Patent Publication Number: US-8983782-B2

Title: Magnetic beacon guidance system

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is the national phase, under 35 U.S.C. §371, of International Application No.: PCT/AU2005/001964, filed Dec. 29, 2005, which designated the United States of America. The present application claims the benefit of priority to and incorporates herein by reference, in its entirety, the disclosure of International Application No.: PCT/AU2005/001964. 
     FIELD OF THE INVENTION 
     This invention relates to guidance systems. More particularly, the invention relates to a method of, and a system for, guiding a probe to a target. The invention has particular, but not necessarily exclusive, application in the field of drilling lateral holes to a vertical borehole in the field of coal bed methane gas extraction. 
     BACKGROUND TO THE INVENTION 
     In a number of applications, it is necessary to guide a probe to a target through a solid medium. An example of such an application is in the field of coal bed methane gas (CBM) extraction. While the invention has been specifically developed for this application, it could be used in other applications with few, if any, modifications. The invention is therefore not limited to such an application and those skilled in the art will readily appreciate the applicability of the invention to other fields of use. 
     In one CBM extraction method, a vertical well is drilled from the surface down through the target coal seam. A pump maintains low pressure in a sump cavity below the seam at the bottom of the well. A lateral hole is drilled horizontally through the coal seam with the intention of intersecting the well. The pump is then used to extract methane from the coal seam. The lateral hole enters the ground from a surface location 300 to 1500 meters in horizontal distance up dip from the vent well. Once in the coal seam the drill string is turned to a more horizontal attitude but following the dip of the coal seam. Due primarily to cumulative systematic errors introduced by the measurement systems, an ellipse of uncertainty is created. In effect, there is a very small chance of the lateral hole intersecting the borehole on a first pass of the drill string. 
     As a result, it a very hit and miss affair to cause the lateral hole to intersect the borehole and, to date, repeated passes of the drill string have been required to achieve this objective. It will be appreciated that it is very costly to operate a drill rig and each pass of the drill string is therefore very costly not to say time-consuming. Each time a further pass of the drill string is required, the drill string needs to be retracted and a new trajectory plotted and drilled. 
     SUMMARY OF THE INVENTION 
     According to a first aspect of the invention, there is provided a method of guiding a probe to a target, the method including 
     placing a magnetic field generator at the target; 
     guiding the probe to a region of the target, the probe carrying a survey sensor pack; 
     using the survey sensor pack to obtain a plurality of survey readings; 
     using the survey sensor pack to obtain a plurality of magnetic beacon readings using a magnetic field generated by the magnetic field generator; 
     comparing a selected number of the survey readings and the magnetic beacon readings and determining a difference between the survey readings and the magnetic beacon readings; and 
     compensating for that difference thereafter to guide the probe to the target. 
     The difference between the survey readings and the magnetic beacon readings may include an angular difference and/or a displacement difference. 
     The method may include selecting the magnetic field generator to be of predetermined dimensions. In particular, the method may include selecting the dimensions of the magnetic field generator in dependence of the distance it is estimated the probe is likely to be from the target. Thus, the method may include implementing the magnetic field generator in segments so that a magnetic field generator of desired length can be used. 
     The method may include initially defining a commencement position and termination position for the probe. In the field of coal bed methane gas extraction, the commencement position of the probe may be a entry collar of a lateral hole to be drilled and the termination position may be the position at which the probe should intersect the target assuming there were no errors. 
     The method may include processing and recording data generated by the probe along its initial trajectory. Due to the fact that some parts of the trajectory may result in dead ends, the method may include excluding data relating to non-completed, unusable portions of the initial trajectory. 
     The method may include taking a predetermined number of magnetic beacon readings when the probe is within range of the magnetic field generator. The method may further include deriving fixes from at least two pairs of predetermined magnetic beacon readings. Thus, the method may include selecting each magnetic beacon reading for use in deriving the fixes by comparing the magnetic beacon reading with a corresponding survey reading and, if the magnetic beacon reading differs from the survey reading by an amount exceeding a predetermined value, disregarding that magnetic beacon reading. The method may then include forming a segment of magnetic beacon readings from the fixes. Further, the method may include comparing the segment of magnetic beacon readings with a segment of corresponding survey readings. 
     Preferably, the method includes taking two measurements for each magnetic beacon reading, one with poles of the magnetic field generator in a first orientation and the other with the poles of the magnetic field generator in an opposite orientation to minimise the effects of earth&#39;s magnetic field. 
     The method may include obtaining a vector representative of a radial component of the magnetic field generated by the magnetic field generator at each magnetic beacon reading. The method may include transforming raw vectors from each magnetic beacon reading to obtain the radial component. 
     The method may include calculating an angular difference between each magnetic beacon reading and its associated survey reading and calculating a difference in displacement between the magnetic beacon reading and its associated survey reading. 
     Further, the method may include calculating a new trajectory and displaying the new trajectory to an operator. In particular, the new trajectory may be displayed to the operator both graphically and numerically. 
     According to a second aspect of the invention there is provided a system for guiding a probe to a target, the system including 
     a magnetic field generator to be located at the target; 
     a survey probe to be guided to the target, the survey probe carrying a survey sensor pack, sensors of the sensor pack being operable to obtain a plurality of survey readings and a plurality of magnetic beacon readings using a magnetic field generated by the magnetic field generator; and 
     processing equipment for processing data relating to a selected number of the measured survey readings and the magnetic beacon readings to determine a difference between the survey readings and the magnetic beacon readings and for compensating for that difference thereafter to guide the probe to the target. 
     The magnetic field generator may have variable dimensions, the dimensions of the magnetic field generator being selected in dependence of the distance it is estimated the probe is likely to be from the target. Preferably, the magnetic field generator comprises a plurality of interconnectable segments so that a magnetic field generator of desired length can be used. The magnetic field generator may be a solenoid having switchable poles. 
     The survey sensor pack may comprise a plurality of magnetometer/accelerometer pairs, the pairs being arranged to take the readings along Cartesian coordinates. 
     The processing equipment may be operable to process and record data generated by the probe along its initial trajectory. 
     The survey pack may be operable to take a predetermined number of magnetic beacon readings when the probe is within range of the magnetic field generator. Then, the processing equipment may be operable to derive fixes from at least two pairs of predetermined magnetic beacon readings. 
     The processing equipment may be operable to select each magnetic beacon reading for use in deriving the fixes by comparing the magnetic beacon reading with a corresponding survey reading and, if the magnetic beacon reading differs from the survey reading by an amount exceeding a predetermined value, disregarding that magnetic beacon reading. 
     Further, the processing equipment may be operable to form a segment of magnetic beacon readings from the fixes and to compare the segment of magnetic beacon readings with a segment of corresponding survey readings. 
     The system may include a switching arrangement for switching the relative orientation of poles of the magnetic field generator to minimise the effects of earth&#39;s magnetic field. 
     The processing equipment may be operable to obtain a vector representative of a radial component of the magnetic field generated by the magnetic field generator at each magnetic beacon reading. Thus, the processing equipment may transform raw vectors from each magnetic beacon reading to obtain the radial component. 
     Further, the processing equipment may be operable to calculate an angular difference between each magnetic beacon reading and its associated survey reading and to calculate a difference in displacement between the magnetic beacon reading and its associated survey reading. From this, the processing equipment may calculate a new trajectory for the probe. 
     The system may include a display arrangement for displaying the new trajectory of the probe to an operator. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       An embodiment of the invention is now described by way of example with reference to the accompanying diagrammatic drawings in which: 
         FIG. 1  shows a schematic representation of a system, in accordance with an embodiment of the invention, for guiding a probe to a target; 
         FIG. 2  shows a schematic plot of a comparison between an original trajectory and an adjusted trajectory of a probe of the system of  FIG. 1 ; 
         FIG. 3  shows a schematic side view of a path of the probe to the target; 
         FIG. 4  shows a schematic plan view of the last part of the path of the probe relative to the target indicating a pullback and intersect operation; 
         FIG. 5  shows a schematic plan view of the last part of the path of the probe relative to the target indicating a part of a method, in accordance with an embodiment of the invention, for guiding a probe to a target; 
         FIG. 6  shows a schematic, sectional side view of the target with a magnetic field generator at the target; 
         FIG. 7  shows a schematic plan view of part of the path with vectors used in the method superimposed thereon; 
         FIG. 8  shows a view similar to that of  FIG. 7  with further information used in the method superimposed thereon; 
         FIG. 9  shows a schematic plan view after transformation of vectors used in the method; 
         FIG. 10  shows a schematic plan view of the part of the path of  FIG. 8  after correction of the trajectory; 
         FIG. 11  shows a screen shot of a display of the system of  FIG. 1 ; and 
         FIG. 12  shows a further screen shot of the display of the system of  FIG. 1 . 
     
    
    
     DETAILED DESCRIPTION OF THE EXEMPLARY EMBODIMENT 
     Referring initially to  FIG. 1  of the drawings, an embodiment of a system for guiding a probe to a target is illustrated and is designated generally by the reference numeral  10 . The system  10  can be used in numerous applications. However, for ease of explanation only, the system  10  will be described with reference to its application in the field of coal bed methane gas (CBM) extraction from a coal seam. 
     In such a system, a lateral hole  12  ( FIG. 3 ) is drilled to a target in the form of a vertically extending borehole  14  to intersect the borehole  14 . The lateral hole  12  is drilled through a coal seam indicated schematically at  16  in  FIG. 6  of the drawings. 
     The system  10  incorporates a magnetic field generator or beacon  18  received in the vertical bore hole  14  to be suspended just within the coal seam  16  as illustrated in  FIG. 6  of the drawings. 
     The system  10  further includes a survey probe  20  arranged in a drill string  22 . More particularly, the survey probe  20  is arranged in a bottom hole assembly  24  carrying a drill bit  26 . The survey probe  20  can be mounted up to 6 to 12 meters rearwardly of the drill bit  26 . The survey probe carries a survey sensor pack  28 . While the survey sensor pack  28  is shown as a separate component in  FIG. 1  of the drawings, this is purely for the sake of illustration. In practice, the survey pack  28  is arranged within the survey probe  20 . The survey pack  28  carries a plurality of sensors. The sensors are operable to obtain a plurality of survey readings. More particularly, the sensors comprise three magnetometers and three accelerometers arranged in magnetometer/accelerometer pairs along Cartesian coordinates  30 . 
     The survey probe  20  and, more particularly, its sensor pack  28  communicate with remotely arranged processing equipment in the form of a processor  32 . The processor  32  displays data generated on a display  34 . 
     The magnetic beacon  18  may be constituted by any suitable magnetic field generator. In a preferred implementation, the magnetic beacon  18  is in the form of an electromagnet or solenoid  36  which can have its poles switched. It will, however, be appreciated that the magnetic beacon  18  could be a permanent magnet although this would require removing the beacon  18  from the borehole  14  and reversing it in order to reverse its polarity. 
     The solenoid  36  generates a magnetic field  38 . The size and shape of the magnetic field  38  is governed by the length of the solenoid  36 . Thus, the solenoid  36  may be arranged in segments which can be secured together in an end-to-end relationship to vary the size and shape of the magnetic field  38  as required. 
     The lateral hole  12  is dug from an entry position or entry collar  40  ( FIG. 2 ) towards the borehole  14  along a predetermined trajectory  42 . The trajectory  42  is plotted relative to a baseline  44 . 
     Due to errors in the sensors of the sensor pack  28  and other factors such as drill string stretch, errors accumulate as the drill string  22  follows the trajectory  42 . Thus, although the original trajectory  42  is shown as extending from the entry collar to intersect the target  14 , in practice, the trajectory as drilled is more often than not likely to miss the target  14  as shown by the trajectory  46  in  FIG. 2  of the drawings. It will be appreciated that the resolution of the sensors in the azimuthal plane is only approximately 0.5°. The entry collar  40  could be up to 1,500 meters away from the target  14  and the target  14  only has a diameter of approximately 15 cm so the likelihood of a trajectory  42  intersecting the target  14  is low. 
     In  FIG. 2  of the drawings, point  48  indicates the last survey point of the original trajectory and point  50  indicates the last survey point of the adjusted trajectory. This shows azimuthal error  52  as well as a base line displacement error  54 . 
     In addition, as shown in  FIG. 3  of the drawings, the lateral hole  14 , being dug from the surface, must be turned from a few degrees from the vertical towards the horizontal as shown at  56  in  FIG. 3  of the drawings. This turning of the lateral hole  12  also introduces significant errors into the trajectory  42 . 
     These errors accumulate over the length of the trajectory  42  and it is necessary to compensate for these errors in order that the target  14  can be intersected by the lateral hole  12 . 
     The entry collar  40  and the target  14  must be accurately defined in grid coordinates before drilling commences as they are important datum points for the operation. Normally the survey calculations resolve position relative to the entry collar  40  so knowing the position of the entry collar  40  in local grid coordinates affects the absolute measurement accuracy of all points along the trajectory  42 . 
     Equally, once a beacon fix has resolved the trajectory&#39;s position relative to the target  14  then, assuming that both the position of the entry collar  40  and the position of the target  14  are already well defined, the absolute grid position of the probe  20  at both ends of the trajectory  42  can be determined with a high degree of accuracy. 
     As an initial step, all data generated from the probe  20  is processed and recorded so that the path of the drill string  22  can be defined within the tolerance limits of the sensors of the sensor pack  28 . The path is, however, usually not just a single continuous hole plotted from the entry collar  40  to the target  14 . In a typical operation, the process of drilling to the target  14  usually entails drilling a series of branched holes, known as sidetracks, which, when strung together, form the final path. A combination of factors such as faults and rolls in the seam  16  make it very difficult to navigate within a seam floor  58  ( FIG. 6 ) and a seam roof  60  over the distance of the planned trajectory  42 . As described above, making navigation even more difficult is the fact that the probe  20  is about 6 m to 12 m back from the bit  26 . This, combined with a very constrained turn radius, means the drill string  22  may be unintentionally steered out the coal seam  16  a number of times during any given operation. Each time the seam  16  is exited, the drill string  22  must be withdrawn back into the coal seam  16  where a branch hole can be initiated. 
     It is a function of the software of the processor  32  of the system  10  to determine the continuous path running from the entry collar  40  to the target  14 . Useable portions of the branch holes are included in the final trajectory  42  and interpolated up to their branch points, while unusable dead end sections are excluded. 
     The processor  32  must obtain all sensor data from the sensor pack  28  of the probe  20  and measured depth interval lengths from the operator or from a sensor attached to the drill string  22 . These data are used to resolve position using raw data from the sensor pack  28  of the probe  20 . The assumption is made that the trajectory  42  interpolates a circular path between any two surveyed points which has an orientation and radius that is defined by the two point segment. Each segment is calculated using 2×azimuth+2×inclination values Pt 1  (az 1 , inc 1 )−Pt 2  (az 2 , inc 2 ) plus the measured distance (Δmd) along that segment.
 
(Δ md=md 2 −md 1).  Equation 1
 
Since
 
cos(θ)={right arrow over (v)} 1 ·{right arrow over (v)} 2  (dot product of any two vectors)
 
where θ is the total angular difference between the two vectors being measured.
 
Then
 
θ=cos −1 ( {right arrow over (v)} 1′ (GCS)   ·{right arrow over (v)} 2′ (GCS) )
 
where
 
{right arrow over (v)} 1 ′ (GCS) ,{right arrow over (v)} 2 ′ (GCS)  are the probe to target unit vectors transformed to the grid coordinate system.
 
 f =(2/θ)*tan(θ/2)(bulge factor)  Equation 2
 
 P·x =( f*Δmd/ 2)*(sin(Inc (i-1) )*sin( Az   (i-1) )+(sin(Inc (i) )*sin( Az   (i) ))  Equation 3
 
 P·y =( f*Δmd/ 2)*(sin(Inc (i-1) )*cos( Az   (i-1) )+sin(Inc (i) *cos( Az   (i) ))  Equation 4
 
 P·z =( f*Δmd/ 2)*cos(Inc (i-1) )+cos(Inc (i) )  Equation 5
 
where:
 
P is the end point of the segment.
 
Δmd=md 2 −md 1 
 
Inc=inclination
 
Az=azimuth
 
i=shot sequence index
 
     The measured depth (md) is the total distance measured along the hole  12  relative to the entry collar  40  which is md=0. A trajectory  42  is traced from an accumulating sum of each consecutive point generated from Equations 3 to 5. Thus, 
                   Trajectory   =       ∑     i   =   1     n     ⁢     p   ⁢           ⁢     t   i                 Equation   ⁢           ⁢   6               
where n is the shot number that needs to be resolved and the index i starting from 1 is the sequence number of any point within the set. From Equation 6, it is clear that the trajectory  42  is formed from the accumulating sum of the points calculated from each consecutive pair of measurements taken along the hole  12 .
 
Substituting Equations 3 to 5 for pt i  in Equation 6 gives:
 
     
       
         
           
             
               
                 
                   
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                   9 
                 
               
             
           
         
       
     
     An operator of the drill rig  22  uses the results of Equations 7 to 9 to steer along the coal seam  16  to intersect the target  14  eventually. Each point in the trajectory  42  is plotted on a chart that shows the trajectory path  42 , entry point at the entry collar  40 , target  14  and baseline  44  projected in both plan and vertical section views. 
     To range the target  14  using the beacon  18 , the solenoid  36  is first lowered down the vertical target hole  14  so the lower pole is sitting just above the roof  60  of the seam  16 . The operator locates the solenoid  36  by performing a cluster of beacon shots out of which there must be at least three good shots  62 ,  64  and  66  ( FIGS. 4 ,  5  and  10 ). As will be described in greater detail below, each beacon shot  62 ,  64  and  66  should produce a large radial vector pointing towards the solenoid. The radial vector is the component of the magnetic field  38  which is perpendicular to the solenoid  36 . In this regard, it will be noted that the shape of the magnetic field  38  is largely toroidal and the part of the field having a large radial component lies above and below the solenoid  36  as shown by arrows  68 . Conversely, the part of the magnetic field  38  alongside the solenoid  36  has flux lines parallel to the longitudinal axis of the solenoid and, therefore, has a large axial component and a small radial component as indicated by arrows  70 . 
     The extracted radial magnetic field vector acts as a pointer to the solenoid  36 . The radial magnetic field vector is obtained by transforming the raw vectors from the sensor pack  28  of the probe  20  as though the probe&#39;s coordinate system (the PCS) was oriented to the solenoid  36  and the grid. 
     Irrespective of the actual orientation of the probe  20 , the processor  32  mathematically counter-rotates each sensor output so it measures the field  38  as though the probe  20  were rolled around its axis and inclined so the X sensors of the probe  20  are parallel with the longitudinal axis of the solenoid  36 . If the solenoid  36  were perfectly vertical then the X sensor would be pointing straight up indicating 1G, the Y sensor would be horizontal and perpendicular to the horizon therefore showing 0G and the Z axis rotated to north on a grid coordinate system (GCS). 
     By performing this manipulation, the Y, Z magnetometers (virtually rotated as a result of the transformation) of the sensor pack  28  of the probe  20  will “see” only the radial component  68  of the magnetic field  38  of the solenoid  36  while the virtual X sensor will see only the axial component  70  of the magnetic field  36  of the solenoid  36 . Therefore, to find the radial component  68  of the magnetic field  36 , the transformation that performs these rotations is applied and Y, Z vectors are obtained. Considering that the horizontal vectors will be rotated to the grid, i.e. the virtual Z axis will be pointing north, then the radial component will be oriented in the GCS in the horizontal plane. 
     In any set of beacon readings, or shots,  62 ,  64  and  66  there will be one less fix than the number of shots taken, so for example, the three beacon shots  62 ,  64  and  66  (obtained from six pole shots) will yield two 2-shot fixes  72 ,  74  (which is one multi-shot fix) as shown in  FIG. 5 . 
     Each fix  72 ,  74  processes shots in pairs—so fix  1  contains shots  1  and  2 , fix  2  contains shots  2  and  3 , fix  3  contains shots  3  and  4  etc. The exceptions are the first shot in the first fix and the last shot in the last fix. This means that there are actually 2*(n−1) shots in total, with common points that may not be exactly aligned with each other as shown at  76  and  78  in  FIG. 5 . The two common points  76  and  78  are averaged so that there are the same number of points as the number of shots taken. Before a point is used however it must pass the misalignment test described below or it is rejected. The misalignment test operates as follows:
         Each segment  80  ( FIG. 4 ) is independently derived and if all measurement data were entirely accurate then each segment would fit seamlessly on to the next without aberration. However, this is usually not the case as the beacon&#39;s magnetic field measurements can be noisy—especially if the measured flux density of the radial component of the field is below approx 100 nt. Thus, each vector is checked for contiguous spatial alignment from each shot to the next, i.e. the system ranks the common point between two 2 point fixes in order of the magnitude of their misalignment.   Any angular deviation between corresponding survey shots (shown, for example, at  82 ,  84  and  86 ) and beacon shots greater than 4 deg is considered unacceptable. If this condition exists, then the processor  32  rejects the beacon point that caused the problem. If a point is rejected, then the next pair is used, e.g. if point  3  is rejected from the series s 1 , s 2 , s 3 , s 4 , fixes f 1  (s 1 , s 2 ), f 2  (s 2 , s 4 ) will remain and then, after averaging the common point s 2  (s 2   (fix1) +s 2   (fix2) )/2, the final fix (s 1 , s 2 , s 4 ) is obtained.   Each permutation containing from 3 to 8 shots is then checked for best fit contiguous spatial alignment against the corresponding survey segment  88 .   If found to be within acceptable limits, the survey to beacon shot misalignment distances are averaged for each permutation and contribute to a weighting factor which is used to determine a cluster position in a ranking order. The weighting factor is stored as a single weighted number then enumerated in a list. The list is sorted in order of the least misaligned to the most misaligned (best first-worst last). The processor  32  presents the list to the user as a set of selectable solutions as shown in  FIG. 12  of the drawings. However, the system  10  will default to the best solution, i.e. the solution with the least survey to beacon misalignment.       

       FIG. 5  shows a simple example using the three beacon shots  62 ,  64  and  66 . As described above, there are two fixes  72  and  74  and fixes  1  and  2  produce slightly different displacements  76  and  78 . To resolve this, the two displaced shots are averaged and the result is shown as the shot  64 . This yields three points which reduces the cluster back to the same number of beacon shots that were actually taken. Although displaced (due to systematic errors), it is to be noted that the segment  80  of beacon shots lines up closely in shape and direction with the segment  88  of survey shots calculated to interpolate the same points. 
     As described above, the dotted trajectory line  42  represents the beacon ranging run. The points  82 ,  84  and  86  represent interpolated survey points along the conventionally surveyed trajectory  42  that are at exactly the same measured distance in the hole  12  as each of the beacon points, e.g. points p 1 , p 2  and p 3  were ranged when the drill string  22  was at md=1210 m, 1216 m and 1222 m along the hole  12  respectively. Theoretically, the survey shots  82 ,  84  and  86  should exactly overlie the beacon shots  62 ,  64  and  66 . The fact that they don&#39;t means that there are errors. It may be assumed that the errors are in the survey data. The errors are unlikely to be in the beacon shot cluster as they pass the fidelity checks. 
     The processor  32  could find the coincident survey points by either using a process of interpolation using a minimum curve algorithm to calculate the coordinates of a point that is in between two known points. Another method of obtaining the survey points is by reversing the process of earth field filtering by isolating and using the earth&#39;s magnetic field instead of the magnetic field  38  of the solenoid  36 . 
     The processor  32  determines the position in the horizontal plane of the probe  20  with respect to the beacon  18 . This is implemented by making magnetic field vector measurements while the solenoid  36  is energized in each pole state as will be described in greater detail below. Accumulated position measurements derived from the survey are compared with the positions derived from beacon. Any deviation component is assumed to be an error and is quantified. 
     The survey points are calculated using the following equations:
 
 G   (total) =√{square root over ( G   .x   2   +G   .y   2   +G   .z   2 )}  Equation 10
 
Inc=tan −1 ( G   .z /(√{square root over ( G   .x   2   +G   .y   2 )})  Equation 11
 
 G   (roll) =tan −1 (− G   .z   /G   .x )  Equation 12
 
 M   (total) =√{square root over ( M   .x   2   +M   .y   2   +M   .z   2 )}
 
 M   (Azimuth) =tan −1 (( M   .y   *G   .x   −M   .x   *G   .y )/( M   .z   ·G   (total)   2   −M   .x   ·G   .x   ·G   .z   −M   .y   ·G   .y   ·G   .z   −M   .z   ·G   .z   2 ))  Equation 13
 
 M   (dip) =tan −1 ( I/K )  Equation 14
 
with
 
 I=M   .x   *G   .x   +M   .y   *G   .y   +M   .z   *G   .z   Equation 15
 
 J=α   (total)   ·G   (total)   Equation 16
 
 K= √{square root over ( J   2   −I   2 )}  Equation 17
 
where
 
G (total) =earth gravity.
 
Inc=Inclination of the survey tool relative to the vertical
 
G (roll) =The radial orientation of the probe (number of degrees of rotation around its longitudinal axis). The datum i.e. the high side of the probe is determined by noting the direction of the G vector which is always pointing toward the center of the earth.
 
M (total) =Total magnetic flux density in nano-teslas
 
M (Azimuth) 0-360 degrees clockwise from magnetic north
 
M (dip) =Dip of earth field relative to the horizon
 
     There are two kinds of errors that require correction: 
     Azimuth Error 
     Azimuth, or horizontal angular, error  52  is the difference in azimuth between the conventional survey segment  88  and the beacon segment  80 . Once this error  52  has been determined, the surveyed trajectory  42  can be adjusted by adding the azimuth error to every point in the trajectory  42  or by rotating all points using a geometrical transformation. Azimuth error is in the horizontal plane and manifests as accumulating horizontal position error tracing an arc pivoting around the entry collar. It can be caused from unknowns such as magnetic earth field perturbations, both global and local, sensor misalignments, running gear and rod string interference etc. Because the target is a long vertical formation, it is not necessary to correct for verticality errors. Also, the resolution of the accelerometers of the sensor pack  28  is much higher compared with the magnetometers, typically in the order of +−0.1 deg. This only translates to a meter or so at &gt;1000 m horizontal displacement. 
     Baseline Displacement Error 
     Baseline error accumulates along the baseline  44  in a backward or forward direction as shown, for example, at  54  in  FIG. 2  of the drawings. Baseline error will have many sources including rod stretch (or rod miscount) but in an operation where the drill hole  12  pitches up from almost vertical to almost horizontal then a very large component will be due to inclination errors accumulating in the vertical to inclined attitude section of the well. This is the catenary section  56  at the beginning of the trajectory  42  in  FIG. 3  of the drawings. 
     To quantify the azimuth error  52  and the baseline displacement error  54 , the processor  32 , firstly, compares the beacon point cluster with the conventional survey point cluster. To enable this to be done, it is required that the beacon shots  62 ,  64  and  66  are taken at a known measured depth in the trajectory  42  (typically at a point where the probe  20  communicates to the processor  32  that it is in the field  38  of the solenoid  36 ). Once a cluster of beacon shots  62 ,  64  and  66  that pass the misalignment tests have been obtained and the common points normalized, every derived beacon shot is tested against its coincident survey point as defined by their measured depth values. It is to be noted in  FIG. 6  of the drawings that only two beacon shots  62  and  64  are illustrated. This is purely for clarity purposes and the processor, in use, requires at least three acceptable (i.e. satisfying the misalignment criteria) to resolve the errors. 
     BR{right arrow over (v)} 1 ,BR{right arrow over (v)} 2  are the two magnetic beacon&#39;s radial unit vectors each associated with their respective measurement points at the time of the fix. BR{right arrow over (v)} 1 ,BR{right arrow over (v)} 2  are unit vectors having a magnitude of one and therefore convey directional information only. Thus, BR{right arrow over (v)} 1  may be thought of as an arrow pointing toward the beacon  18  at the first location of the fix and BR{right arrow over (v)} 2  as an arrow also pointing toward the beacon  18  but from the second location. 
     Each beacon shot consists of two measurements or pole shots. The first measurement is made by the sensor pack  28  of the probe  20  when it is within the magnetic field  38  of the solenoid  36  while the solenoid  36  is energized with a positive (north) pole on the top and negative (south) pole on the bottom. The second measurement is made by the sensor pack  28  of the probe  20  at the same location relative to the solenoid  36  but with the field of the solenoid  36  reversed, i.e. negative (south) pole on top and positive (north) pole on the bottom. 
     The gravity vector will not fluctuate significantly as the probe  20  is not moved when the measurement procedure is performed at each location (two pole shots are taken at each measurement point to resolve beacon position) so the processor  32  arbitrarily uses the gravity vector from only one of the two pole shots. 
     If the probe  20  is not moved between shots, then 
     
       
         
           
             
               
                 
                   
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                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   18 
                 
               
             
           
         
       
     
     BM and BG are raw magnetic and raw gravity vectors, respectively, taken directly from the probe  20 . They are raw output from the analogue to digital converters (ADC) of the probe  20 . Each ADC serves one of the six orientation sensors in the probe—magnetic (x, y, z) and gravity (x, y, z). 
     In order to remove the influence of the earth&#39;s magnetic field from the measurement, the earth magnetic vector in the second pole shot is subtracted from the earth magnetic vector in the first pole shot. This cancels all unchanged magnetic quantities including earth&#39;s magnetic field. Conversely the two switched magnetic field vectors from the beacon  18  will be additive so that the total intensity of the beacon magnetic field vectors will be twice that of a single measurement as shown by Equation 19 below. 
                     (             BM     (   i   )       .   x                 BM     (   i   )       .   y                 BM     (   i   )       .   z           )     =       (     1   /   2     )     ⁢     (       (             M     (   i   )       .   x                 M     (   i   )       .   y                 M     (   i   )       .   z           )     -     (             M     (     i   -   1     )       .   x                 M     (     i   -   1     )       .   y                 M     (     i   -   1     )       .   z           )       )               Equation   ⁢           ⁢   19               
Conversely to expose the earth field:
 
     
       
         
           
             
               
                 
                   
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                   Equation 
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                   20 
                 
               
             
           
         
       
     
     As described above, the system  10  only uses the radial component  68  of the magnetic field  38  of the beacon  18 . To extract the radial component  68 , the measured field is transformed into the coordinate system of the solenoid  36 . To enable this to be done, it is necessary to know the attitude of the solenoid  36  in the borehole  14  in order to be able define a geometric transformation matrix. 
     The attitude and roll angle of the probe  20  also need to be taken into account. To do so, a 3D transform, S, starting with the attitude of the solenoid  36  needs to be constructed. S could be constructed either using the direction vector of the solenoid  36  or by multiplying two separate rotation matrices (azimuth and inclination of the solenoid  36 ). For example, one could start with +z axis that is oriented to point positive north. The +z axis is first rotated it around the inclined direction (if it is) of the solenoid  36 . Then, the +z axis is rotated again around the Y axis by (INC-90). 
     To find a transform matrix, T, matrix S must be multiplied by three other matrices being PR (probe roll), PI (probe inclination) and then PA (probe azimuth) to give:
 
 T=PR*PI*PA*S   Equation 21
 
where
 
S is the composite rotation matrix of the solenoid  36  and is the same as T only the roll matrix, PR, is not relevant for the solenoid;
 
PA is the azimuth rotation matrix of the probe  20 ;
 
PI is the inclination rotation matrix of the probe  20 ; and
 
PR is the roll rotation matrix of the probe  20 .
 
     This rotates the sensor outputs so that the probe&#39;s X sensor axis is virtually aligned with the longitudinal axis of the solenoid  36  (which may be vertical). First, the probe  20  must be rotated around its Z axis (rolled) so in effect the Y sensor is pointing horizontally and the X sensor is pointing straight down so gravity is felt only on the X, Z sensors with zero G on the Y axis sensor. Then, the coordinate system should be rotated up by the same amount that the probe is inclined. Finally, the coordinate system should be rotated to grid north. The simplest example would be if the solenoid  36  were vertical and the probe  20  were horizontal (90 deg inclination) with the roll orientation of 0 (oriented toward high side) and moving due north. In that case, T would be an identity matrix. 
     The orientation vector of the probe  20  would look like PG below if it were rolled to its high side around its Z axis which would make the Y axis of the probe  20  parallel with the horizon and then rotated around its Y axis until Z is also parallel with the horizon. In this configuration, the accelerometers of the sensor pack  28  on the Y and Z axes will read 0 G force and therefore the Y axis accelerometer would read the total 1G. 
     PG[1 0 0] 
     It is necessary to transform the vectors from the probe coordinate system (PCS) (also referred to as the sensor coordinate system (SCS)) by rotation using Equations 22 23 below. Points are rotated using Equations 22, 23 and 24 below. Since the calculated heading of the probe  20 , is already known, the following general rotation functions can be used:
 
 BRv 2 .y′   (GCS)   =BRv 2 .x   (SCS) *sin( Az )+ BRv 2 .y   (SCS) *cos( Az )  Equation 22
 
 BRv 2 .x′   (GCS)   =BRv 2 .x   (SCS) *cos( Az )− BRv 2 .y   (SCS) *sin( Az )  Equation 23
 
where Az=Probe  20  Magnetic Heading+Declination
 
     When transforming a segment of two or more points using the above transformations, the first point is translated to the origin and all other points translated equally so BP 1   (SCS) =[0 0] before performing the rotation in Equation 24 below.
 
 BPn   (SCS)   =BPn   (SCS)   −BP 1 (SCS)   Equation 24
 
Sometimes it may also require, after performing the transformation, that:
 
 BPn   (SCS)   =BPn   (SCS)   −BP 1 (SCS)   Equation 25
 
     As shown most clearly in  FIG. 9  of the drawings, in order to reconstruct the true beacon fix geometry, it is necessary to find scalars s and tv. A convenient way of doing this is to first perform a temporary rotation using a transform constructed from BR{right arrow over (v)} 1 ′ so that BR{right arrow over (v)} 1 ′ becomes the X axis of a temporary coordinate system. This is done by taking the triangle defined by vertices BP 1 ′, BP 2 ′ and the beacon B and rotating it into the X axis and translating P 1  to the origin to give: 
                   Equation   ⁢           ⁢   22   ⁢           ⁢   and   ⁢           ⁢   23                             p   ⁢           ⁢     1   ″       =     [         0       0         ]             Equation   ⁢           ⁢   26                 p   ⁢           ⁢     2   ″       =     A   ⋆     (       BP   ⁢           ⁢     2     (   GCS   )     ′       -     BP   ⁢           ⁢     1     (   GCS   )     ′         )               Equation   ⁢           ⁢   27                       ⁢   A                             where   ⁢           ⁢   A     =     (           BR   ⁢           ⁢   v   ⁢           ⁢       1   →         (   GCS   )     .   x     ′             BR   ⁢           ⁢   v   ⁢           ⁢       1   →         (   GCS   )     .   y     ′                   -   BR     ⁢           ⁢   v   ⁢           ⁢       1   →         (   GCS   )     .     y   ′       ′             BR   ⁢           ⁢   v   ⁢           ⁢       1   →         (   GCS   )     .     x   ′       ′             )             Equation   ⁢           ⁢   28               
In A above, v 1 ′ is the unit vector pointing to the beacon but rotated into the GCS i.e. BRv 1 ′ (GCS) . It is also to be noted that there is a transposition of y and x between the rows in A.
 
To find s we already know p 2 ″ from above and
 
     
       
         
           
             
               
                 
                   
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                   29 
                 
               
             
             
               
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                   31 
                 
               
             
           
         
       
     
     However, because the segment has been translated to a temporary origin, i.e. p 1 ″=[0 0], and v{right arrow over ( 1 )} is rotated to the x axis, i.e. v{right arrow over ( 1 )}″.x=1, tv can be simplified as follows:
 
 tv=s* {right arrow over (v2)}″ .x   −p 2″ .x   Equation 32
 
     Because, s, {right arrow over (v 2 )}″ .x  and p 2 ″ .x  have already been calculated, tv can be determined. 
     A segment as defined by the minimum curvature algorithms is created using Equations 1 to 5 above to compare the survey data with the beacon fix to establish the systematic errors. 
     In  FIG. 7  of the drawings, the horizontal radial vectors BRv 1 ′ (GCS)  and BRv 2 ′ (GCS)  are BRv 1   (SCS)  and BRv 2   (SCS)  rotated or transformed to align with the grid coordinate system by an amount equal to the heading of the probe  20  in GCS but relative to the field generated by the beacon  18 . In order to differentiate between survey measurements and beacon measurements below, the point or vector in question is prefixed with S and B respectively. Thus, for example, BP 2 ′ (GCS)  is in GCS coordinates but relative to the beacon whereas SP 2 ′ (GCS)  is in GCS coordinates but relative to the survey. It must also be borne in mind that the survey accumulates errors relative to the entry collar  40 . Not shown in  FIG. 7  are vectors BRv 1   (SCS)  and BRv 2   (SCS)  which point to the beacon from the raw survey sensor data but not fixed to the grid. Bv 3 ′ (GCS)  is the straight path measured between P 1  and P 2  relative to the beacon  18 . 
     Radial vectors BRv 1   (SCS)  and BRv 2   (SCS)  point to the beacon  18  with respect to the longitudinal axis of the probe  20 . In  FIG. 7 , vectors BRv 1 ′ (GCS)  and BRv 2 ′ (GCS)  are transformed from BRv 1   (SCS)  and BRv 2   (SCS) . Vectors BRv 1   (SCS)  and BRv 2   (SCS)  are each individually rotated by the amount dictated by the azimuthal heading of the probe  20  in the horizontal plane. This rotates the vectors so they are pointing in a direction relative to the grid rather than to the probe  20  which itself could be pointing anywhere. Because of this, it is necessary to look at the geometry of the system  10  in terms of the fixed grid, i.e. it must be independent of the heading of the probe  20 . If, for example, a beacon survey were taken at p 1  and the probe&#39;s heading was 275 deg GCS and a heading of 265 deg GCS at P 2  then this would clearly add a 10 degrees rotational discrepancy to BRv 2   (SCS)  in addition to the change in angle due to the translation (displacement from one point to the next). Therefore,
 
 SRv 3′ (GCS)   =BP 2′ (GCS)   −BP 1′ (GCS)   Equation 33
 
     It is assumed that:
 
 SRv 3′ (GCS)   =BRv 3′ (GCS)   Equation 34
 
     This is a reasonable assumption to make as the errors introduced by the survey have accumulated over a great distance but they will be insignificant over the small distance measured over a fix v 3 ′ (SCS) =Bv 3 ′ (SCS)    
     It is known in which directions BRv 1 ′ (GCS) , BRv 2 ′ (GCS)  and v 3   (GCS)  are pointing in GCS. The processor  32  now needs to scale BRv 1 ′ (GCS) , BRv 2 ′ (GCS)  by the calculated scalars tv and s, respectively. Once the scalars have been applied and since the position of the target  14  is already known to a high degree of certainty in absolute GCS terms, it is possible to anchor the scaled vectors −BRv 1   (GCS)  and −BRv 2   (GCS)  to the target  14 . Since the scaled vectors are pointing in exactly the opposite direction −BRv 1   (GCS)  will point back to Bp 1 ′ (GCS)  and −BRv 2   (GCS)  will point back to Bp 2 ′ (GCS) . To find p 1 ′ (GCS) , it is necessary to translate the scaled vectors to the known beacon point and invert the scaled vectors to provide:
 
 p 1′ (GCS)   =Bp   (GCS)   −tv*BRv 1 (GCS)   Equation 35
 
 p 2′ (GCS)   =Bp   (GCS)   −s*BRv 2 (GCS)   Equation 36
 
where
 
p 1 ′ (GCS) , p 2 ′ (GCS)  are the final recalculated positions; and
 
Bp (GCS)  are the target beacon coordinates in GCS
 
     In order to determine the difference in angle and position between the surveyed point and the ranged point, the processor  32  first calculates the centra of the beacon shot clusters and the centra of the equivalent survey point clusters. Angular error can be found by applying Equation 37 below. After the angular error correction has been applied to the trajectory, either by use of an appropriate transform or by simply adding the error to the azimuth parameter, both beacon shots and survey shots should line up in angle but not necessarily in baseline displacement. Displacement is calculated by simply subtracting as shown in Equation 38 below. 
                     Δ   ⁢           ⁢   Angle     =       tan     -   1       ⁡     (       B   ⁢     1   n     ⁢       ∑     l   =   1     n     ⁢     a   l         -     S   ⁢     1   n     ⁢       ∑     l   =   1     n     ⁢     a   l           )               Equation   ⁢           ⁢   37               
where B indicates the cluster of beacon shots and S indicates the cluster of equivalent survey derived shots at the same location.
 
ΔDisplacement= CSpt 2 ′−CBpt 1′  Equation 38
 
where
 
CS is the centrum point of the cluster of survey derived shots; and
 
CB is the centrum point of the cluster of beacon shots.
 
     Once the angular error and the baseline displacement error have been calculated, the processor  32  re-calculates the trajectory  46  which the drill string  22  is now to follow. Thus, once the new trajectory  46  has been calculated, the drill string  22  is withdrawn along the lateral hole  12  towards the entry collar  40 . The processor  32  indicates to what position the drill string  22  must be withdrawn. This is communicated to the operator in a discernible manner, for example, by the use of a lighting arrangement. A red light indicates that the drill string  22  needs to be withdrawn and the light remains red until the new pull back position has been reached. At this position, the light turns green indicating that drilling along the new trajectory  46  can commence. 
     It is therefore an advantage of the invention that only one pass of the target borehole  14  needs be made by the drill string  22 . Once the errors have been calculated, the second trajectory should result in an intersection of the borehole  14 . This considerably reduces the amount of time and effort required to intersect the borehole  14  as, in the past, numerous approaches to a borehole have needed to be made in order, eventually, to intersect the target. Thus, the cost of intersecting the target using the system  10  is considerably reduced. This has major cost benefits and time benefits for an operator of the drill string  22 . 
     Additionally, the system  10  is simple to operate as movement of the magnetic beacon is not required in order to develop an adjusted trajectory. The system  10  is largely implemented in software so no hardware modifications need be made to existing drill strings  22 . Once again, this has resultant cost benefits. 
     It will be appreciated by persons skilled in the art that numerous variations and/or modifications may be made to the invention as shown in the specific embodiments without departing from the spirit or scope of the invention as broadly described. The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive.