using uphole and/or downhole processors. The disclosed embodiments are not limited in this regard. For example, magnetic field measurements may be transmitted to the surface (using any suitable telemetry techniques). The distance may then be computed at the surface and further used to compute a new drilling direction which may then be transmitted back to the tool. Alternatively, the magnetic field measurements may be processed downhole to obtain the distance, for example, using one or more look up tables to correlate the computed ratio(s) to distance. The obtained distance may then be used to compute a new drilling direction downhole which may be implemented as part of a closed loop well twinning methodology. 
     While the aforementioned examples make use of a target well is magnetization having axially spaced opposing magnetic poles it will be understood that the disclosed embodiments are not so limited. The use of first spatial derivatives and second spatial derivatives of the magnetic field and ratios including those derivatives may be used with substantially any suitable target well magnetization. 
     Although a method for magnetic gradient and curvature based ranging and certain advantages thereof have been described in detail, it should be understood that various changes, substitutions and alternations can be made herein without departing from the spirit and scope of the disclosure as defined by the appended claims.