Patent Publication Number: US-7219749-B2

Title: Single solenoid guide system

Description:
BACKGROUND OF THE INVENTION 
   The present invention relates, in general, to a method and apparatus for tracking and guiding the drilling of generally horizontal boreholes below the earth&#39;s surface, and more particularly to an improved system and apparatus for tracking a borehole being drilled generally horizontally under an obstacle such as a river, where access to the surface of the ground immediately above the borehole is difficult. Measurements of borehole location and direction are made for use in guiding the borehole to a specified location. 
   Horizontal directional drilling techniques are well known, and have long been used to drill boreholes which cross under areas where trenching is not permitted or is impractical. For example, such techniques are used to drill boreholes under manmade or natural obstacles such as rivers, lakes, or other bodies of water, under highways, airport runways, housing developments or the like. These boreholes may be used to position pipelines, underground transmission lines, communications lines such as optical fibers and other utilities, for example, and often must be drilled within defined areas, must travel long distances, and must exit the ground at predetermined locations. The borehole typically is tunneled from an entry point on the earth&#39;s surface at the near side of an obstacle, travels under the obstacle, and exits the ground at a predetermined location on its far side. In drilling such boreholes it is important to maintain them on a carefully controlled track following a prescribed drilling path, or proposal, for often the borehole must remain within a right of way as it passes under the obstacle and its entry and exit points on opposite sides of the obstacle must often be within precisely defined areas. 
   Conventional directional drilling apparatus for drilling such boreholes commonly incorporates a steering tool which measures the borehole inclination, magnetic azimuth, and tool roll angle with respect to the earth&#39;s gravity and magnetic field at each station where measurements are made. The borehole coordinates are computed and tabulated from these steering tool data as a function of the measured distance along the borehole, which may be referred to as the measured depth of the steering tool. These borehole coordinates suffer from serious cumulative effects produced by inclination and azimuth determinations made at spaced locations along the borehole, and by the lateral errors generated by conventional borehole surveying techniques. The inherent imprecision of these techniques is the reason for turning to electromagnetic methods for directly determining drill bit location. 
   Prior systems such as those illustrated in U.S. Pat. Nos. 4,875,014 and 3,712,391 provide guidance for the drilling of boreholes, but in some circumstances present problems to the user since they require access to the land above the path to be followed by the borehole. These systems utilize surface grids or other guidance systems on the earth&#39;s surface, but the access they require often is not available. 
   U.S. Pat. Nos. 5,513,710 and 6,626,252 overcome the foregoing problem by providing drilling guidance methods and systems for drilling boreholes under rivers and under obstacles, the &#39;710 patent utilizing a direct current powered solenoid at a known location with respect to the target exit for the borehole, and the &#39;252 patent utilizing two horizontal AC solenoids near a borehole path on the surface of the earth at the far side of the obstacle. 
   The foregoing systems require precise location and orientation of the solenoids used to provide the magnetic fields, and this can be an inconvenience in some circumstances and impossible in others, where there may not be access to an appropriate location for the solenoids or where there is not sufficient time to carry out the required orientation procedure. Thus, there is a need to provide a simple, yet accurate system for detecting and tracking the drill stem used to produce an underground borehole, where a magnetic field source such as a solenoid can be deployed in a body of water, for example, above the path of the borehole being drilled, and where it is not necessary to determine the orientation of the source. 
   SUMMARY OF THE INVENTION  
   In accordance with the present invention, the precise location of a drill bit while drilling under an obstacle such as a body of water is determined by the use of a single solenoid at a known location above the borehole path, wherein the solenoid has an unknown orientation. For example, when a borehole is being drilled under a river, a single solenoid may be carried to an appropriate location in the river above the desired path of the borehole, and the solenoid lowered to the bottom of the river and energized to produce an alternating current magnetic field. The location of the solenoid can be accurately determined, as by triangulation from the shore and from a measurement of its depth below the surface of the river. However, the direction of the axis of the solenoid when it rests on the bottom of the river is unknown. 
   The drill head to be located and tracked is in the borehole beneath the riverbed, and includes standard measurement while drilling (MWD) sensors so that the direction of the drill head with respect to the Earth&#39;s magnetic field and its inclination with respect to the earth&#39;s gravity can be used to determine the direction of the borehole. In addition, the depth of the drill head along the borehole is precisely measured. These measurements allow the location of the drill head with respect to the entry point of the borehole on the near side of the river to be determined only approximately, i.e., to the precision given by the standard methods of integrating MWD measurements of the Earth&#39;s magnetic field and gravity along the borehole. In accordance with the present invention, the precise drill head location can be determined using the apparatus and method described herein. 
   During the initial phase of drilling, the process of the present invention is normally used in conjunction with another borehole tracking process which provides insitu measurement of the relative direction of the Earth&#39;s magnetic field with respect to an “away” direction from a surface reference, defined by land surveys. Accordingly, during an initial phase of drilling, a tracking method such as that disclosed in U.S. Pat. No. 6,466,020, U.S. Pat. No. 6,626,252B1, or U.S. Pat. No. 4,875,014, for example, is used to determine the borehole coordinates precisely with respect to land survey coordinates; i.e., the away, elevation and right distances (aer coordinates), and to determine the relative direction between the local Earth&#39;s magnetic field and the away direction. After this initial phase of drilling, standard tracking methods using the Earth&#39;s magnetic field and gravity measurements in the gne coordinate system along the borehole provide an approximate determination of the location where the present invention is to be used. 
   To provide a precise location of the drill head sensors, and thus the location of the borehole, after the initial phase, in accordance with the invention the solenoid, which is at a known location but at an unknown orientation, is energized and measurements of its field are made at the MWD sensors in the drill head at a first location along the borehole. The drill head is then advanced along the borehole, as by drilling, to a second location and field measurements again are made. The measurements made at these two locations are then mathematically analyzed to determine the location of the drill head relative to the solenoid. Using the precisely known solenoid location, the drill head location can then be related to the overall coordinate system defined by land surveys. 
   The first step in analyzing the measured AC magnetic field data for determining drill head location is the generation of time reference waveforms, which are synchronized with solenoid switching circuitry for controlling the excitation current. The resulting magnetic field data are measured at two locations and the data are signal averaged with respect to these time reference waveforms to evaluate the solenoid magnetic field vector at each location. These magnetic field vectors, together with the known approximate location vector between the measuring locations are used to compute the relative location vector from the solenoid to the measuring locations in the “gravity, magnetic north, east” (gne) coordinate system which is defined by the measuring instruments in the down hole tool. The relative vector from the solenoid to the current drilling location that is found in this way is then transformed from the tool&#39;s gravity, magnetic north, east coordinate system to the land survey system of “away, elevation, right” (aer). This relative drill head location is then readily combined with the land survey defined location of the solenoid to find the location vector of the current drilling location relative to the borehole entry point in the desired land survey coordinates. 

   
     BRIEF DESCRIPTION OF DRAWINGS 
     The foregoing, and additional objects, features and advantages of the present invention will become apparent to those of skill in the art from a consideration of the following detailed description of a preferred embodiment thereof, taken in conjunction with the accompanying drawings, in which: 
       FIG. 1  is a perspective view of a borehole location and guidance system in accordance with the present invention; 
       FIG. 2  is a cross-sectional view of the system of  FIG. 1 ; 
       FIG. 3  is a block diagram of the downhole detector circuitry and uphole computer of the system of  FIG. 1 ; 
       FIG. 4A  illustrates the clock signals controlling the power to the magnetic field generating solenoid; 
       FIG. 4B  illustrates the resulting solenoid current; and 
       FIG. 5  is a schematic diagram showing the relationship between the location vectors and angles in the aer and gne coordinate systems and the direction of the solenoid with respect to the gne system. 
   

   DESCRIPTION OF PREFERRED EMBODIMENT 
   One embodiment of the apparatus utilized in the method of the present invention for drilling a borehole under an obstacle is generally illustrated at  10  in  FIGS. 1 and 2 . A borehole  12  is illustrated as being drilled using an industry standard drilling motor in a drill head  14  connected to a drill rig  16 . Drilling under an obstacle such as river  18  involves drilling along a planned path  20  at a depth of, for example, 20 meters below the bed  22  of the river to a planned exit location, such as a borehole punch-out point  24 , on the far side  26  of the river. This exit location may be 1,000 to 1,500 meters away from a borehole entry point  28  on the near side  30  of the river. 
   During the initial stage of drilling the borehole  12 , the drill enters the earth at the entry point  28 , and progresses along the planned path  20  under the guidance of a survey system  32  of the type described, for example, in U.S. Pat. No. 6,466,020, the disclosure of which is hereby incorporated herein by reference. This system includes a loop  34  of wire on the near side  30  of the river, and at least one loop  36  of wire on the far side  26  of the river. As described in the &#39;020 patent, the surface elevation, northing and easting coordinates of multiple points specifying the surface loop configurations for each of loops  34  and  36  are determined using standard land surveying techniques. Logical reference points for each of the loops are the specified borehole entry point  28  and exit point  24  locations associated with each. The entry side loop  34  is powered by a source  38  which may be an alternating current (AC) source or may be a direct current (DC) source which can be turned on and off preferably with reversed current flow polarity, to enable separation of the electromagnetic field generated by the loop from the Earth&#39;s magnetic field. Similarly, the loop  36  is powered by a current source  40  which may be an alternating current source or a direct current source that can be turned on and off, preferably with reversed current flow polarity. 
   The borehole  12  is drilled using drilling apparatus which includes a drill stem  42  of precisely known length, a control unit  44  at the entry end for controlling the direction of drilling, and a drill head  14  at the down hole end of the drill stem  42 . The drill head includes a drilling bit  46  driven by a drill motor  47 , and an electronic steering tool  48  together with conventional apparatus for communicating steering tool measurements to the Earth&#39;s surface. Steering tools, which are standard to the drilling industry, normally incorporate three Earth&#39;s magnetic field sensors and three accelerometers. Traditionally, the axial gravity or the axial magnetic field vector component sensors are designated as z axis sensors, while those measuring vector components perpendicular to the borehole axis are perpendicular to each other and are designated as x and y sensors. These sensors are used to determine the drilling direction and the roll angle of the “tool face” for changing the direction of drilling. 
   In accordance with one form of the present invention, in the initial phase of drilling at the near side of the river  18  or other similar obstacle, the entry magnetic field source loop  34  is energized by a reversible direct current from source  38 . This excitation produces a corresponding magnetic field in the Earth in the region of the steering tool, and x, and y, and z electromagnetic field components generated by the loop at a measuring station are found by making two sequential measurements with known positive and negative currents. Usually, currents of approximately 50 amperes in each direction are appropriate. The apparent Earth field values are fractionally weighted by the positive and negative current values, with the sum of these values giving the normally measured Earth field x, y and z components, and the difference of these fields giving the x, y and z electromagnetic components. This method of separating the Earth field and electromagnetic field is simple, well known and straightforward and can be used with any standard steering tool. 
   The location of the drill head is determined by the process described in detail in the &#39;020 patent, and the drilling of the borehole is guided until it starts to pass under the obstacle  18  and the system  32  loses its effectiveness. At this point, the system and method of the present invention is utilized to guide further drilling of the borehole under the obstacle. To accomplish this, a magnetic field source such as solenoid  50  is located on the riverbed  22  ( FIG. 2 ) generally above the proposed path  20  of the borehole. The solenoid is energizable to produce an alternating current magnetic field that will provide the information needed to guide the drill head  14  as it moves along the path  20  under the river. The electronic steering tool, or instrument package  48  ( FIG. 3 ) incorporates a three-component accelerometer  52  to measure the direction of gravity and a three-component magnetometer  54  to measure alternating magnetic fields. The instrument package preferably is mounted on the drill stem  42  just above the drill head motor  47  and may or may not be part of a conventional measurement while drilling (MWD) package. 
   As illustrated in  FIGS. 2 and 3 , the solenoid  50 , which may include a conventional core  56  and coil  58 , may be suspended by a cable  60  from a floating platform  62  such as a boat, barge or the like, on the surface  64  of river  18 . The location of the solenoid is measured by, for example, conventional surveying equipment  66  on the shore of the river using a marker pole  68  on barge  62  so that the solenoid can be located in azimuth and distance with respect to the location of the entry point  28 . By measuring the length of cable  60  and the vertical distance  70  between the connection point  72  of the cable and the horizontal location of the surveying equipment, the elevation of the solenoid can be determined with respect to entry point  28 . However, because the shape or slope of the riverbed  22  is unknown, neither the inclination nor the direction of the axis  74  of solenoid  50  is known. 
   The solenoid  50  illustrated in  FIG. 3  may have, for example, a 23 kilogram laminated core  56  that, in a preferred embodiment, is 1.25 meters long. To provide the desired magnetic field, this solenoid may require 40 watts of power, for example, and this may be supplied by a portable power supply  76  which may be a small, 12 volt lead acid battery  78  connected to a polarity reversing FET (field effect transistor) switch circuit  80  connected across the solenoid winding  58 . The direction of electric current flow in the solenoid winding is periodically reversed by the switch circuit to produce a reference square wave with a precise cycle period of 0.5 seconds derived from clock signals  82  ( FIG. 4A ) generated by a crystal oscillator  84  having a frequency that is precise to a few parts per million. The solenoid current vs. time waveform illustrated at  86  in  FIG. 4B  produces a magnetic dipole field  88  of alternating polarity. Although the principles of physics governing the behavior of the magnetic fields used in the analysis to be described are those appropriate to time independent magnetic fields, it is desirable to repeatedly reverse the direction of current flow in the solenoid to allow precise separation of the solenoid field from the Earth&#39;s magnetic field and from instrument and magnetic field noise. The method is thus readily adapted to manually switching the field of a solenoid and appropriately analyzing the results. 
   The electromagnetic field  88  generated by the solenoid ( FIG. 3 ) is detected by the downhole instrument package  48 . This package is connected by way of a borehole telemetry link  90  to the uphole drilling control unit  44  located at the drilling rig  16  on the earth&#39;s surface. The control unit  44  includes a computer  92  for processing data received from the downhole electronics and a controller  94  ( FIG. 1 ) for operating the drill. An instrument power supply and telemetry circuit  96  is connected by way of link  90  to supply power to the downhole measuring instruments and to permit them to transmit data uphole and to convert the data to computer input signals on line  98 . The power supply link  90  may be a wire inside the drill stem  42  leading to the downhole instrument package  48 . 
   The package  48  ( FIG. 3 ) includes the three-vector component magnetometer  54  and the three-vector component accelerometer  52 , each of which generates output signals with respect to an XYZ set of axes. The Z axis of the instrument package  48  is aligned with the axis of the borehole  12  being drilled, and the perpendicular X and Y axes have a known orientation alignment to the drill face; that is, to the direction of a conventional bent housing in the drilling motor which controls the direction of drilling. Direct current is received from the power supply  96  on the surface to power the instruments. The magnetometer AC outputs are passed through band pass filters and amplifiers  100  and are multiplexed with the magnetometer DC outputs and the accelerometer outputs at a multiplexer  102 , where the signals are converted from analog to digital form and put into a form suitable for telemetry to the surface. The timing for digitization and telemetry is generated by a downhole clock  104  controlled by a quartz crystal whose frequency is precise to a few parts per million. 
   In accordance with the invention, measurements are taken at two locations along the borehole  12  in order to determine the actual path of the borehole being drilled. Thus, for example, a first measurement is taken at a first position generally indicated at  110  in  FIG. 2 , and thereafter the drill stem is advanced (for example, by drilling) along the borehole to a position indicated at  112 . After drilling has been stopped for positioning the drill head at each of the measurement stations  110  and  112  along the proposed borehole path  20 , the solenoid  50  is energized from the source  76 , which may be located on the platform  62 , to produce the reversing field  88 . This field is detected by magnetometers  54  and the resulting output signals from the magnetometer are sampled by multiplexer  102  and are transmitted uphole. A few minutes of data are recorded, as indicated at  114  in computer  92 , and data files are generated at  116 . The drill head is then moved to the second location  112 , the solenoid  50  is again energized to create a reversing field which is detected by magnetometers  54 , a second set of data are received at  114 , and a second set of data files  116  is generated. During each set of measurements the downhole multiplexer circuitry  102  also sequentially samples the output voltages of the accelerometers  52  at fixed time intervals and telemeters the results to the surface computer  92 , which receives the gravity measurements at  118  and creates a data file  120 . Measurements of the Earth&#39;s field are also made by magnetometers  54 , are sampled by multiplexer  102 , are transmitted uphole to computer  92 , where the data is received at  122  and a data file is created at  124 . The relative time at which each measurement is made is precisely preserved in the data files by the position it has in the serial data stream being telemetered. 
   Data Acquisition and Processing 
   After drilling has been stopped at the first measurement station  110  along the proposed borehole path, the solenoid  50  is energized as described with respect to  FIG. 3 . The resulting reversing field  88  with an alternating polarity component is detected by magnetometers  54  and the resulting output voltages are transmitted up hole by way of multiplexer  102 . The AC field measurements are separated at  114 , a few minutes of data are recorded, and an AC field data file is recorded at  116 . The earth&#39;s field measurements are separated at  122 , and the earth&#39;s field data is recorded at file  124 . During these measurements the down hole multiplexer circuitry  102  also sequentially samples the output voltages of the accelerometers  52  at fixed time intervals and telemeters the results to the surface computer  92 , which separates the gravity measurements at  118  from the Earth&#39;s field measurements and the AC field measurements, and gravity data is recorded at file  120 . 
   The computer  92  generates from the gravity data in file  120  a 3-row, single column matrix gxyz with elements gx, gy and gz, which are the representation of the measured gravity g in the xyz coordinate system, and from the Earth&#39;s field data file  124  a 3-row single column matrix of the Earth&#39;s field components Efxyz is generated. From the AC magnetometer measurement data in file  116 , a 3-column matrix h 1  is generated. It has three columns h 1 x, h 1 y, and h 1 z, which are tabulations of the time sequence of the digitized magnetometer measurement data from the solenoid. The matrix h 1  is signal averaged with respect to time to find the solenoid magnetic field vector H 1  at location  110 , i.e., the three vector components H 1 x, H 1 y, and H 1 z. 
   Data taken at a second measurement location  112  along the proposed borehole path are analyzed using a similar procedure to compute the magnetic field vector components H 2 x, H 2 y, and H 2 z of the solenoid&#39;s field and the matrix vector H 2 xyz at the second measurement station. 
   Generation of Reference Signal and Signal Averaging 
   The first part of the digital analysis procedure includes generating in computer  92  a symmetric reference waveform which is time-synchronized with the uphole solenoid source  76  to determine an optimal time shift from the AC field signals recorded at  114  for a given measuring station. The strongest signal of the 3 magnetic field vector components is selected and processed to find an optimal time shift for location  110 . For this purpose, a reference waveform is defined, against which all 3 magnetic field components can be signal averaged. To choose the magnetic field component with the strongest signal, the average square of the three data columns of h 1  is computed, using the MATLAB operation “mean(h 1 .*h 1 ).” The largest of the three numbers found defines the largest vector component of the AC field received, i.e., the column “h 1 max” which is the appropriate column of h 1  from which the time shift between the source clock and the downhole clock is found. The serial telemetry data stream locations assign a time to each of the measurements of h 1  max, and those times are put into a single column matrix called Timeh 1 max. The functional form of the reference wave form to be used is cos(w*t), where w is the fundamental radian frequency of the source, i.e., w=2*pi/SrcdPer, where SrcPer is the source period, i.e., 0.5 sec. 
   A two-column reference test matrix RefTest is defined with the first column being Reftest 1  and the second as Reftest 2 :
 
RefTest1=cos( w *Time h 1max)
 
RefTest2=cos( w *Time h 1max−SrcPer/4))  Eq. (1)
 
   RefTest 1  is a single column matrix evaluating cos(w*t) at values of t equal to the times Timeh 1 max, i.e., the times at which the measurements of h 1 max were made according to the downhole clock. RefTest 2  is a second cosine reference waveform evaluated at times delayed by a quarter time period of the solenoid clock from RefTest 1 .
 
 H maxRef12=[RefTest1 RefTest2 ones(size(Time h 1max))]\ h 1max  Eq. (2)
 
   HmaxRef 12  is a 3-row, 1 column matrix. The first row is the least squares fit of evaluating h 1 max with respect to RefTest 1 , the second row is the least squares fit with respect to RefTest 2 , and the third row is the zero offset of h 1 max. The optimum time shift (TShft) indicated by HmaxRef 12  is:
 
 T Shift=(ScrPer/4)* a  tan 2( H maxRef12(2), H maxRef12(1))  Eq. (3)
 
   All three columns of the data are signal averaged with the time reference matrix to give least squares fits for H 1 x, H 1 y and H 1 z:
 
 H 1 x =cos( w *(Time h 1 x−T shift))\ h 1 x 
 
 H 1 y =cos( w *(Time h 1 y−T shift))\ h 1 y 
 
 H 1 z =cos( w *(Time h 1 z−T shift))\ h 1 z   Eq. (4)
 
   Timeh 1 x is a column matrix of the times at which the h 1 x measurements were made, Timeh 1 y is a column matrix of the h 1 y measurements, and Timeh 1 z is a column matrix of the h 1 z measurements. Since the reference function cos(w*t) used is symmetric with respect to positive and negative values, there is an intrinsic sign ambiguity in the values of H 1 x, H 1 y and H 1 z and in the sign of the magnetic moment m. This ambiguity in the sign will be addressed below. 
   This signal averaging method optimally extracts the time variation of all three components, which is in phase with the single reference signal. The method thus gives no information of the relative phases of the three vector components with respect to each other. Since the further analysis of the fields assumes DC behavior of the fields, finding and including quadrature components, i.e., phase information, has the effect of adding random noise into the analysis and degrading the final results obtained. 
   Fitting the Magnetic Field Measurements to Find Location 
   A linear least squares fitting procedure is used to find the optimum value of the location vector r 1  of the drilling head  14  when it is at location  110  relative to the solenoid  50 , as illustrated in diagram  130  in  FIG. 5 . To apply this method, it is necessary to know at the outset an approximate value of the unit vector direction m 1 Uv of solenoid  50 . This vector can be computed analytically from measurement data at each of the locations  110  and  112 . 
   Start by noting that the approximate value of the location vector r 1  from the solenoid to measurement location  110  is known, since Rsol is known and R 1 , the location vector of the measurement location  110 , is approximately known in the aer (away, elevation and right) coordinate system. Since the angle Aan between magnetic north and the away direction is also known, the representation of r 1 gne in the gne (gravity, magnetic north, east) coordinate system is also known. 
   The general theoretical value for H 1 , i.e., the solenoid field  88  at location  110 , is given by the expression:
 
 H 1 32  ( M mag/(4 *pi*r 1Mag^3))*(3*dot( m 1 Uv, r 1 Uv )* r 1 Uv−m 1 Uv )  Eq. (4)
 
   At the outset, the value of Mmag, the magnitude of the solenoid magnetic moment, is known and the approximate value of the magnitude of r 1  is known. Taking the vector dot product of r 1 Uv and H 1 , the value of the vector dot product dot(m 1 Uv, r 1 Uv) is readily computed to be:
 
dot( m 1 Uv, r 1 Uv )=dot( H 1,  rUv )/( M mag/(8 *pi*r 1Mag^3))  Eq. (5)
 
   The value of dot(m 1 Uv, r 1 Uv) is readily computed from the known approximate value of r 1 Uv and the measured value of H 1 , using their representations in the gne (gravity, magnetic north) coordinate system. The gne representation of the approximate value of r 1 Uv is readily found using the known angle between the away axis and magnetic north Aan using standard means. To find the transformation matrix from the xyz coordinate system of the downhole tool to the gne system we use the measurements of the Earth&#39;s field Efxyz and the gravity gxyz vectors at location  110 . The measured unit vector of the magnetic north direction NUvxyz is:
 
 NUvxyz =( Efxyz −dot( Efxyz, gxyz )* gxyz )/mag( Efxyz −dot( Efxyz, gxyz ))  Eq. (6)
 
   The unit vector in the East direction is given by the vector cross product:
 
 EUvxyz =cross( gxyz, NUvxyz )  Eq. (7)
 
Thus, the transformation matrix converting from the xyz coordinate system the gne coordinate system is:
 
xyztogne=[gxyz′; NUvxyz′; EUvxyz′]  Eq. (8)
 
Thus:
 
 H 1 gne=xyztogne*H 1 xyz   Eq. (9)
 
Thus, a first approximation to the unit vector of the solenoid direction, in the gne system representation, from measurements at location  110  is given by
 
   
     
       
         
           
             
               
                 
                   m 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                   ⁢ 
                   Uvgne 
                 
                 = 
                 
                   
 
                 
                 ⁢ 
                 
                   = 
                   
                     
                       ( 
                       
                         
                           
                             ( 
                             
                               3 
                               / 
                               2 
                             
                             ) 
                           
                           * 
                           
                             dot 
                             ( 
                             
                               
                                 H 
                                 ⁢ 
                                 1 
                                 ⁢ 
                                 gne 
                               
                               , 
                               
                                 r 
                                 ⁢ 
                                 1 
                                 ⁢ 
                                 Uvgne 
                               
                             
                             ) 
                           
                           * 
                           
                             r 
                             ⁢ 
                             1 
                             ⁢ 
                             Uvgne 
                           
                         
                         - 
                         
                           H 
                           ⁢ 
                           1 
                           ⁢ 
                           gne 
                         
                       
                       ) 
                     
                     / 
                     
                       
 
                     
                     ⁢ 
                     
                       ( 
                       
                         Mmag 
                         / 
                         
                           ( 
                           
                             4 
                             * 
                             pi 
                             * 
                             r 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                             ⁢ 
                             
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                               3 
                             
                           
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                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
   
   Measurements made at a second location  112  defined by the vector r 2  from the solenoid  50  to the drill head location are analyzed in the same way to determine H 2 xyz and a first approximation unit vector: 
   
     
       
         
           
             
               
                 
                   m 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2 
                   ⁢ 
                   Uvgne 
                 
                 = 
                 
                   
 
                 
                 ⁢ 
                 
                   = 
                   
                     
                       ( 
                       
                         
                           
                             ( 
                             
                               3 
                               / 
                               2 
                             
                             ) 
                           
                           * 
                           
                             dot 
                             ( 
                             
                               
                                 H 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 gne 
                               
                               , 
                               
                                 r 
                                 ⁢ 
                                 1 
                                 ⁢ 
                                 Uvgne 
                               
                             
                             ) 
                           
                           * 
                           r2Uvgne 
                         
                         - 
                         
                           H2 
                           ⁢ 
                           gne 
                         
                       
                       ) 
                     
                     / 
                     
                       
 
                     
                     ⁢ 
                     
                       ( 
                       
                         Mmag 
                         / 
                         
                           ( 
                           
                             4 
                             * 
                             pi 
                             * 
                             r 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                             ⁢ 
                             
                               Mag 
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                 Eq 
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                   11 
                   ) 
                 
               
             
           
         
       
     
   
   Because of the double valued nature of the TimeShft parameter at each location, the directions of the field derived at each location H 1  and H 2  have an ambiguity of sign, with a corresponding ambiguity in the signs of m 1 Uv and m 2 Uvgne. The sign of m 1 Uv at location  110  is taken as the defining sign and the direction of m 1  gne is assigned to be equal to solenoid direction MUvgne. The sign of H 2  is adjusted by noting whether dot(m 1 Uvgne,m 2 Uvgne) is greater than or less than zero (ideally this dot product should be either +1 or −1). If it is &gt;0 then H 2  is not changed; if it is &lt;0 the sign of H 2  is changed. 
   After making this adjustment, the first approximation to the solenoid direction is taken to be the average of m 1 Uvgne and m 2 Uvgne, i.e.:
 
 mUvgne =( m 1 Uvgne+m 2 Uvgne )/2  Eq. (12)
 
   The angle from magnetic north to the solenoid axis Anm and the angle from g to the solenoid axis Agm are given by:
 
 Anm=a  tan 2( mUvgne (3), mUvgne (2))
 
 Agm=a  tan(sqrt( mUvgne (2)^2+sqrt( mUvgne (3)^2),  mUvgne (1))  Eq. (13)
 
   The final step in the analysis is to do a linear least squares fit to find the best values for r 2 , and the direction of the solenoid unit vector mUvgne. Thus, 5 parameters must be found: 3 for the vector r 2 gne, and 2 for the direction of mUvgne, to be determined from the six component values H 1 gne and H 2 gne. The relationship between r 1 gne and r 2 gne is known from the usual method of borehole surveying using the Earth&#39;s magnetic field and gravity measurements and the along-the-borehole distance R 12  between the locations  110  and  112 . 
   The analysis procedure is to find the values of the parameters defining the solenoid direction, i.e., mUvgne, and the directions of the drill head r 2 gne and r 1 gne relative to the solenoid. As indicated, this analysis is done in the gne coordinate system that is the logical one since it is the Earth&#39;s field magnetometers and the gravity sensors in the sensor tool  48  which define the “local” coordinate system around the solenoid. The vector R 12  connecting locations  110  and  112 , shown in  FIG. 5 , is determined by integrating the measured depth and borehole direction found from Earth&#39;s field and gravity measurements, as is standard in the drilling industry. Thus r 2  is found from:
 
 r 2= r 1+ R 12  Eq. (14)
 
R 12  is a known constant vector, thus differential vectors dr 1  are equal to differential vectors dr 2 .
 
   The five parameters to be determined, the solenoid azimuth angle (a) between magnetic north and the solenoid axis, the solenoid inclination with respect to the gravity direction (b), and the 3 vector components of r 2  (cde), which is the vector between the solenoid location and the second measurement location  112 , referred to as the parameters a, b, c, d, e, will be combined into a 5-parameter column vector abcde. A differential column vector dabcde is the difference between neighboring values of abcde in the usual spirit of differential calculus. All will be done in the gne coordinate system; thus, the gne identifiers will be dropped in the display of the method. Thus:
 
 abcde (1)= Anm 
 
 abcde (2)= Agm 
 
 abcde (3)= r 2(1)
 
 abcde (4)= r 2(2)
 
 abcde (5)= r 2(3)  Eq. (15)
 
   The procedure is to start with the known approximate value of the column vector abcde, i.e., Eq. 15, and to evaluate the theoretical values of the solenoid electromagnetic fields H 1  and H 2  in the vicinity of the value of abcde in a 5-dimensional Taylor expansion. The differential column vector dabcde relating the value of abcde 1  at parameter vector neighboring abcde 0  is:
 
 abcde 1 =abcde 0 +dabcde   Eq. (16)
 
   The measured values of the field in the gne coordinate system are H 1 meas and H 2 meas; they define a six-parameter column vector H 12 meas, i.e.:
 
H12meas=[H1meas; H2meas]  Eq. (17)
 
   Likewise, the theoretical value of the fields H 1  and H 2  define a six-parameter column vector H 12 th, i.e.:
 
H12th=[H1th; H12th]  Eq. (18)
 
   The value of H 12 th at parameter location  0  is designated H 12 th 0 , that at parameter location  1  as H 12 th 1 . The Taylor expansion relating H 12 th 1  to H 12 th 0  can be written as:
 
 H 12 th 1= H 12 th 0+ dH 12 dabcde*dabcde   Eq. (19)
 
following the usual procedures of differential calculus. The derivative matrix dH 12 dabcde has 6 rows and 5 columns. It can be evaluated around the parameter value abcde 0  using the partial derivative expressions (using a “delta” value of 0.001):
 
 dH 12 dabcd (:,1)=( H 12 th ( abcde   0+[0.001 0 0 0 0]′)−   H 12 th ( abcde 0))/0.001
 
 dH 12 dabcd (:,2)=( H 12 th ( abcde   0+[0 0.001 0 0 0]′)−   H 12 th ( abcde 0))/0.001
 
 dH 12 dabcd (:,3)=( H 12 th ( abcde   0+[0 0 0.001 0 0]′)−   H 12 th ( abcde 0))/0.001
 
 dH 12 dabcd (:,4)=( H 12 th ( abcde   0+[0 0 0 0.001 0]′)−   H 12 th ( abcde 0))/0.001
 
 dH 12 dabcd (:,5)=( H 12 th ( abcde   0+[0 0 0 0 0.001]′)−   H 12 th ( abcde 0))/0.001  Eq. (20)
 
   In expression 20 the quantities between the brackets, e.g. (:,1), denote all the rows of column  1  following the MATLAB convention. Between the brackets on the right side of each expression, the quantity between “( 0 )” is taken to follow the standard mathematical convention, i.e., (H 12 th(abcde 0 +[0.001 0 0 0 0])]) means to evaluate H 12 th at abcde 0 +[0.001 0 0 0 0]′. The best “linear least squares” value of the differential column vector dabcde is found by equating the value of H 12 th 1  to H 12 meas. 
   Starting with the approximate value of abcde 0 , a better value abcde 1  is found from:
 
 dabcde=dH 12 dabcde \( H 12meas− H 12 th 0)  Eq. (21)
 
and the new value abcde 1  is then given by
 
 abcde 1 =abcde 0 +dabcde   Eq. (22)
 
This new value of abcde 1  is now used as a new abcde 0  and the process is repeated a few times to produce an optimum value for abcde and thus for the solenoid orientation and drill bit position vector r 2 .
 
   The desired location r 2  of the drill bit with respect to the solenoid, expressed in the gne coordinate system of the land survey, is found from the components of the final value of abcde 1  using the expression:
 
 r 2 gne=abcde 1([ 3 4 5])   Eq. (23)
 
while the desired location of the drill bit R 2 , expressed in the aer coordinate system of the land survey, is found from the components of the final value of abcde 1  using the expression:
 
 R 2 aer=Rmaer+gnetoaer*r 2 gne 
 
 gnetoaer=[ 0 cos( Aan )−sin( Aan );−1 0 0; 0 sin( Aan )]  Eq. (24)
 
   The error in R 2 aer due to imprecision of the direction of the Earth&#39;s magnetic field relative to the away direction is minimal because in practice RSol is much larger than R 2 , the distance between the solenoid and the drill bit. On a 1500 meter river crossing project a typical distance between the entry point and the Solenoid is 750 meters and the distance R 2  is 30 meters or less so that the effect of error in the true value of the Earth&#39;s magnetic field direction used in finding R 2  is reduced by a factor of 25. Thus, a 2-degree difference in the Earth&#39;s magnetic north direction determined during the initial drilling phase and that at the locations  110  and  112  leads to an error of less than 1 meter for location  112 . 
   The foregoing measurements are repeated at additional measuring points along the borehole as the drilling progresses, with the measured relative drill head locations being used to control further drilling beneath the obstacle  18 . When the borehole reaches the far side of the obstacle, represented here by the far side  26  of the river, further drilling of the borehole to the exit point  24  is controlled by measurements of magnetic fields produced by loop  36 , again in the manner described in U.S. Pat. No. 6,466,020, for example. 
   Although the present invention has been described in terms of a preferred embodiment, it will be apparent that modifications and variations may be made without departing from the true spirit and scope thereof. For example, although the process has been described in the context of guiding the drilling of a borehole along a proposed path under an obstacle, it will be understood that it is equally applicable to surveys of existing boreholes. In the latter case, the measuring tool is simply moved along the existing borehole and the measurements are made as described above. If the motion of the measuring tool does not cause the tool to rotate between measuring locations, then it is not necessary to measure the earth&#39;s magnetic field or to measure gravity after the first such measurements are made; the original determination of the orientation and direction of the tool can be used at subsequent locations. Accordingly, the scope of the invention is limited only by the accompanying claims.