Patent Publication Number: US-7725263-B2

Title: Gravity azimuth measurement at a non-rotating housing

Description:
RELATED APPLICATIONS 
     None. 
     FIELD OF THE INVENTION 
     The present invention relates generally to downhole tools, for example, including directional drilling tools having one or more steering blades. More particularly, embodiments of this invention relate to a surveying method in which gravity measurement sensors are utilized to determine a change in borehole azimuth between first and second longitudinally spaced positions in a borehole. 
     BACKGROUND OF THE INVENTION 
     The use of accelerometers in conventional surveying techniques is well known. The use of magnetometers or gyroscopes in combination with one or more accelerometers to determine direction is also known. Deployments of such sensor sets are well known to determine borehole characteristics such as inclination, azimuth, positions in space, gravity toolface, magnetic toolface, and magnetic azimuth (i.e., an azimuth value determined from magnetic field measurements). While magnetometers and gyroscopes may provide valuable information to the surveyor, their use in borehole surveying, and in particular measurement while drilling (MWD) applications, tends to be limited by various factors. For example, magnetic interference, such as from magnetic steel or ferrous minerals in formations or ore bodies, tends to cause errors in the azimuth values obtained from a magnetometer. Motors, stabilizers, and bits used in directional drilling applications are typically permanently magnetized during magnetic particle inspection processes, and thus magnetometer readings obtained low in the bottom hole assembly (BHA) are often unreliable. Gyroscopes are sensitive to high temperature and vibration and thus tend to be difficult to utilize in drilling applications. Gyroscopes also require a relatively long time interval (as compared to accelerometers and magnetometers) to obtain accurate readings. Furthermore, at low angles of inclination (i.e., near vertical); it becomes very difficult to obtain accurate azimuth values from gyroscopes. 
     U.S. Pat. No. 6,480,119 to McElhinney and commonly assigned U.S. Pat. No. 7,080,460 to Illfelder disclose techniques for determining borehole azimuth via tri-axial accelerometer measurements made at first and second longitudinal positions on a drill string. Using gravity as a primary reference, the disclosed methods make use of the inherent bending of the structure between the accelerometer sets in order to calculate a change in borehole azimuth between the first and second positions. The disclosed methods assume that the tri-axial accelerometer sets are spaced by a known distance via a rigid structure, such as a drill collar, that prevents relative rotation between the sets. Gravity based methods for determining borehole azimuth, including the McElhinney and Illfelder methods, as well as exemplary embodiments of the present invention, are referred to herein as Gravity MWD. 
     While the Gravity MWD techniques disclosed by McElhinney and Illfelder are known to be commercially serviceable, there is yet room for further improvement. For example, the physical constraint that the accelerometer sets be rotationally fixed relative to one another imposes a constraint on the structure of the BHA. It would be highly advantageous to extend Gravity MWD methods to eliminate this constraint and thereby allow relative rotation between the first and second accelerometer sets. 
     The Illfelder patent further discloses that the change in borehole azimuth can be determined from borehole inclination and gravity toolface measurements using numerical root finding algorithms, graphical methods, and/or look-up tables. Such methods are readily available and easily utilized at the surface, e.g., via a conventional PC using software routines available in MathCad® and/or Mathematica®. However, it is difficult to apply such numerical and/or graphical methods using on-board, downhole processors due to their limited processing power. This is particularly so in smaller diameter tools which require physically smaller processors (which therefore typically have lower processing power). Furthermore, surface processing tends to be disadvantageous in that it requires transmission of multiple high resolution (e.g., 12 bit) gravity measurement values or inclination and tool face angles to the surface. Such downhole to surface transmission is often accomplished via bandwidth limited mud pulse telemetry techniques. 
     Therefore there also exists a need for a simplified method for determining the change in borehole azimuth, preferably including calculations that can be readily achieved using a low-processing-power downhole processor. 
     SUMMARY OF THE INVENTION 
     The present invention addresses one or more of the above-described drawbacks of prior art gravity surveying techniques. Exemplary embodiments of the present invention advantageously remove the above described rotational constraint between longitudinally spaced Gravity MWD sensors. One exemplary aspect of this invention includes a method for surveying a subterranean borehole. A change in borehole azimuth between first and second longitudinally spaced gravity measurement sensors may be determined directly from gravity measurements made by the sensors and a measured angular position between the sensors. The gravity measurement sensors are typically disposed to rotate freely with respect to one another about a longitudinal axis of the borehole. Relative rotation is accounted via measurements of the relative angular position between the first and second sensors. The change in azimuth is typically processed downhole (in a downhole processor) via a simplified algorithm (simplified as compared to prior art Gravity MWD algorithms). 
     Exemplary embodiments of the present invention may advantageously provide several technical advantages. For example, Gravity MWD measurements in accordance with the present invention may be advantageously made without imposing any rotational constraints between the first and second gravity sensor sets. Elimination of the prior art rotational constraints advantageously provides for improved flexibility in BHA design. For example, in one exemplary embodiment of the invention, a first gravity sensor may be rotationally coupled with the drill string (e.g., in a conventional MWD tool) while the second gravity sensor may be deployed in a substantially non-rotating housing (e.g., a conventional rotary steerable tool blade housing). Such deployments advantageously enable near-bit borehole azimuth measurements to be made free from the effects of magnetic interference. 
     The present invention also advantageously provides for downhole processing of the change in azimuth between the first and second gravity sensor sets. As such, Gravity MWD measurements in accordance with this invention may be advantageously utilized in closed-loop steering control methods. 
     In one aspect the present invention includes a method for surveying a subterranean borehole. The method includes providing a string of downhole tools including first and second gravity measurement devices at corresponding first and second longitudinal positions in the borehole. The first and second gravity measurement devices are substantially free to rotate with respect to one another about a substantially cylindrical borehole axis. The string of tools further includes an angular position sensor disposed to measure a relative angular position between the first and second gravity measurement devices. The method further includes causing the first and second gravity measurement devices to measure corresponding first and second gravity vector sets and causing the angular position sensor to measure a corresponding relative angular position between the first and second gravity measurement devices. The method still further includes processing the first and second gravity vector sets and the angular position to calculate a change in borehole azimuth between the first and second positions in the borehole. 
     In another aspect this invention includes a method for surveying a subterranean borehole. The method includes providing first and second gravity measurement devices at corresponding first and second longitudinal positions in the borehole and causing the first and second gravity measurement devices to measure corresponding first and second gravity vector sets. The method further includes processing downhole the first and second gravity vector sets to calculate a change in borehole azimuth between the first and second positions in the borehole. 
     The foregoing has outlined rather broadly the features of the present invention in order that the detailed description of the invention that follows may be better understood. Additional features and advantages of the invention will be described hereinafter which form the subject of the claims of the invention. It should be appreciated by those skilled in the art that the conception and the specific embodiments disclosed may be readily utilized as a basis for modifying or designing other methods, structures, and encoding schemes for carrying out the same purposes of the present invention. It should also be realized by those skilled in the art that such equivalent constructions do not depart from the spirit and scope of the invention as set forth in the appended claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       For a more complete understanding of the present invention, and the advantages thereof, reference is now made to the following descriptions taken in conjunction with the accompanying drawings, in which: 
         FIG. 1  depicts a drilling rig on which exemplary embodiments of the present invention may be deployed. 
         FIG. 2  is a perspective view of the steering tool shown on  FIG. 1 . 
         FIG. 3  depicts, in cross section, another portion of the steering tool shown on  FIG. 2  showing an exemplary angular sensor deployment in accordance with the present invention. 
         FIG. 4A  depicts a plot of magnetic field strength versus angular position emanating from the magnets in the angular sensor deployment shown on  FIG. 4 . 
         FIG. 4B  depicts a plot of exemplary magnetic field strength measurements made by each of the magnetic sensors in the angular sensor deployment shown on  FIG. 4 . 
         FIG. 5  depicts, in cross section, another exemplary angular sensor deployment in accordance with the present invention. 
         FIG. 6  depicts a perspective view of an exemplary eyebrow magnet utilized in the angular sensor deployment shown on  FIG. 6 . 
         FIG. 7A  depicts a plot of magnetic field strength versus angular position emanating from the magnets in the angular sensor deployment shown on  FIG. 7 . 
         FIG. 7B  depicts a plot of exemplary magnetic field strength measurements made by each of the magnetic sensors in the angular sensor deployment shown on  FIG. 7 . 
         FIG. 8  depicts a bottom hole assembly suitable for use with Gravity MWD embodiments of the present invention. 
     
    
    
     DETAILED DESCRIPTION 
     Before proceeding with a discussion of the present invention, it is necessary to make clear what is meant by “azimuth” as used herein. The term azimuth has been used in the downhole drilling arts in two contexts, with a somewhat different meaning in each context. In a general sense, an azimuth angle is a horizontal angle from a fixed reference position. Mariners performing celestial navigation used the term, and it is this use that apparently forms the basis for the generally understood meaning of the term azimuth. In celestial navigation, a particular celestial object is selected and then a vertical circle, with the mariner at its center, is constructed such that the circle passes through the celestial object. The angular distance from a reference point (usually magnetic north) to the point at which the vertical circle intersects the horizon is the azimuth. As a matter of practice, the azimuth angle was usually measured in the clockwise direction. 
     In this traditional meaning of azimuth, the reference plane is the horizontal plane tangent to the earth&#39;s surface at the point from which the celestial observation is made. In other words, the mariner&#39;s location forms the point of contact between the horizontal azimuthal reference plane and the surface of the earth. This context can be easily extended to a downhole drilling application. A borehole azimuth in the downhole drilling context is the relative bearing direction of the borehole at any particular point in a horizontal reference frame. Just as a vertical circle was drawn through the celestial object in the traditional azimuth calculation, a vertical circle may also be drawn in the downhole drilling context with the point of interest within the borehole being the center of the circle and the tangent to the borehole at the point of interest being the radius of the circle. The angular distance from the point at which this circle intersects the horizontal reference plane and the fixed reference point (e.g., magnetic north) is referred to as the borehole azimuth. And just as in the celestial navigation context, the borehole azimuth is typically measured in a clockwise direction. 
     It is this meaning of “azimuth” that is used to define the course of a drilling path. The borehole inclination is also used in this context to define a three-dimensional bearing direction of a point of interest within the borehole. Inclination is the angular separation between a tangent to the borehole at the point of interest and vertical. The azimuth and inclination values are typically used in drilling applications to identify bearing direction at various points along the length of the borehole. A set of discrete inclination and azimuth measurements along the length of the borehole is further commonly utilized to assemble a well survey (e.g., using the minimum curvature assumption). Such a survey describes the three-dimensional location of the borehole in a subterranean formation. 
     A somewhat different meaning of “azimuth” is found in some borehole imaging art. In this context, the azimuthal reference plane is not necessarily horizontal (indeed, it seldom is). When a borehole image of a particular formation property is desired at a particular point in the borehole, measurements of the property are taken at points around the circumference of the measurement tool. The azimuthal reference plane in this context is the plane centered at the measurement tool and perpendicular to the longitudinal direction of the borehole at that point. This plane, therefore, is fixed by the particular orientation of the borehole measurement tool at the time the relevant measurements are taken. 
     An azimuth in this borehole imaging context is the angular separation in the azimuthal reference plane from a reference point to the measurement point. The azimuth is typically measured in the clockwise direction, and the reference point is frequently the high side of the borehole or measurement tool, relative to the earth&#39;s gravitational field, though magnetic north may be used as a reference direction in some situations. Though this context is different, and the meaning of azimuth here is somewhat different, this use is consistent with the traditional meaning and use of the term azimuth. If the longitudinal direction of the borehole at the measurement point is equated to the vertical direction in the traditional context, then the determination of an azimuth in the borehole imaging context is essentially the same as the traditional azimuthal determination. 
     Another important label used in the borehole imaging context is “toolface angle”. When a measurement tool is used to gather azimuthal imaging data, the point of the tool with the measuring sensor is identified as the “face” of the tool. The toolface angle, therefore, is defined as the angular separation from a reference point to the radial direction of the toolface. The assumption here is that data gathered by the measuring sensor will be indicative of properties of the formation along a line or path that extends radially outward from the toolface into the formation. The toolface angle is an azimuth angle, where the measurement line or direction is defined for the position of the tool sensors. The oilfield services industry uses the term “gravitational toolface” when the toolface angle has a gravity reference (e.g., the high side of the borehole) and “magnetic toolface” when the toolface angle has a magnetic reference (e.g., magnetic north). 
     In the remainder of this document, when referring to the course of a drilling path (i.e., a drilling direction), the term “borehole azimuth” will be used. Thus, a drilling direction may be defined, for example, via a borehole azimuth and an inclination (or borehole inclination). The terms toolface and azimuth will be used interchangeably, though the toolface identifier will be used predominantly, to refer to an angular position about the circumference of a downhole tool (or about the circumference of the borehole). Thus, an LWD sensor, for example, may be described as having an azimuth or a toolface. 
     Referring first to  FIGS. 1 to 10 , it will be understood that features or aspects of the embodiments illustrated may be shown from various views. Where such features or aspects are common to particular views, they are labeled using the same reference numeral. Thus, a feature or aspect labeled with a particular reference numeral on one view in  FIGS. 1 to 10  may be described herein with respect to that reference numeral shown on other views. 
       FIG. 1  illustrates a drilling rig  10  suitable for utilizing exemplary downhole tool and method embodiments of the present invention. In the exemplary embodiment shown on  FIG. 1 , a semisubmersible drilling platform  12  is positioned over an oil or gas formation (not shown) disposed below the sea floor  16 . A subsea conduit  18  extends from deck  20  of platform  12  to a wellhead installation  22 . The platform may include a derrick  26  and a hoisting apparatus  28  for raising and lowering the drill string  30 , which, as shown, extends into borehole  40  and includes a drill bit  32  and a directional drilling tool  100  (such as a three-dimensional rotary steerable tool). In the exemplary embodiment shown, steering tool  100  includes one or more, usually three, blades  150  disposed to extend outward from the tool  100  and apply a lateral force and/or displacement to the borehole wall  42 . The extension of the blades deflects the drill string  30  from the central axis of the borehole  40 , thereby changing the drilling direction. Drill string  30  may further include a downhole drilling motor, a mud pulse telemetry system, and one or more additional sensors, such as LWD and/or MWD tools for sensing downhole characteristics of the borehole and the surrounding formation. The invention is not limited in these regards. 
     It will be understood by those of ordinary skill in the art that methods and apparatuses in accordance with this invention are not limited to use with a semisubmersible platform  12  as illustrated in  FIG. 1 . This invention is equally well suited for use with any kind of subterranean drilling operation, either offshore or onshore. Moreover, while the invention is described with respect to exemplary three-dimensional rotary steerable (3DRS) tool embodiments, it will also be understood that the present invention is not limited in this regard. The invention is equally well suited for use in substantially any downhole tool requiring an angular position measurement of one component (e.g., a shaft) with respect to another (e.g., a sleeve deployed about the shaft). 
     Turning now to  FIG. 2 , one exemplary embodiment of rotary steerable tool  100  from  FIG. 1  is illustrated in perspective view. In the exemplary embodiment shown, rotary steerable tool  100  is substantially cylindrical and includes threaded ends  102  and  104  (threads not shown) for connecting with other bottom hole assembly (BHA) components (e.g., connecting with the drill bit at end  104 ). The rotary steerable tool  100  further includes a housing  110  deployed about a shaft (not shown on  FIG. 2 ). The shaft is typically configured to rotate relative to the housing  110 . The housing  110  further includes at least one blade  150  deployed, for example, in a recess (not shown) therein. Directional drilling tool  100  further includes hydraulics  130  and electronics  140  modules (also referred to herein as control modules  130  and  140 ) deployed in the housing  110 . In general, the control modules  130  and  140  are configured for sensing and controlling the relative positions of the blades  150 . As described in more detail below, electronic module also typically includes a tri-axial arrangement of accelerometers with one of the accelerometer having a known orientation relative to the longitudinal axis of the tool  100 . 
     To steer (i.e., change the direction of drilling), one or more of blades  150  are extended and exert a force against the borehole wall. The rotary steerable tool  100  is moved away from the center of the borehole by this operation, thereby altering the drilling path. In general, increasing the offset (i.e., increasing the distance between the tool axis and the borehole axis via extending one or more of the blades) tends to increase the curvature (dogleg severity) of the borehole upon subsequent drilling. The tool  100  may also be moved back towards the borehole axis if it is already eccentered. It will be understood that the drilling direction (whether straight or curved) is determined by the positions of the blades with respect to housing  110  as well as by the angular position (i.e., the azimuth) of the housing  110  in the borehole. 
     Angular Sensor Embodiments 
     With reference now to  FIG. 3 , one exemplary embodiment of an angular sensor  200  in accordance with the present invention is depicted in cross section. Angular sensor  200  is disposed to measure the relative angular position between shaft  115  and housing  110  and may be deployed, for example, in control module  140  ( FIG. 2 ). In the exemplary embodiment shown, angular sensor  200  includes first and second magnets  220 A and  220 B deployed on the shaft  115  and a plurality of magnetic field sensors  210 A-H deployed about the circumference of the housing  110 . The invention is not limited in this regard, however, as the magnets  220 A and  220 B may be deployed on the housing  110  and magnetic field sensors  210 A-H on the shaft  115 . 
     Magnets  220 A and  220 B are angularly offset about the circumference of the shaft  115  by an angle θ. In the exemplary embodiment shown, magnets  220 A and  220 B are angularly offset by an angle of 90 degrees, however, the invention is not limited in this regard. Magnets  220 A and  220 B may be angularly offset by substantially any suitable angle. Angles in the range from about 30 to about 180 degrees are generally advantageous. Magnets  220 A and  220 B also typically have substantially equal magnetic pole strengths and opposite polarity, although the invention is expressly not limited in this regard. In the exemplary embodiment shown on  FIG. 3 , magnet  220 A includes an approximately cylindrical magnet having a magnetic north pole facing radially outward from the tool axis while magnetic  220 B includes an approximately cylindrical magnet having a magnetic south pole facing radially outward towards the tool axis. It will be appreciated that other more complex magnetic arrangements may be utilized. Certain other arrangements are described in more detail below with respect to  FIGS. 5-8B . In one other alternative arrangement, magnets  220 A and  220 B may each include first and second magnets having opposing magnetic poles facing one another such that magnetic flux emanates radially outward from the tool axis (or inward towards the tool axis depending upon the polarity of the magnets). In such an embodiment, magnet  220 A may include north-north opposing poles, for example, while magnet  220 B may include south-south opposing poles. 
     With continued reference to  FIG. 3 , magnetic field sensors  210 A-H are deployed about the circumference of the tool  100  such that at least two of the sensors  210 A-H are within sensory range of magnetic flux emanating from the magnets  220 A and  220 B. In the exemplary embodiment shown, at least sensors  210 A and  210 C are in sensory range of the magnetic flux. Magnetic field sensors  210 A-H may include substantially any type of magnetic sensor, e.g., including magnetometers, reed switches, magnetoresistive sensors, and/or Hall-Effect sensors, however magnetoresistive sensors and Hall-Effect sensors are generally preferred. Moreover, each sensor may have either a ratiometric (analog) or digital output. While  FIG. 3  shows eight magnetic field sensors  210 A-H, it will be appreciated by those of ordinary skill on the art that this invention may equivalently utilize substantially any suitable plurality of magnetic field sensors. Typically from about four to about sixteen sensors are preferred. Too few sensors tend to result in a degradation of angular sensitivity (although degraded angular sensitivity may be acceptable, for example, in certain LWD imaging applications in which the LWD sensor has poor angular sensitivity). The use of sixteen or more sensors, while providing excellent angular sensitivity, increases wiring and power requirements while also tending to negatively impact system reliability. 
     In the exemplary embodiment shown on  FIG. 3 , each magnetic field sensor  210 A-H is deployed so that its axis of sensitivity is substantially radially aligned (i.e., pointing towards the center of the shaft  115 ), although the invention is not limited in this regard. It will be appreciated by those of ordinary skill in the art that a magnetic sensor is typically sensitive only to the component of the magnetic flux that is aligned (parallel) with the sensor&#39;s axis of sensitivity. It will also be appreciated that the exemplary embodiment shown on  FIG. 3  results in magnetic flux lines that are substantially radially aligned adjacent magnets  220 A and  220 B. Therefore, the magnetic sensor  210 A-H located closest to magnet  220 A tends to sense the highest positive magnetic flux (magnetic flux directed outward for the tool axis) and the sensor closest to magnet  220 B tends to sense the highest negative magnetic flux (magnetic flux directed inward towards the tool axis). For example, in the exemplary embodiment shown, magnetic sensor  210 A tends to measure the highest positive magnetic flux while sensor  210 C tends to measure the highest negative magnetic flux. The invention is not limited by the exemplary sensor orientation depicted on  FIG. 3 . 
     With reference now to  FIG. 4A , a plot of the radial flux emanating from magnets  220 A and  220 B versus angular position about the shaft  115  is depicted. Note that the radial flux includes positive  510  and negative  520  maxima. As described above, the positive maximum  510  is located radially outward from magnet  220 A (i.e., at about 15 degrees in the exemplary embodiment shown). The negative maximum  520  is located radially outward from magnet  220 B (i.e., at about 105 degrees in the exemplary embodiment shown). A magnetic flux null  530  (also referred to as a zero-crossing) is located between the positive  510  and negative  520  maxima (i.e., at about 60 degrees in the exemplary embodiment shown). The radial flux depicted in  FIG. 4A  is for an exemplary embodiment in which the shaft  115  and housing  110  are fabricated from a non-magnetic steel. For embodiments in which the shaft and/or housing are fabricated from a magnetic steel (or other magnetically permeable material), the positive and negative maxima  510  and  520  typically become more sharply defined with respect to angular position. Notwithstanding, it will be appreciated that the relative rotational position of the magnets  220 A and  220 B (and therefore the shaft) with respect to the magnetic sensors  210 A-H (and therefore the housing  110 ) may be determined by locating the positive and/or negative maxima  510  and  520  or the zero-crossing  530 . 
     With reference now to  FIG. 4B , a graphical representation of one exemplary mathematical technique for determining the angular position is illustrated. Data points  450  represent the magnetic field strength as measured by each of sensors  210 A-H on  FIG. 3 . In this exemplary sensor embodiment, the angular position half way between magnets  220 A and  220 B is indicated by zero-crossing  430 , the location on the circumferential array of magnetic field sensors at which the magnetic flux is substantially null and at which the polarity of the magnetic field changes from positive to negative (or negative to positive). In the exemplary embodiment shown, zero-crossing  430  is at an angular position of about 60 degrees (as described above with respect to  FIG. 3 ). Note that the position of the zero crossing  430  (and therefore the angular position half way between the magnets  220 A and  220 B) is located between sensors  210 B and  210 C. In one exemplary method embodiment, a processor (such as processor  255 ) first selects adjacent sensors (e.g., sensors  210 B and  210 C) between which the sign of the magnetic field changes (from positive to negative or negative to positive). The position of the zero crossing  430  may then be determined, for example, by fitting a straight line  470  through the data points on either side of the zero crossing (e.g., between the measurements made by sensors  210 B and  210 C in the embodiment shown on  FIG. 4B ). The location of the zero crossing  820  may then be determined mathematically from the magnetic field measurements, for example, as follows: 
     
       
         
           
             
               
                 
                   P 
                   = 
                   
                     L 
                     ( 
                     
                       x 
                       + 
                       
                         A 
                         
                           A 
                           + 
                           B 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
             
           
         
       
     
     where P represents the angular position of the zero crossing, L represents the angular distance interval between adjacent sensors in degrees (e.g., 45 degrees in the exemplary embodiment shown on  FIGS. 3 and 5 ), A and B represent the absolute values of the magnetic field measured on either side of the zero crossing (A and B are shown on  FIGS. 4B and 7B ), and x is a counting variable having an integer value representing the first of the two adjacent sensors positioned on either side of the zero crossing (such that x=1 for sensor  210 A, x=2 for sensor  210 B, x=3 for sensor  210 C, and so on). In the exemplary embodiments shown on  FIGS. 4B and 7B , x=2 (sensor  210 B). 
     It will be appreciated that the magnet arrangement shown on  FIG. 3  (including magnets  220 A and  220 B) tends to result angular position values having small, systematic errors at certain angular positions due to the non-linearly of the magnetic flux profile as a function of angular position. This error is readily corrected, when necessary, using known calibration methods (e.g., look-up tables or polynomial fitting). It will also be appreciated that the magnet arrangement shown on  FIG. 3  advantageously makes use of inexpensive and readily available off-the-shelf magnets (e.g., square, rectangular or cylindrical magnets). 
     Turning now to  FIG. 5 , an alternative embodiment of an angular sensor  200 ′ in accordance with the present invention is depicted in cross section. Angular sensor  200 ′ is also disposed to measure the relative angular position between shaft  115  and housing  110  and may be deployed, for example, in control module  140  ( FIG. 2 ). Sensor  200 ′ is substantially identical to sensor  200  with the exception that it includes first and second tapered, arc-shaped magnets  240 A and  240 B (also referred to herein as eyebrow magnets) deployed on the shaft  115 . One exemplary embodiment of eyebrow magnet  240 A is also shown on  FIG. 6 . Eyebrow magnets  240 A and  240 B include inner and outer faces  242  and  244 , with the outer face  244  having a radius of curvature approximately equal to that of the outer surface of the shaft  115 . Eyebrow magnets  240 A and  240 B also include relatively thick  246  and relatively thin  248  ends. While the invention is not limited in this regard, the thickness of end  246  is at least four times greater than that of end  248  in one exemplary embodiment. 
     In the exemplary embodiment shown, magnets  240 A and  240 B are substantially identical in shape and have substantially equal and opposite magnetic pole strengths. Magnet  240 A includes a magnetic north pole on its outer face  244  and a magnetic south pole on its inner face  242  ( FIG. 6 ). Magnet  240 B has the opposite polarity with a magnetic south pole on its outer face  244  and a magnetic north pole on its inner face  242 . Magnets  240 A and  240 B are typically deployed adjacent to one another about the shaft  115  such that their thin ends  248  are in contact (or near contact) with one another. While  FIG. 5  shows an exemplary embodiment in which the magnets  240 A and  240 B are deployed in a tapered recess in the outer surface of the shaft, it will be appreciated that magnets  240 A and  240 B may be equivalently deployed on the outer surface of the shaft  115 . The invention is not limited in these regards. In the exemplary embodiment shown, magnets  240 A and  240 B each span a circular arc of about 55 degrees about the circumference of the shaft. Thus magnets  240 A and  240 B in combination span a circular arc θ′ of about 110 degrees. The invention is also not limited in these regards (as described in more detail below). 
     With reference now to  FIG. 7A , a plot of the radial flux emanating from magnets  240 A and  240 B versus angular position about shaft  115  is depicted. Similar to the embodiment described above with respect to  FIGS. 3-4B , the radial flux includes positive  710  and negative  720  maxima. The positive maximum  710  is located radially outward from and near the thick end  246  of magnet  240 A (i.e., at an angle of about 5-10 degrees in the exemplary embodiment shown). The negative maximum  720  is located radially outward from and near the thick end of magnet  240 B (i.e., at about 100-105 degrees in the exemplary embodiment shown). A magnetic flux null  730  (also referred to as a zero-crossing) is located between the positive  710  and negative  720  maxima (i.e., at about 55 degrees in the exemplary embodiment shown). Moreover, as shown at  740 , the radial flux is advantageously substantially linear with angular position between the maxima  710  and  720 , which typically eliminates the need for correction algorithms. As described above with respect to angular sensor  200 , the relative rotational position of the magnets  240 A and  240 B (and therefore the shaft) with respect to the magnetic sensors  210 A-H (and therefore the housing  110 ) may be determined from the positive and/or negative maxima  710  and  720  or the zero-crossing  730 . 
     With continued reference to  FIG. 7A , and with reference again to  FIGS. 5 and 6 , eyebrow magnets  240 A and  240 B may be advantageously sized and shaped to generate a magnetic flux that varies linearly  740  with angular position between the positive and negative maxima  710  and  720 . In the exemplary embodiment shown, this linear region  740  spans approximately 95 degrees in angular position. The invention is not limited in this regard, however, as the angular expanse of the linear region  740  may be increased by increasing the arc-length of magnets  240 A and  240 B and decreased by decreasing the arc-length of magnets  240 A and  240 B. In general, it is desirable for substantially linear region  740  to have an angular expanse of at least twice the angular interval between adjacent ones of magnetic sensors  210 A-H. In this way at least two of the magnetic sensors  210 A-H are located in the linear region  740  at all relative angular positions. It will thus be understood that embodiments of the invention utilizing fewer magnetic field sensors desirably utilize eyebrow magnets having a longer arc-length (e.g., about 90 degrees each for an embodiment including five magnetic field sensors). Likewise, embodiments of the invention utilizing more magnetic field sensors may optionally utilize eyebrow magnets having a shorter arc-length (e.g., about 30 degrees each for an embodiment including 16 magnetic field sensors). 
     Eyebrow magnets  240 A and  240 B are also advantageously sized and shaped to generate the above described magnetic flux profile (as a function of angular position) for tool embodiments in which both the shaft  115  and the housing  110  are fabricated from a magnetic material such as 4145 low alloy steel. It will be readily understood by those of ordinary skill in the art that the use of magnetic steel is advantageous in that it tends to significantly reduce manufacturing costs (due to the increased availability and reduced cost of the steel itself) and also tends to increase overall tool strength. Notwithstanding, magnets  240 A and  240 B may also be sized and shaped to generate the above described magnetic profile for tool embodiments in which either one or both of the shaft  115  and the housing  110  are fabricated from nonmagnetic steel. 
     With reference now to  FIG. 7B , a graphical representation of one exemplary mathematical technique for determining the angular position is illustrated. The technique illustrated in  FIG. 7B  is similar to that described above with respect to  FIG. 4B . Data points  750  represent the magnetic field strength values measured by sensors  210 A-H on  FIG. 5 . In this embodiment, the angular position of the contact point  245  between magnets  240 A and  240 B is indicated by zero-crossing  730 , which as described above is the location on the circumferential array of magnetic field sensors  210 A-H at which the magnetic flux is substantially null and at which the polarity of the magnetic field changes from positive to negative (or negative to positive). In the exemplary embodiment shown, zero-crossing  730  is at an angular position of about 55 degrees (as described above with respect to  FIGS. 5 and 7A ). Note that the position of the zero crossing  730  (and therefore the angular position of contact point  245 ) is located between sensors  210 B and  210 C. Thus, as described above, a processor may first select adjacent sensors (e.g., sensors  210 B and  210 C) between which the sign of the magnetic field changes (from positive to negative or negative to positive). The position of the zero crossing  730  may then be determined, for example, by fitting a straight line  770  through the data points on either side of the zero crossing (e.g., between the measurements made by sensors  210 B and  210 C in the embodiment shown on  FIG. 7B ). The location of the zero crossing  730  may then be determined mathematically from the magnetic field measurements, for example, via Equation 1 as described above. 
     The exemplary angular position sensor embodiments shown on  FIGS. 3 and 5  include magnetic sensors  210 A-H deployed at equal angular intervals about the circumference of housing  110 . It will be appreciated that the invention is not limited in this regard. Magnetic sensors  210 A-H may alternatively be deployed at unequal intervals. For example, more sensors may be deployed on a one side of the housing  110  than on an opposing side to provide better angular sensitivity on that side of the tool. It will also be appreciated that angular position sensors  200  and  200 ′ are not limited to embodiments in which the magnets are deployed on the shaft  115  and the magnetic sensors  210 A-H in the housing. The magnets may be equivalently deployed in the housing  110  and the magnetic sensors  210 A-H on the shaft. 
     It will be appreciated that angular position sensing methods described above with respect to  FIGS. 3 through 7B  and Equation 1 advantageously require minimal computational resources (minimal processing power), which is critical in downhole applications in which 8-bit microprocessors are commonly used. These methods also provide accurate angular position determination about substantially the entire circumference of the tool. The zero-crossing method tends to be further advantageous in that a wider sensor input range is available (from the negative to positive saturation limits of the sensors). 
     It will also be appreciated that downhole tools must typically be designed to withstand shock levels in the range of 1000 G on each axis and vibration levels of 50 G root mean square. Moreover, downhole tools are also typically subject to pressures ranging up to about 25,000 psi and temperatures ranging up to about 200 degrees C. With reference again to  FIGS. 3 and 5 , magnetic field sensors  210 A-H are shown deployed in a pressure resistant housing  205 . Such an arrangement is preferred for downhole applications utilizing solid state magnetic field sensors such as Hall-Effect sensors and magnetoresistive sensors. In the exemplary embodiment shown, pressure housing  205  includes a sealed ring that is configured to resist downhole pressures which can damage sensitive electronic components. The pressure housing  205  is also configured to accommodate the magnetic field sensors  210 A-H and other optional electronics, such as processor  255 . Advantageous embodiments of the pressure housing  205  are fabricated from nonmagnetic material, such as P550 (austenitic manganese chromium steel). In the exemplary embodiment shown, magnetic field sensors  210 A-H are deployed on a circumferential circuit board array  250 , which is fabricated, for example from a flexible, temperature resistant material, such as PEEK (polyetheretherketone). The circumferential array  250 , including the magnetic field sensors  210 A-H and processor  255 , is also typically encapsulated in a potting material to improve resistance to shocks and vibrations. 
     The magnets utilized in this invention are also typically selected in view of demanding downhole conditions. For example, suitable magnets must posses a sufficiently high Curie Temperature to prevent demagnetization at downhole temperatures. Samarium cobalt (SaCo 5 ) magnets are typically preferred in view of their high Curie Temperatures (e.g., from about 700 to 800 degrees C.). To provide further protection from downhole conditions, the magnets may also be deployed in a shock resistant housing, for example, including a non-magnetic sleeve deployed about the magnets and shaft  115 . 
     In the exemplary embodiments shown on  FIGS. 3 and 5 , the output of each magnetic sensor may be advantageously electronically coupled to the input of a local microprocessor. The microprocessor serves to process the data received by the magnetic sensors (e.g., according to Equation 1 as described above). In preferred embodiments, the microprocessor (such as processor  255 ) is embedded with the magnetic field sensors  210 A-H in the circumferential array  250 , for example, as shown on  FIGS. 3 and 5  and therefore located close to the magnetic sensors. In such an embodiment, the microprocessor output (rather than the signals from the individual magnetic sensors) is typically electronically coupled with a main processor which is deployed further away from the magnetic field sensors (e.g., deployed in control module  140  as shown on  FIG. 2 ). This configuration advantageously reduces wiring and feed-through requirements in the body of the downhole tool, which is particularly important in smaller diameter tool embodiments (e.g., tools having a diameter of less than about 12 inches). Digital output from the embedded microprocessor also tends to advantageously reduce electrical interference in wiring to the main processor. Embedded microprocessor output may also be combined with a voltage source line to further reduce the number of wires required, e.g., one wire for combined power and data output and one wire for ground (or alternatively, the use of a chassis ground). This may be accomplished, for example, by imparting a high frequency digital signal to the voltage source line or by modulating the current draw from the voltage source line. Such techniques are known to those of ordinary skill in the art. 
     In preferred embodiments of this invention, microprocessor  255  ( FIGS. 3 and 5 ) includes processor-readable or computer-readable program code embodying logic, including instructions for calculating a precise angular position of the shaft  115  relative to the housing  110  from the received magnetic sensor measurements. While substantially any logic routines may be utilized, it will be appreciated that logic routines requiring minimal processing power (e.g., as described above with respect to Equation 1) are advantageous for downhole applications (particularly for small-diameter LWD, MWD, and directional drilling embodiments of the invention in which both electrical and electronic processing power are often severely limited). 
     While the above described exemplary embodiments pertain to rotary steerable tool embodiments including hydraulically actuated blades, it will be understood that the invention is not limited in this regard. The artisan of ordinary skill will readily recognize other downhole uses of angular position sensors in accordance with the present invention. For example, angular position sensors in accordance with this invention may be deployed in conventional and/or steerable drilling fluid (mud) motors and utilized to determine the angular position of drill string components (e.g., MWD or LWD sensors) deployed below the motor with respect to those deployed above the motor. In one exemplary embodiment, the angular position sensor may be disposed, for example, to measure the relative angular position between the rotor and stator in the mud motor. 
     Near-Bit Gravity Azimuth Measurements 
     As described above in the Background Section, U.S. Pat. No. 6,480,119 to McElhinney and commonly assigned U.S. Pat. No. 7,080,460 to Illfelder disclose Gravity MWD techniques for determining borehole azimuth via tri-axial accelerometer measurements made at first and second longitudinal positions on a drill string. Using gravity as a primary reference, the disclosed methods make use of the inherent bending of the structure between the accelerometer sets in order to calculate a change in borehole azimuth between the first and second positions. 
     As also described above, it would be highly advantageous to extend Gravity MWD methods to eliminate the rotational constraint and thereby allow relative rotation between the first and second accelerometer sets. This would advantageously enable conventional tool deployments to be utilized in making Gravity MWD measurements. For example, as described in more detail below, a first (upper) accelerometer set may be deployed in a conventional MWD tool coupled to the drill string and a second accelerometer set may be deployed in the non rotating housing of a rotary steerable tool (e.g., in housing  110  of steering tool  100  shown on  FIG. 2 ). It will be understood that in such a tool configuration the upper set will rotate (with the drill string) with respect to the lower set (which is substantially non-rotating in the borehole during drilling). 
     Referring now to  FIG. 8 , one exemplary embodiment of a BHA suitable for Gravity MWD method embodiments in accordance with the present invention is illustrated. In  FIG. 8 , the BHA includes a drill bit assembly  32  coupled with a steering tool  100 . Steering tool  100  includes a lower accelerometer set  180  deployed in the substantially non-rotating housing  110 . The BHA also includes an MWD tool  75  including an upper accelerometer set  80 . The upper and lower accelerometer sets  80  and  180  each typically include three mutually perpendicular (tri-axial) gravity sensors, one of which is oriented substantially parallel with the borehole axis  50  and measures gravity vectors denoted as Gz 1  and Gz 2  for the upper and lower sensor sets, respectively. The invention is not limited in this regard, however. Each accelerometer set shown on  FIG. 8  may thus be considered as determining a plane (Gx and Gy) and a pole (Gz) as shown. The upper  80  and lower  180  accelerometer sets are typically disposed at a known longitudinal spacing in the BHA. The spacing may be, for example, in a range of from about 10 to about 30 meters (i.e., from about 30 to about 100 feet) or more, but the invention is not limited in this regard. Moreover, it will be understood that this invention is not limited to a known or fixed separation between the upper and lower sensor sets  80  and  180 . 
     It will be understood that in the exemplary BHA embodiment shown, MWD tool  75  is rotationally coupled with the drill string  30 . As such accelerometer set  80  is free to rotate with respect to accelerometer set  180  about the longitudinal axis  50  of the BHA. During drilling accelerometer set  80  rotates with the drill string  30  in the borehole  42 , while accelerometer set  180  is substantially non-rotating with respect to the borehole in housing  110  while blades  150  engage the borehole wall. 
     With continued reference to  FIG. 8 , steering tool  100  further includes an angular sensor  200 ,  200 ′ ( FIGS. 3 and 5 ) disposed to measure an angular position of the housing  110  relative to the drill string  30  (which is rotationally coupled to shaft  115 ). It will thus be appreciated that angular sensor  200 ,  200 ′ is also disposed to measure the relative angular position between the upper and lower accelerometer sets  80  and  180  (since set  80  is deployed in MWD tool  75  and set  180  is deployed in housing  110 ). While the exemplary embodiment shown utilizes angular sensor  200 ,  200 ′, it will be appreciated that Gravity MWD embodiments of the present invention are not limited to any particular angular sensor embodiments. Any suitable angular sensor may be utilized. 
     It will also be understood that the invention is not limited to steering tool and/or rotary steerable embodiments, such as that shown on  FIG. 8 . Rather, Gravity MWD measurements in accordance with this invention may be made using substantially any suitable BHA configuration. In advantageous configurations the upper and lower accelerometer sets  80  and  180  are free to rotate about cylindrical axis  50  with respect to one another. In one alternative configuration enabling such rotational freedom, the upper and lower accelerometer sets  80  and  180  are deployed respectively above and below a conventional and/or steerable mud motor. An angular position sensor may be deployed in the mud motor, e.g., as described above, and utilized to determine the relative angular position between the upper and lower accelerometer sets  80  and  180 . 
     In order to determine the change in borehole azimuth between the upper and lower accelerometer sets  80  and  180  the relative rotation between the sets needs to be accounted. This may be accomplished, for example, by measuring the angular position of housing  110  relative to the drill string  30  concurrently while making accelerometer measurements at sets  80  and  180 . The accelerometer measurements at set  180  may then be corrected for the angular offset, for example as follows: 
     
       
         
           
             
               
                 
                   
                     
                       Gx 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         2 
                         ′ 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               Gx 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 2 
                                 2 
                               
                             
                             + 
                             
                               Gy 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 2 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         cos 
                         ( 
                         
                           
                             arc 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               tan 
                               ( 
                               
                                 
                                   Gx 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 
                                   Gy 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                               
                               ) 
                             
                           
                           - 
                           A 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       Gy 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         2 
                         ′ 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               Gx 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 2 
                                 2 
                               
                             
                             + 
                             
                               Gy 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 2 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         sin 
                         ( 
                         
                           
                             arc 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               tan 
                               ( 
                               
                                 
                                   Gx 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 
                                   Gy 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                               
                               ) 
                             
                           
                           - 
                           A 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       Gz 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         2 
                         ′ 
                       
                     
                     = 
                     
                       Gz 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     Where Gx 2 , Gy 2 , and Gz 2  represent the accelerometer measurements made at the lower accelerometer set  180 , Gx 2 ′, Gy 2 ′, and Gz 2 ′ represent the corrected accelerometer measurements, and A represents the measured angular position (the angular offset) between the first and second accelerometer sets  80  and  180 . The artisan of ordinary skill in the art will readily recognize that the accelerometer measurements made at the upper set  80  may alternatively be corrected for angular offset (by an angle of −A degrees). 
     The accelerometer measurements made at the first set  80  and the corrected accelerometer measurements for the second set  180  may then be utilized to calculate the change in borehole azimuth between the first and second sets  80  and  180 . This may be accomplished, for example, by substituting Gx2′, Gy2′, and Gz2′ for Gx2, Gy2, and Gz2 in Equations 4 and 5 of U.S. Pat. No. 7,002,484 to McElhinney and solving for the change in borehole azimuth. Alternatively, Gx2′, Gy2′, and Gz2′ may be substituted for Gx2, Gy2, and Gz2 in Column 6 of U.S. Pat. No. 7,028,409 to Engebretson et al. and solving for the change in borehole azimuth. 
     The relative rotation between the accelerometer sets  80  and  180  may also be accounted by recognizing that such rotation changes the toolface angle of one sensor set with respect to the other. As such, the toolface angle at the lower accelerometer set  180  may be corrected, for example, as follows:
 
 TF 2 ′=TF 2 −A   Equation 3
 
     where TF 2  represents the toolface angle of the lower accelerometer set  180  (e.g., of housing  110 ), TF 2 ′ represents the corrected toolface angle, and A represents the measured angular position (the angular offset) between the first and second accelerometer sets  80  and  180 . It will of course be understood that the toolface angle at the upper accelerometer may alternatively be corrected (e.g., by the equation: TF 1 ′=TF 1 +A). 
     The corrected toolface angle may also be utilized to calculate the change in borehole azimuth between the first and second sets  80  and  180 . The Illfelder patent discloses that the change in borehole azimuth may be determined directly from borehole inclination and gravity toolface measurements made at each of the first and second positions according to the following equation (Equation 7 in the Illfelder patent): 
     
       
         
           
             
               
                 
                   
                     
                       TF 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                     
                     - 
                     
                       TF 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                   
                   = 
                   
                     
                       arc 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         tan 
                         [ 
                         
                           
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   Inc 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 DeltaAzi 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       sin 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           Inc 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           2 
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       cos 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           Inc 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           1 
                                         
                                         ) 
                                       
                                     
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   
                                     sin 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         Inc 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     cos 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         Inc 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         2 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     cos 
                                     ⁡ 
                                     
                                       ( 
                                       DeltaAzi 
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     - 
                     
                       arc 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         tan 
                         [ 
                         
                           
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   inc 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 DeltaAzi 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       sin 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           Inc 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           2 
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       cos 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           Inc 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           1 
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       cos 
                                       ⁡ 
                                       
                                         ( 
                                         DeltaAzi 
                                         ) 
                                       
                                     
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   
                                     sin 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         Inc 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     cos 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         Inc 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         2 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   4 
                 
               
             
           
         
       
     
     where Inc 1  and Inc 2  represent the borehole inclination angles at the first and second positions, TF 1  and TF 2  represent the gravity toolface angles at the first and second positions, and DeltaAzi represents the change in borehole azimuth between the first and second positions. Those of ordinary skill in the art will readily be able to calculate the borehole inclination and gravity toolface angles directly from the accelerometer measurements (e.g., using Equations 1 through 4 disclosed in the Illfelder patent). The change in borehole azimuth may then be determined, for example, by substituting TF 2 ′ for TF 2  in Equation 4 and solving for the change in borehole azimuth (DeltaAzi) as described in the Illfelder patent. 
     The Illfelder patent further discloses that the change in borehole azimuth, DeltaAzi, can be determined from Equation 4 using numerical root finding algorithms, graphical methods, and/or look-up tables. Such methods are readily available and easily utilized at the surface, e.g., via a conventional PC using software routines available in MathCad® and/or Mathematica®. However, it is difficult to apply such numerical and/or graphical methods using on-board, downhole processors due to their limited processing power. Therefore there also exists a need for a simplified method for determining DeltaAzi, preferably including an equation that can be readily solved using a low-power, downhole processor. 
     Using linear regression techniques and trigonometric function fitting techniques Equation 4 may be rewritten in simplified form as follows: 
     
       
         
           
             
               
                 
                   DeltaAzi 
                   = 
                   
                     
                       
                         TF 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                       - 
                       
                         TF 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     
                       
                         0.008759 
                         ⁢ 
                         
                           ( 
                           
                             
                               Inc 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                             - 
                             
                               Inc 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               Inc 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                             ) 
                           
                         
                       
                       - 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             Inc 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   5 
                 
               
             
           
         
       
     
     where Inc 1 , Inc 2 , TF 1 , TF 2 , and DeltaAzi are defined above with respect to Equation 4. In Equation 5, the numerical coefficient 0.008759 is selected for use with input parameters Inc 1 , Inc 2 , TF 1 , and TF 2  being in units of degrees. Equivalent equations can be readily derived by those of ordinary skill in the art for other angular units, e.g. radians. Equation 5 has been found to provide a highly accurate approximation of Equation 4, with a resulting DeltaAzi error of less than 0.03 degrees over nearly the entire range of possible borehole inclination, borehole azimuth, and gravity toolface values. Those of ordinary skill in the art will readily recognize that an error of less than 0.03 degrees is negligible in comparison, for example, to errors in the inclination and gravity toolface angles used to compute DeltaAzi. Those of ordinary skill in the art will also readily recognize that Equation 5 may be rewritten to express DeltaAzi as a function of Gx 1 , Gy 1 , Gz 1 , Gx 2 , Gy 2 , and Gz 2 . 
     It will be appreciated that the present invention advantageously provides for downhole determination of a near-bit borehole azimuth that is substantially free from magnetic interference. For example, in the exemplary embodiment shown on  FIG. 8 , the lower sensor set  180  is deployed in steering tool  100  just above the drill bit. Such a near-bit borehole azimuth may be determined, for example, via the following equation: 
     
       
         
           
             
               
                 
                   
                     Azi 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                   = 
                   
                     
                       
                         Azi 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                       + 
                       DeltaAzi 
                     
                     = 
                     
                       
                         Azi 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                       + 
                       
                         
                           
                             TF 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           - 
                           A 
                           - 
                           
                             TF 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                         
                           
                             
                               
                                 0.008759 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       Inc 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                     
                                     - 
                                     
                                       Inc 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       Inc 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       Inc 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   6 
                 
               
             
           
         
       
     
     where Azi 2  represents the near-bit borehole azimuth in degrees (i.e., the borehole azimuth at the lower accelerometer set), Azi 1  represents the borehole azimuth in degrees at the upper accelerometer set (e.g., determined via concurrent magnetometer measurements made at the upper set), and Inc 1 , Inc 2 , TF 1 , TF 2 , and DeltaAzi are defined above in degrees with respect to Equation 4. 
     Due to their simplicity, Equations 5 and 6 are especially well suited for use with downhole microcontrollers having limited processing power. Equation 6, for example, advantageously includes only 5 subtractions/additions, 2 multiplies, 1 division, and 2 trigonometry functions. It will be appreciated that Azi 2  (or DeltaAzi) may be advantageously computed at substantially any downhole microcontroller deployed substantially anywhere in the BHA. For example, Azi 2  may be computed at a microcontroller located in MWD tool  75 . To facilitate such computations, Inc 2  and TF 2  may be transmitted (e.g., via relatively high-speed communication bus among downhole tools) from accelerometer set  180  to MWD tool  75 . Alternatively and/or additionally Azi 2  may be computed at a microcontroller located in housing  110 . To facilitate such computations, Inc 1 , TF 1 , and Azi 1  may be transmitted from accelerometer set  80  to the microcontroller in housing  110 . However, the invention is not limited in this regard. In some high-technology rigs, raw data may be telemetered to the surface via wired drill pipe connections providing high speed communication (e.g., 56 Kbps or 1 M bps). Those of ordinary skill in the art will readily recognize that the measurement of near-bit borehole azimuth may be advantageously utilized for several purposes. For example, the combination of near-bit borehole azimuth and near-bit borehole inclination provides a substantially real time indication of the bearing direction of a borehole during drilling, which enables errors in bearing to be quickly recognized and corrected. 
     Near-bit azimuth measurements may also be advantageously utilized in closed-loop methods for controlling the direction of drilling. For example, the drilling direction may be controlled such that predetermined borehole inclination and borehole azimuth values are maintained. Alternatively, a predetermined borehole curvature (e.g., build rate, turn rate, or other dogleg) may be maintained. The build and turn rates of the borehole may be expressed mathematically, for example, as follows: 
     
       
         
           
             
               
                 
                   BuildRate 
                   = 
                   
                     
                       
                         Inc 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                       - 
                       
                         Inc 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     d 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   7 
                 
               
             
             
               
                 
                   TurnRate 
                   = 
                   
                     
                       
                         Azi 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                       - 
                       
                         Azi 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     d 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   8 
                 
               
             
           
         
       
     
     where Inc 1 , Inc 2 , Azi 1  and Azi 2  are defined above with respect to Equations 4 and 6 and d is the axial distance between the first and second accelerometer sets  80  and  180 . As is known to those of ordinary skill in the art, the combination of build rate and turn rate fully define the curvature of the borehole (both the direction and severity of the curve). Thus, an exemplary closed-loop control method may advantageously control the curvature of the borehole during drilling by controlling the build rate and turn rate (as determined in Equations 7 and 8) to be within predetermined limits. One such closed-loop method is disclosed in commonly assigned U.S. Patent Publication No. 2005/0269082. 
     Although the present invention and its advantages have been described in detail, it should be understood that various changes, substitutions and alternations may be made herein without departing from the spirit and scope of the invention as defined by the appended claims.