Patent Publication Number: US-2009229888-A1

Title: Methods and systems for designing and/or selecting drilling equipment using predictions of rotary drill bit walk

Description:
RELATED APPLICATIONS 
     This application is a U.S. Continuation-In-Part of U.S. patent application Ser. No. 11/462,898 filed Aug. 7, 2006, and entitled “Methods and Systems for Designing and/or Selecting Drilling Equipment Using Predictions of Rotary Drill Bit Walk,” which claims the benefit of U.S. provisional patent application entitled “Methods and Systems of Rotary Drill Bit Steerability Prediction, Rotary Drill Bit Design and Operation,” Application Ser. No. 60/706,321 filed Aug. 8, 2005. 
     This application claims the benefit of provisional patent application entitled “Methods and Systems of Rotary Drill Bit Walk Prediction, Rotary Drill Bit Design and Operation,” Application Ser. No. 60/738,431 filed Nov. 21, 2005. 
     This application claims the benefit of provisional patent application entitled “Methods and Systems of Rotary Drill Bit Walk Prediction, Rotary Drill Bit Design and Operation,” Application Ser. No. 60/706,323 filed Aug. 8, 2005. 
     This application claims the benefit of provisional patent application entitled “Methods and Systems of Rotary Drill Bit Steerability Prediction, Rotary Drill Bit Design and Operation,” Application Ser. No. 60/738,453 filed Nov. 21, 2005. 
    
    
     TECHNICAL FIELD 
     The present disclosure is related to wellbore drilling equipment and more particularly to designing rotary drill bits and/or bottom hole assemblies with desired bit walk characteristics or selecting a rotary drill bit and/or components for an associated bottom hole assembly with desired bit walk characteristics from existing designs. 
     BACKGROUND 
     Various types of rotary drill bits have been used to form wellbores or boreholes in downhole formations. Such wellbores are often formed using a rotary drill bit attached to the end of a generally hollow, tubular drill string extending from an associated well surface. Rotation of a rotary drill bit progressively cuts away adjacent portions of a downhole formation by contact between cutting elements and cutting structures disposed on exterior portions of the rotary drill bit. Examples of rotary drill bits include fixed cutter drill bits or drag drill bits and impregnated diamond bits. Various types of drilling fluids are often used in conjunction with rotary drill bits to form wellbores or boreholes extending from a well surface through one or more downhole formations. 
     Various types of computer based systems, software applications and/or computer programs have previously been used to simulate forming wellbores including, but not limited to, directional wellbores and to simulate the performance of a wide variety of drilling equipment including, but not limited to, rotary drill bits which may be used to form such wellbores. Some examples of such computer based systems, software applications and/or computer programs are discussed in various patents and other references listed on Information Disclosure Statements filed during prosecution of this patent application. 
     SUMMARY 
     In accordance with teachings of the present disclosure, rotary drill bits including fixed cutter drill bits may be designed with bit walk characteristics and/or controllability optimized for a desired wellbore profile and/or anticipated downhole drilling conditions. Alternatively, a rotary drill bit including a fixed cutter drill bit with desired bit walk and/or controllability may be selected from existing drill bit designs. 
     Rotary drill bits designed or selected to form a straight hole or vertical wellbore may require approximately zero or neutral bit walk. Rotary drill bits designed or selected for use with a directional drilling system may have an optimum bit walk rate for a desired wellbore profile and/or anticipated downhole drilling conditions. 
     One aspect of the present disclosure may include procedures to evaluate walk tendency of a rotary drill bit under a combination of bit motions including, but not limited to, rotation, axial penetration, side penetration, tilt rate and/or transition drilling. For example, methods and systems incorporating teachings of the present disclosure may be used to simulate drilling through inclined formation interfaces and complex formations with hard stringers disposed in softer formation materials and/or alternating layers of hard and soft formation materials. 
     Drilling a wellbore profile, trajectory, or path using a wide variety of rotary drill bits and bottom hole assemblies may be simulated in three dimensions (3D) using methods and systems incorporating teachings of the present disclosure. Such simulations may be used to design rotary drill bits and/or bottom hole assemblies with optimum bit walk characteristics for drilling a wellbore profile. Such simulation may also be used to select a rotary drill bit and/or components for an associated bottom hole assembly from existing designs with optimum bit walk characteristics for drilling a wellbore profile. 
     Systems and methods incorporating teachings of the present disclosure may be used to simulate drilling various types of wellbores and segments of wellbores using both push-the-bit directional drilling systems and point-the-bit directional drilling systems. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A more complete and thorough understanding of the present disclosure and advantages thereof may be acquired by referring to the following description taken in conjunction with the accompanying drawings, in which like reference numbers indicate like features, and wherein: 
         FIG. 1A  is a schematic drawing in section and in elevation with portions broken away showing one example of a directional wellbore which may be formed by a drill bit designed in accordance with teachings of the present disclosure or selected from existing drill bit designs in accordance with teachings of the present disclosure; 
         FIG. 1B  is a schematic drawing showing a graphical representation of a directional wellbore having a constant bend radius between a generally vertical section and a generally horizontal section which may be formed by a drill bit designed in accordance with teachings of the present disclosure or selected from existing drill bit designs in accordance with teachings of the present disclosure; 
         FIG. 1C  is a schematic drawing showing one example of a system and associate apparatus operable to simulate drilling a complex, directional wellbore in accordance with teachings of the present disclosure; 
         FIG. 2A  is a schematic drawing showing an isometric view with portions broken away of a rotary drill bit with six (6) degrees of freedom which may be used to describe motion of the rotary drill bit in three dimensions in a bit coordinate system; 
         FIG. 2B  is a schematic drawing showing forces applied to a rotary drill bit while forming a substantially vertical wellbore; 
         FIG. 3A  is a schematic representation showing a side force applied to a rotary drill bit at an instant in time in a two dimensional Cartesian bit coordinate system. 
         FIG. 3B  is a schematic representation showing a trajectory of a directional wellbore and a rotary drill bit disposed in a tilt plane at an instant of time in a three dimensional Cartesian hole coordinate system; 
         FIG. 3C  is a schematic representation showing the rotary drill bit in  FIG. 3B  at the same instant of time in a two dimensional Cartesian hole coordinate system; 
         FIG. 4A  is a schematic drawing in section and in elevation with portions broken away showing one example of a push-the-bit directional drilling system adjacent to the end of a wellbore; 
         FIG. 4B  is a graphical representation showing portions of a push-the-bit directional drilling system forming a directional wellbore; 
         FIG. 4C  is a schematic drawing showing an isometric view of a rotary drill bit having various design features which may be optimized for use with a push-the-bit directional drilling system in accordance with teachings of the present disclosure; 
         FIG. 5A  is a schematic drawing in section and in elevation with portions broken away showing one example of a point-the-bit directional drilling system adjacent to the end of a wellbore; 
         FIG. 5B  is a graphical representation showing portions of a point-the-bit directional drilling system forming a directional wellbore; 
         FIG. 5C  is a schematic drawing showing an isometric view of a rotary drill bit having various design features which may be optimized for use with a point-the-bit directional drilling system in accordance with teachings of the present disclosure; 
         FIG. 5D  is a schematic drawing showing an isometric view of a rotary drill bit having various design features which may be optimized for use with a point-the-bit directional drilling system in accordance with teachings of the present disclosure; 
         FIG. 6A  is a schematic drawing in section with portions broken away showing one simulation of forming a directional wellbore using a simulation model incorporating teachings of the present disclosure; 
         FIG. 6B  is a schematic drawing in section with portions broken away showing one example of parameters used to simulate drilling a direction wellbore in accordance with teachings of the present disclosure; 
         FIG. 6C  is a schematic drawing in section with portions broken away showing one simulation of forming a direction wellbore using a prior simulation model; 
         FIG. 6D  is a schematic drawing in section with portions broken away showing one example of forces used to simulate drilling a directional wellbore with a rotary drill bit in accordance with the prior simulation model; 
         FIG. 7A  is a schematic drawing in section with portions broken away showing another example of a rotary drill bit disposed within a wellbore; 
         FIG. 7B  is a schematic drawing showing various features of an active gage and a passive gage disposed on exterior portions of the rotary drill bit of  FIG. 7A ; 
         FIG. 8A  is a schematic drawing in elevation with portions broken away showing one example of interaction between an active gage element and adjacent portions of a wellbore; 
         FIG. 8B  is a schematic drawing taken along lines  8 B- 8 B of  FIG. 8A ; 
         FIG. 8C  is a schematic drawing in elevation with portions broken away showing one example of interaction between a passive gage element and adjacent portions of a wellbore; 
         FIG. 8D  is a schematic drawing taken along lines  8 D- 8 D of  FIG. 8C ; 
         FIG. 9  is a graphical representation of forces used to calculate a walk angle of a rotary drill bit at a downhole location within a wellbore; 
         FIG. 10  is a graphical representation of forces used to calculate a walk angle of a rotary drill bit at a respective downhole location in a wellbore; 
         FIG. 11  is a schematic drawing in section with portions broken away of a rotary drill bit showing changes in dogleg severity with respect to side forces applied to a rotary drill bit during drilling of a directional wellbore; 
         FIG. 12  is a schematic drawing in section with portions broken away of a rotary drill bit showing changes in torque on bit (TOB) with respect to revolutions of a rotary drill bit during drilling of a directional wellbore; 
         FIG. 13A  is a graphical representation of various dimensions associated with a push-the-bit directional drilling system; 
         FIG. 13B  is a graphical representation of various dimensions associated with a point-the-bit directional drilling system; 
         FIG. 14A  is a schematic drawing in section with portions broken away showing interaction between a rotary drill bit and two inclined formations during generally vertical drilling relative to the formation; 
         FIG. 14B  is a schematic drawing in section with portions broken away showing a graphical representation of a rotary drill bit interacting with two inclined formations during directional drilling relative to the formations; 
         FIG. 14C  is a schematic drawing in section with portions broken away showing a graphical representation of a rotary drill bit interacting with two inclined formations during directional drilling of the formations; 
         FIG. 14D  shows one example of a three dimensional graphical simulation incorporating teachings of the present disclosure of a rotary drill bit penetrating a first rock layer and a second rock layer; 
         FIG. 15A  is a schematic drawing showing a graphical representation of a spherical coordinate system which may be used to describe motion of a rotary drill bit and also describe the bottom of a wellbore in accordance with teachings of the present disclosure; 
         FIG. 15B  is a schematic drawing showing forces operating on a rotary drill bit against the bottom and/or the sidewall of a bore hole in a spherical coordinate system; 
         FIG. 15C  is a schematic drawing showing forces acting on a cutter of a rotary drill bit in a cutter local coordinate system; 
         FIG. 16  is a graphical representation of one example of calculations used to estimate cutting depth of a cutter disposed on a rotary drill bit in accordance with teachings of the present disclosure; 
         FIGS. 17A-17G  is a block diagram showing one example of a method for simulating or modeling drilling of a directional wellbore using a rotary drill bit in accordance with teachings of the present disclosure; and 
         FIG. 18  is a graphical representation showing examples of the results of multiple simulations incorporating teachings of the present disclosure of using a rotary drill bit and associated downhole equipment to form a wellbore. 
     
    
    
     DETAILED DESCRIPTION OF THE DISCLOSURE 
     Preferred embodiments of the present disclosure and their advantages may be understood by referring to  FIGS. 1A-17G  of the drawings, like numerals may be used for like and corresponding parts of the various drawings. 
     The term “bottom hole assembly” or “BHA” may be used in this application to describe various components and assemblies disposed proximate to a rotary drill bit at the downhole end of a drill string. Examples of components and assemblies (not expressly shown) which may be included in a bottom hole assembly or BHA include, but are not limited to, a bent sub, a downhole drilling motor, a near bit reamer, stabilizers and down hole instruments. A bottom hole assembly may also include various types of well logging tools (not expressly shown) and other downhole instruments associated with directional drilling of a wellbore. Examples of such logging tools and/or directional drilling equipment may include, but are not limited to, acoustic, neutron, gamma ray, density, photoelectric, nuclear magnetic resonance and/or any other commercially available logging instruments. 
     The term “cutter” may be used in this application to include various types of compacts, inserts, milled teeth, welded compacts and gage cutters satisfactory for use with a wide variety of rotary drill bits. Impact arrestors, which may be included as part of the cutting structure on some types of rotary drill bits, sometimes function as cutters to remove formation materials from adjacent portions of a wellbore. Impact arrestors or any other portion of the cutting structure of a rotary drill bit may be analyzed and evaluated using various techniques and procedures as discussed herein with respect to cutters. Polycrystalline diamond compacts (PDC) and tungsten carbide inserts are often used to form cutters for rotary drill bits. A wide variety of other types of hard, abrasive materials may also be satisfactorily used to form such cutters. 
     The terms “cutting element” and “cutlet” may be used to describe a small portion or segment of an associated cutter which interacts with adjacent portions of a wellbore and may be used to simulate interaction between the cutter and adjacent portions of a wellbore. As discussed later in more detail, cutters and other portions of a rotary drill bit may also be meshed into small segments or portions sometimes referred to as “mesh units” for purposes of analyzing interaction between each small portion or segment and adjacent portions of a wellbore. 
     The term “cutting structure” may be used in this application to include various combinations and arrangements of cutters, face cutters, impact arrestors and/or gage cutters formed on exterior portions of a rotary drill bit. Some fixed cutter drill bits may include one or more blades extending from an associated bit body with cutters disposed of the blades. Various configurations of blades and cutters may be used to form cutting structures for a fixed cutter drill bit. 
     The term “rotary drill bit” may be used in this application to include various types of fixed cutter drill bits, drag bits and matrix drill bits operable to form a wellbore extending through one or more downhole formations. Rotary drill bits and associated components formed in accordance with teachings of the present disclosure may have many different designs and configurations. 
     Simulating drilling a wellbore in accordance with teachings of the present disclosure may be used to optimize the design of various features of a rotary drill bit including, but not limited to, the number of blades or cutter blades, dimensions and configurations of each cutter blade, configuration and dimensions of junk slots disposed between adjacent cutter blades, the number, location, orientation and type of cutters and gages (active or passive) and length of associated gages. The location of nozzles and associated nozzle outlets may also be optimized. 
     Various teachings of the present disclosure may also be used with other types of rotary drill bits having active or passive gages similar to active or passive gages associated with fixed cutter drill bits. For example, a stabilizer (not expressly shown) located relatively close to a roller cone drill bit (not expressly shown) may function similar to a passive gage portion of a fixed cutter drill bit. A near bit reamer (not expressly shown) located relatively close to a roller cone drill bit may function similar to an active gage portion of a fixed cutter drill bit. 
     For fixed cutter drill bits one of the differences between a “passive gage” and an “active gage” is that a passive gage will generally not remove formation materials from the sidewall of a wellbore or borehole while an active gage may at least partially cut into the sidewall of a wellbore or borehole during directional drilling. A passive gage may deform a sidewall plastically or elastically during directional drilling. Mathematically, if we define aggressiveness of a typical face cutter as one (1.0), then aggressiveness of a passive gage is nearly zero (0) and aggressiveness of an active gage may be between 0 and 1.0, depending on the configuration of respective active gage elements. 
     Aggressiveness of various types of active gage elements may be determined by testing and may be inputted into a simulation program such as represented by  FIGS. 17A-17G . Similar comments apply with respect to near bit stabilizers and near bit reamers contacting adjacent portions of a wellbore. Various characteristics of active and passive gages will be discussed in more detail with respect to  FIGS. 7A-8D . 
     The term “straight hole” may be used in this application to describe a wellbore or portions of a wellbore that extends at generally a constant angle relative to vertical. Vertical wellbores and horizontal wellbores are examples of straight holes. 
     The terms “slant hole” and “slant hole segment” may be used in this application to describe a straight hole formed at a substantially constant angle relative to vertical. The constant angle of a slant hole is typically less than ninety (90) degrees and greater than zero (0) degrees. 
     Most straight holes such as vertical wellbores and horizontal wellbores with any significant length will have some variation from vertical or horizontal based in part on characteristics of associated drilling equipment used to form such wellbores. A slant hole may have similar variations depending upon the length and associated drilling equipment used to form the slant hole. 
     The term “directional wellbore” may be used in this application to describe a wellbore or portions of a wellbore that extend at a desired angle or angles relative to vertical. Such angles are greater than normal variations associated with straight holes. A directional wellbore sometimes may be described as a wellbore deviated from vertical. 
     Sections, segments and/or portions of a directional wellbore may include, but are not limited to, a vertical section, a kick off section, a building section, a holding section and/or a dropping section. A vertical section may have substantially no change in degrees from vertical. Holding sections such as slant hole segments and horizontal segments may extend at respective fixed angles relative to vertical and may have substantially zero rate of change in degrees from vertical. Transition sections formed between straight hole portions of a wellbore may include, but are not limited to, kick off segments, building segments and dropping segments. Such transition sections generally have a rate of change in degrees greater than zero. Building segments generally have a positive rate of change in degrees. Dropping segments generally have a negative rate of change in degrees. The rate of change in degrees may vary along the length of all or portions of a transition section or may be substantially constant along the length of all or portions of the transition section. 
     The term “kick off segment” may be used to describe a portion or section of a wellbore forming a transition between the end point of a straight hole segment and the first point where a desired DLS or tilt rate is achieved. A kick off segment may be formed as a transition from a vertical wellbore to an equilibrium wellbore with a constant curvature or tilt rate. A kick off segment of a wellbore may have a variable curvature and a variable rate of change in degrees from vertical (variable tilt rate). 
     A building segment having a relatively constant radius and a relatively constant change in degrees from vertical (constant tilt rate) may be used to form a transition from vertical segments to a slant hole segment or horizontal segment of a wellbore. A dropping segment may have a relatively constant radius and a relatively constant change in degrees from vertical (constant tilt rate) may be used to form a transition from a slant hole segment or a horizontal segment to a vertical segment of a wellbore. See  FIG. 1A . For some applications a transition between a vertical segment and a horizontal segment may only be a building segment having a relatively constant radius and a relatively constant change in degrees from vertical. See  FIG. 1B . Building segments and dropping segments may also be described as “equilibrium” segments. 
     The terms “dogleg severity” or “DLS” may be used to describe the rate of change in degrees of a wellbore from vertical during drilling of the wellbore. DLS is often measured in degrees per one hundred feet (°/100 ft). A straight hole, vertical hole, slant hole or horizontal hole will generally have a value of DLS of approximately zero. DLS may be positive, negative or zero. 
     Tilt angle (TA) may be defined as the angle in degrees from vertical of a segment or portion of a wellbore. A vertical wellbore has a generally constant tilt angle (TA) approximately equal to zero. A horizontal wellbore has a generally constant tilt angle (TA) approximately equal to ninety degrees (90°). 
     Tilt rate (TR) may be defined as the rate of change of a wellbore in degrees (TA) from vertical per hour of drilling. Tilt rate may also be referred to as “steer rate.” 
     
       
         
           
             TR 
             = 
             
               
                  
                 
                   ( 
                   TA 
                   ) 
                 
               
               
                  
                 t 
               
             
           
         
       
         
         
           
             Where t=drilling time in hours 
           
         
       
    
     Tilt rate (TR) of a rotary drill bit may also be defined as DLS times rate of penetration (ROP). 
       TR=DLS×ROP/100=(degrees/hour) 
     Bit tilting motion is often a critical parameter for accurately simulating drilling directional wellbores and evaluating characteristics of rotary drill bits and other downhole tools used with directional drilling systems. Prior two dimensional (2D) and prior three dimensional (3D) bit models and hole models are often unable to consider bit tilting motion due to limitations of Cartesian coordinate systems or cylindrical coordinate systems used to describe bit motion relative to a wellbore. The use of spherical coordinate system to simulate drilling of directional wellbore in accordance with teachings of the present disclosure allows the use of bit tilting motion and associated parameters to enhance the accuracy and reliability of such simulations. 
     Various aspects of the present disclosure may be described with respect to modeling or simulating drilling a wellbore or portions of a wellbore. Dogleg severity (DLS) of respective segments, portions or sections of a wellbore and corresponding tilt rate (TR) may be used to conduct such simulations. Appendix A lists some examples of data including parameters such as simulation run time and simulation mesh size which may be used to conduct such simulations. 
     Various features of the present disclosure may also be described with respect to modeling or simulating drilling of a wellbore based on at least one of three possible drilling modes. See for example,  FIG. 17A . A first drilling mode (straight hole drilling) may be used to simulate forming segments of a wellbore having a value of DLS approximately equal to zero. A second drilling mode (kick off drilling) may be used to simulate forming segments of a wellbore having a value of DLS greater than zero and a value of DLS which varies along portions of an associated section or segment of the wellbore. A third drilling mode (building or dropping) may be used to simulate drilling segments of a wellbore having a relatively constant value of DLS (positive or negative) other than zero. 
     The terms “downhole data” and “downhole drilling conditions” may include, but are not limited to, wellbore data and formation data such as listed on Appendix A. The terms “downhole data” and “downhole drilling conditions” may also include, but are not limited to, drilling equipment operating data such as listed on Appendix A. 
     The terms “design parameters,” “operating parameters,” “wellbore parameters” and “formation parameters” may sometimes be used to refer to respective types of data such as listed on Appendix A. The terms “parameter” and “parameters” may be used to describe a range of data or multiple ranges of data. The terms “operating” and “operational” may sometimes be used interchangeably. 
     Directional drilling equipment may be used to form wellbores having a wide variety of profiles or trajectories. Directional drilling system  20  and wellbore  60  as shown in  FIG. 1A  may be used to describe various features of the present disclosure with respect to simulating drilling all or portions of a wellbore and designing or selecting drilling equipment such as a rotary drill bit based at least in part on such simulations. 
     Directional drilling system  20  may include land drilling rig  22 . However, teachings of the present disclosure may be satisfactorily used to simulate drilling wellbores using drilling systems associated with offshore platforms, semi-submersible, drill ships and any other drilling system satisfactory for forming a wellbore extending through one or more downhole formations. The present disclosure is not limited to directional drilling systems or land drilling rigs. 
     Drilling rig  22  and associated directional drilling equipment  50  may be located proximate well head  24 . Drilling rig  22  also includes rotary table  38 , rotary drive motor  40  and other equipment associated with rotation of drill string  32  within wellbore  60 . Annulus  66  may be formed between the exterior of drill string  32  and the inside diameter of wellbore  60 . 
     For some applications drilling rig  22  may also include top drive motor or top drive unit  42 . Blow out preventors (not expressly shown) and other equipment associated with drilling a wellbore may also be provided at well head  24 . One or more pumps  26  may be used to pump drilling fluid  28  from fluid reservoir or pit  30  to one end of drill string  32  extending from well head  24 . Conduit  34  may be used to supply drilling mud from pump  26  to the one end of drilling string  32  extending from well head  24 . Conduit  36  may be used to return drilling fluid, formation cuttings and/or downhole debris from the bottom or end  62  of wellbore  60  to fluid reservoir or pit  30 . Various types of pipes, tube and/or conduits may be used to form conduits  34  and  36 . 
     Drill string  32  may extend from well head  24  and may be coupled with a supply of drilling fluid such as pit or reservoir  30 . Opposite end of drill string  32  may include bottom hole assembly  90  and rotary drill bit  100  disposed adjacent to end  62  of wellbore  60 . As discussed later in more detail, rotary drill bit  100  may include one or more fluid flow passageways with respective nozzles disposed therein. Various types of drilling fluids may be pumped from reservoir  30  through pump  26  and conduit  34  to the end of drill string  32  extending from well head  24 . The drilling fluid may flow through a longitudinal bore (not expressly shown) of drill string  32  and exit from nozzles formed in rotary drill bit  100 . 
     At end  62  of wellbore  60  drilling fluid may mix with formation cuttings and other downhole debris proximate drill bit  100 . The drilling fluid will then flow upwardly through annulus  66  to return formation cuttings and other downhole debris to well head  24 . Conduit  36  may return the drilling fluid to reservoir  30 . Various types of screens, filters and/or centrifuges (not expressly shown) may be provided to remove formation cuttings and other downhole debris prior to returning drilling fluid to pit  30 . 
     Bottom hole assembly  90  may include various components associated with a measurement while drilling (MWD) system that provides logging data and other information from the bottom of wellbore  60  to directional drilling equipment  50 . Logging data and other information may be communicated from end  62  of wellbore  60  through drill string  32  using MWD techniques and converted to electrical signals at well surface  24 . Electrical conduit or wires  52  may communicate the electrical signals to input device  54 . The logging data provided from input device  54  may then be directed to a data processing system  56 . Various displays  58  may be provided as part of directional drilling equipment  50 . 
     For some applications printer  59  and associated printouts  59   a  may also be used to monitor the performance of drilling string  32 , bottom hole assembly  90  and associated rotary drill bit  100 . Outputs  57  may be communicated to various components associated with operating drilling rig  22  and may also be communicated to various remote locations to monitor the performance of directional drilling system  20 . 
     Wellbore  60  may be generally described as a directional wellbore or a deviated wellbore having multiple segments or sections. Section  60   a  of wellbore  60  may be defined by casing  64  extending from well head  24  to a selected downhole location. Remaining portions of wellbore  60  as shown in  FIG. 1A  may be generally described as “open hole” or “uncased.” 
     Teachings of the present disclosure may be used to simulate drilling a wide variety of vertical, directional, deviated, slanted and/or horizontal wellbores. Teachings of the present disclosure are not limited to simulating drilling wellbore  60 , designing drill bits for use in drilling wellbore  60  or selecting drill bits from existing designs for use in drilling wellbore  60 . 
     Wellbore  60  as shown in  FIG. 1A  may be generally described as having multiple sections, segments or portions with respective values of DLS. The tilt rate for rotary drill bit  100  during formation of wellbore  60  will be a function of DLS for each segment, section or portion of wellbore  60  times the rate of penetration for rotary drill bit  100  during formation of the respective segment, section or portion thereof. The tilt rate of rotary drill bit  100  during formation of straight hole sections or vertical section  80   a  and horizontal section  80   c  will be approximately equal to zero. 
     Section  60   a  extending from well head  24  may be generally described as a vertical, straight hole section with a value of DLS approximately equal to zero. When the value of DLS is zero, rotary drill bit  100  will have a tile rate of approximately zero during formation of the corresponding section of wellbore  60 . 
     A first transition from vertical section  60   a  may be described as kick off section  60   b . For some applications the value of DLS for kick off section  60   b  may be greater than zero and may vary from the end of vertical section  60   a  to the beginning of a second transition segment or building section  60   c . Building section  60   c  may be formed with relatively constant radius  70   c  and a substantially constant value of DLS. Building section  60   c  may also be referred to as third section  60   c  of wellbore  60 . 
     Fourth section  60   d  may extend from build section  60   c  opposite from second section  60   b . Fourth section  60   d  may be described as a slant hole portion of wellbore  60 . Section  60   d  may have a DLS of approximately zero. Fourth section  60   d  may also be referred to as a “holding” section. 
     Fifth section  60   e  may start at the end of holding section  60   d . Fifth section  60   e  may be described as a “drop” section having a generally downward looking profile. Drop section  60   e  may have relatively constant radius  70   e.    
     Sixth section  60   f  may also be described as a holding section or slant hole section with a DLS of approximately zero. Section  60   f  as shown in  FIG. 1A  is being formed by rotary drill bit  100 , drill string  32  and associated components of drilling system  20 . 
       FIG. 1B  is a graphical representation of a specific type of directional wellbore represented by wellbore  80 . For this example wellbore  80  may include three segments or three sections—vertical section  80   a , building section  80   b  and horizontal section  80   c . Vertical section  80   a  and horizontal section  80   c  may be straight holes with a value of DLS approximately equal to zero. Building section  80   b  may have a constant radius corresponding with a constant rate of change in degrees from vertical and a constant value of DLS. Tilt rate during formation building section  80   b  may be constant if ROP of a drill bit forming build section  80   b  remains constant. 
     Movement or motion of a rotary drill bit and associated drilling equipment in three dimensions (3D) during formation of a segment, section or portion of a wellbore may be defined by a Cartesian coordinate system (X, Y, and Z axes) and/or a spherical coordinate system (two angles φ and θ and a single radius ρ) in accordance with teachings of the present disclosure. Examples of Cartesian coordinate systems are shown in FIGS.  2 A and  3 A- 3 C. Examples of spherical coordinate systems are shown in  FIGS. 15A and 16 . Various aspects of the present disclosure may include translating the location of downhole drilling equipment and adjacent portions of a wellbore between a Cartesian coordinate system and a spherical coordinate system.  FIG. 15A  shows one example of translating the location of a single point between a Cartesian coordinate system and a spherical coordinate system. 
       FIG. 1C  shows one example of a system operable to simulate drilling a complex, directional wellbore in accordance with teachings of this present disclosure. System  300  may include one or more processing resources  310  operable to run software and computer programs incorporating teaching of the present disclosure. A general purpose computer may be used as a processing resource. All or portions of software and computer programs used by processing resource  310  may be stored one or more memory resources  320 . One or more input devices  330  may be operate to supply data and other information to processing resources  310  and/or memory resources  320 . A keyboard, keypad, touch screen and other digital input mechanisms may be used as an input device. Examples of such data are shown on Appendix A. 
     Processing resources  310  may be operable to simulate drilling a directional wellbore in accordance with teachings of the present disclosure. Processing resources  310  may be operate to use various algorithms to make calculations or estimates based on such simulations. 
     Display resources  340  may be operable to display both data input into processing resources  310  and the results of simulations and/or calculations performed in accordance with teachings of the present disclosure. A copy of input data and results of such simulations and calculations may also be provided at printer  350 . 
     For some applications, processing resource  310  may be operably connected with communication network  360  to accept inputs from remote locations and to provide the results of simulation and associated calculations to remote locations and/or facilities such as directional drilling equipment  50  shown in  FIG. 1A . 
     A Cartesian coordinate system generally includes a Z axis and an X axis and a Y axis which extend normal to each other and normal to the Z axis. See for example  FIG. 2A . A Cartesian bit coordinate system may be defined by a Z axis extending along a rotational axis or bit rotational axis of the rotary drill bit. See  FIG. 2A . A Cartesian hole coordinate system (sometimes referred to as a “downhole coordinate system” or a “wellbore coordinate system”) may be defined by a Z axis extending along a rotational axis of the wellbore. See  FIG. 3B . In  FIG. 2A  the X, Y and Z axes include subscript (b)  to indicate a “bit coordinate system”. In  FIGS. 3A ,  3 B and  3 C the X, Y and Z axes include subscript (h)  to indicate a “hole coordinate system”. 
       FIG. 2A  is a schematic drawing showing rotary drill bit  100 . Rotary drill bit  100  may include bit body  120  having a plurality of blades  128  with respective junk slots or fluid flow paths  140  formed therebetween. A plurality of cutting elements  130  may be disposed on the exterior portions of each blade  128 . Various parameters associated with rotary drill bit  100  including, but not limited to, the location and configuration of blades  128 , junk slots  140  and cutting elements  130 . Such parameters may be designed in accordance with teachings of the present disclosure for optimum performance of rotary drill bit  100  in forming portions of a wellbore. 
     Each blade  128  may include respective gage surface or gage portion  154 . Gage surface  154  may be an active gage and/or a passive gage. Respective gage cutter  130   g  may be disposed on each blade  128 . A plurality of impact arrestors  142  may also be disposed on each blade  128 . Additional information concerning impact arrestors may be found in U.S. Pat. Nos. 6,003,623, 5,595,252 and 4,889,017. 
     Rotary drill bit  100  may translate linearly relative to the X, Y and Z axes as shown in  FIG. 2A  (three (3) degrees of freedom). Rotary drill bit  100  may also rotate relative to the X, Y and Z axes (three (3) additional degrees of freedom). As a result movement of rotary drill bit  100  relative to the X, Y and Z axes as shown in  FIGS. 2A and 2B , rotary drill bit  100  may be described as having six (6) degrees of freedom. 
     Movement or motion of a rotary drill bit during formation of a wellbore may be fully determined or defined by six (6) parameters corresponding with the previously noted six degrees of freedom. The six parameters as shown in  FIG. 2A  include rate of linear motion or translation of rotary drill bit  100  relative to respective X, Y and Z axes and rotational motion relative to the same X, Y and Z axes. These six parameters are independent of each other. 
     For straight hole drilling these six parameters may be reduced to revolutions per minute (RPM) and rate of penetration (ROP). For kick off segment drilling these six parameters may be reduced to RPM, ROP, dogleg severity (DLS), bend length (B L ) and azimuth angle of an associated tilt plane. See tilt plane  170  in  FIG. 3B . For equilibrium drilling these six parameters may be reduced to RPM, ROP and DLS based on the assumption that the rotational axis of the associated rotary drill bit will move in the same vertical plane or tilt plane. 
     For calculations related to steerability only forces acting in an associated tilt plane are considered. Therefore an arbitrary azimuth angle may be selected usually equal to zero. For calculations related to bit walk forces in the associated tilt plane and forces in a plane perpendicular to the tilt plane are considered. 
     In a bit coordinate system, rotational axis or bit rotational axis  104   a  of rotary drill bit  100  corresponds generally with Z axis  104  of the associated bit coordinate system. When sufficient force from rotary drill string  32  has been applied to rotary drill bit  100 , cutting elements  130  will engage and remove adjacent portions of a downhole formation at bottom hole or end  62  of wellbore  60 . Removing such formation materials will allow downhole drilling equipment including rotary drill bit  100  and associated drill string  32  to tilt or move linearly relative to adjacent portions of wellbore  60 . 
     Various kinematic parameters associated with forming a wellbore using a rotary drill bit may be based upon revolutions per minute (RPM) and rate of penetration (ROP) of the rotary drill bit into adjacent portions of a downhole formation. Arrow  110  may be used to represent forces which move rotary drill bit  100  linearly relative to rotational axis  104   a . Such linear forces typically result from weight applied to rotary drill bit  100  by drill string  32  and may be referred to as “weight on bit” or WOB. 
     Rotational force  112  may be applied to rotary drill bit  100  by rotation of drill string  32 . Revolutions per minute (RPM) of rotary drill bit  100  may be a function of rotational force  112 . Rotation speed (RPM) of drill bit  100  is generally defined relative to the rotational axis of rotary drill bit  100  which corresponds with Z axis  104 . 
     Arrow  116  indicates rotational forces which may be applied to rotary drill bit  100  relative to X axis  106 . Arrow  118  indicates rotational forces which may be applied to rotary drill bit  100  relative to Y axis  108 . Rotational forces  116  and  118  may result from interaction between cutting elements  130  disposed on exterior portions of rotary drill bit  100  and adjacent portions of bottom hole  62  during the forming of wellbore  60 . Rotational forces applied to rotary drill bit  100  along X axis  106  and Y axis  108  may result in tilting of rotary drill bit  100  relative to adjacent portions of drill string  32  and wellbore  60 . 
       FIG. 2B  is a schematic drawing showing rotary drill bit  100  disposed within vertical section or straight hole section  60   a  of wellbore  60 . During the drilling of a vertical section or any other straight hole section of a wellbore, the bit rotational axis of rotary drill bit  100  will generally be aligned with a corresponding rotational axis of the straight hole section. The incremental change or the incremental movement of rotary drill bit  100  in a linear direction during a single revolution may be represented by AZ in  FIG. 2B . 
     Rate of penetration (ROP) of a rotary drill bit is typically a function of both weight on bit (WOB) and revolutions per minute (RPM). For some applications a downhole motor (not expressly shown) may be provided as part of bottom hole assembly  90  to also rotate rotary drill bit  100 . The rate of penetration of a rotary drill bit is generally stated in feet per hour. 
     The axial penetration of rotary drill bit  100  may be defined relative to bit rotational axis  104   a  in an associated bit coordinate system. A side penetration rate or lateral penetration rate of rotary drill bit  100  may be defined relative to an associated hole coordinate system. Examples of a hole coordinate system are shown in  FIGS. 3A ,  3 B and  3 C.  FIG. 3A  is a schematic representation of a model showing side force  114  applied to rotary drill bit  100  relative to X axis  106  and Y axis  108 . Angle  72  formed between force vector  114  and X axis  106  may correspond approximately with angle  172  associated with tilt plane  170  as shown in  FIG. 3B . A tilt plane may be defined as a plane extending from an associated Z axis or vertical axis in which dogleg severity (DLS) or tilting of the rotary drill bit occurs. 
     Various forces may be applied to rotary drill bit  100  to cause movement relative to X axis  106  and Y axis  108 . Such forces may be applied to rotary drill bit  100  by one or more components of a directional drilling system included within bottom hole assembly  90 . See  FIGS. 4A ,  4 B,  5 A and  5 B. Various forces may also be applied to rotary drill bit  100  relative to X axis  106  and Y axis  108  in response to engagement between cutting elements  130  and adjacent portions of a wellbore. 
     During drilling of straight hole segments of wellbore  60 , side forces applied to rotary drill bit  100  may be substantially minimized (approximately zero side forces) or may be balanced such that the resultant value of any side forces will be approximately zero. Straight hole segments of wellbore  60  as shown in  FIG. 1A  include, but are not limited to, vertical section  60   a , holding section or slant hole section  60   d , and holding section or slant hole section  60   f.    
     One of the benefits of the present disclosure may include the ability to design a rotary drill bit having either substantially zero side forces or balanced sided forces while drilling a straight hole segment of a wellbore. As a result, any side forces applied to a rotary drill bit by associated cutting elements may be substantially balanced and/or reduced to a small value such that rotary drill bit  100  will have either substantially zero tendency to walk or a neutral tendency to walk relative to a vertical axis. 
     During formation of straight hole segments of wellbore  60 , the primary direction of movement or translation of rotary drill bit  100  will be generally linear relative to an associated longitudinal axis of the respective wellbore segment and relative to associated bit rotational axis  104   a . See  FIG. 2B . During the drilling of portions of wellbore  60  having a DLS with a value greater than zero or less than zero, a side force (F s ) or equivalent side force may be applied to rotary drill bit to cause formation of corresponding wellbore segments  60   b ,  60   c  and  60   e.    
     For some applications such as when a push-the-bit directional drilling system is used with a rotary drill bit, an applied side force may result in a combination of bit tilting and side cutting or lateral penetration of adjacent portions of a wellbore. For other applications such as when a point-the-bit directional drilling system is used with an associated rotary drill bit, side cutting or lateral penetration may generally be very small or may not even occur. When a point-the-bit directional drilling system is used with a rotary drill bit, directional portions of a wellbore may be formed primarily as a result of bit penetration along an associated bit rotational axis and tilting of the rotary drill bit relative to a vertical axis. 
       FIGS. 3A ,  3 B and  3 C are graphical representations of various kinematic parameters which may be satisfactorily used to model or simulate drilling segments or portions of a wellbore having a value of DLS greater than zero.  FIG. 3A  shows a schematic cross section of rotary drill bit  100  in two dimensions relative to a Cartesian bit coordinate system. The bit coordinate system is defined in part by X axis  106  and Y axis  108  extending from bit rotational axis  104   a .  FIGS. 3B and 3C  show graphical representations of rotary drill bit  100  during drilling of a transition segment such as kick off segment  60   b  of wellbore  60  in a Cartesian hole coordinate system defined in part by Z axis  74 , X axis  76  and Y axis  78 . 
     A side force is generally applied to a rotary drill bit by an associated directional drilling system to form a wellbore having a desired profile or trajectory using the rotary drill bit. For a given set of drilling equipment design parameters and a given set of downhole drilling conditions, a respective side force must be applied to an associated rotary drill bit to achieve a desired DLS or tilt rate. Therefore, forming a directional wellbore using a point-the-bit directional drilling system, a push-the-bit directional drilling system or any other directional drilling system may be simulated using substantially the same model incorporating teachings of the present disclosure by determining a required bit side force to achieve an expected DLS or tilt rate for each segment of a directional wellbore. 
       FIG. 3A  shows side force  114  extending at angle  72  relative to X axis  106 . Side force  114  may be applied to rotary drill bit  100  by directional drilling system  20 . Angle  72  (sometimes referred to as an “azimuth” angle) extends from rotational axis  104   a  of rotary drill bit  100  and represents the angle at which side force  114  will be applied to rotary drill bit  100 . For some applications side force  114  may be applied to rotary drill bit  100  at a relatively constant azimuth angle. 
     Side force  114  will typically result in movement of rotary drill bit  100  laterally relative to adjacent portions of wellbore  60 . Directional drilling systems such as rotary drill bit steering units shown in  FIGS. 4A and 5A  may be used to either vary the amount of side force  114  or to maintain a relatively constant amount of side force  114  applied to rotary drill bit  100 . Directional drilling systems may also vary the azimuth angle at which a side force is applied to correspond with a desired wellbore trajectory. 
     Side force  114  may be adjusted or varied to cause associated cutting elements  130  to interact with adjacent portions of a downhole formation so that rotary drill bit  100  will follow profile or trajectory  68   b , as shown in  FIG. 3B , or any other desired profile. Profile  68   b  may correspond approximately with a longitudinal axis extending through kick off segment  60   b . Rotary drill bit  100  will generally move only in tilt plane  170  during formation of kickoff segment  60   b  if rotary drill bit  100  has zero walk tendency or neutral walk tendency. Tilt plane  170  may also be referred to as an “azimuth plane”. 
     Respective tilting angles (not expressly shown) of rotary drill bit  100  will vary along the length of trajectory  68   b . Each tilting angle of rotary drill bit  100  as defined in a hole coordinate system (Z h , X h , Y h ) will generally lie in tilt plane  170 . As previously noted, during the formation of a kickoff segment of a wellbore, tilting rate in degrees per hour as indicated by arrow  174  will also increase along trajectory  68   b . For use in simulating forming kickoff segment  60   b , side penetration rate, side penetration azimuth angle, tilting rate and tilt plane azimuth angle may be defined in a hole coordinate system which includes Z axis  74 , X axis  76  and Y axis  78 . 
     Arrow  174  corresponds with the variable tilt rate of rotary drill bit  100  relative to vertical at any one location along trajectory  68   b . During movement of rotary drill bit  100  along profile or trajectory  68   a , the respective tilt angle at each location on trajectory  68   a  will generally increase relative to Z axis  74  of the hole coordinate system shown in  FIG. 3B . For embodiments such as shown in  FIG. 3B , the tilt angle at each point on trajectory  68   b  will be approximately equal to an angle formed by a respective tangent extending from the point in question and intersecting Z axis  74 . Therefore, the tilt rate will also vary along the length of trajectory  168 . 
     During the formation of kick off segment  60   b  and any other portions of a wellbore in which the value of DLS is either greater than or less than zero and is not constant, rotary drill bit  100  may experience side cutting motion, bit tilting motion and axial penetration in a direction associated with cutting or removing of formation materials from the end or bottom of a wellbore. 
     For embodiments such as shown in  FIGS. 3A ,  3 B and  3 C directional drilling system  20  may cause rotary drill bit  100  to move in the same azimuth plane  170  during formation of kick off segment  60   b .  FIGS. 3B and 3C  show relatively constant azimuth plane angle  172  relative to the X axis  76  and Y axis  78 . Arrow  114  as shown in  FIG. 3B  represents a side force applied to rotary drill bit  100  by directional drilling system  20 . Arrow  114  will generally extend normal to rotational axis  104   a  of rotary drill bit  100 . Arrow  114  will also be disposed in tilt plane  170 . A side force applied to a rotary drill bit in a tilt plane by an associate rotary drill bit steering unit or directional drilling system may also be referred to as a “steer force.” 
     During the formation of a directional wellbore such as shown in  FIG. 3B , without consideration of bit walk, rotational axis  104   a  of rotary drill bit  100  and a longitudinal axis of bottom hole assembly  90  may generally lie in tilt plane  170 . Rotary drill bit  100  will experience tilting motion in tilt plane  170  while rotating relative to rotational axis  104   a . The tilting motion may result from a side force or steer force applied to rotary drill bit  100  by a directional steering unit such as shown in  FIGS. 4A AND 4B  or  5 A and  5 B of an associated directional drilling system. The tilting motion results from a combination of side forces and/or axial forces applied to rotary drill bit  100  by directional drilling system  20 . 
     If rotary drill bit  100  walks, either left or right, bit  100  will generally not move in the same azimuth plane or tilt plane  170  during formation of kickoff segment  60   b . As discussed later in more detail with respect to  FIGS. 9 and 10  rotary drill bit  100  may also experience a walk force (F w ) as indicated by arrow  177 . Arrow  177  as shown in  FIGS. 3B and 3C  represents a walk force which will cause rotary drill bit  100  to “walk” left relative to tilt plane  170 . Simulations of forming a wellbore in accordance with teachings of the present disclosure may be used to modify cutting elements, bit face profiles, gages and other characteristics of a rotary drill bit to substantially reduce or minimize the walk force represented by arrow  177  or to provide a desired right walk rate or left walk rate. 
     Various features of the present disclosure will be discussed with respect to directional drilling equipment including rotary drills such as shown in  FIGS. 4A ,  4 B,  51  and  5 B. These features may be described with respect to vertical axis  74  or Z axis  74  of a Cartesian hole coordinate system such as shown in  FIG. 3B . During drilling of a vertical segment or other types of straight hole segments, vertical axis  74  will generally be aligned with and correspond to an associate longitudinal axis of the vertical segment or straight hole segment. Vertical axis  74  will also generally be aligned with and correspond to an associate bit rotational axis during such straight hole drilling. 
       FIG. 4A  shows portions of bottom hole assembly  90   a  disposed in a generally vertical portion  60   a  of wellbore  60  as rotary drill bit  100   a  begins to form kick off segment  60   b . Bottom hole assembly  90   a  may include rotary drill bit steering unit  92   a  operable to apply side force  114  to rotary drill bit  100   a . Steering unit  92   a  may be one portion of a push-the-bit directional drilling system. 
     Push-the-bit directional drilling systems generally require simultaneous axial penetration and side penetration in order to drill directionally. Bit motion associated with push-the-bit directional drilling systems is often a combination of axial bit penetration, bit rotation, bit side cutting and bit tilting. Simulation of forming a wellbore using a push-the-bit directional drilling system based on a 3D model operable to consider bit tilting motion may result in a more accurate simulation. Some of the benefits of using a 3D model operable to consider bit tilting motion in accordance with teachings of the present disclosure will be discussed with respect to  FIGS. 6A-6D . 
     Steering unit  92   a  may extend arm  94   a  to apply force  114   a  to adjacent portions of wellbore  60  and maintain desired contact between steering unit  92   a  and adjacent portions of wellbore  60 . Side forces  114  and  114   a  may be approximately equal to each other. If there is no weight on rotary drill bit  100   a , no axial penetration will occur at end or bottom hole  62  of wellbore  60 . Side cutting will generally occur as portions of rotary drill bit  100   a  engage and remove adjacent portions of wellbore  60   a.    
       FIG. 4B  shows various parameters associated with a push-the-bit directional drilling system. Steering unit  92   a  will generally include bent subassembly  96   a . A wide variety of bent subassemblies (sometimes referred to as “bent subs”) may be satisfactorily used to allow drill string  32  to rotate drill bit  100   a  while steering unit  92   a  pushes or applies required force to move rotary drill bit  100   a  at a desired tilt rate relative to vertical axis  74 . Arrow  200  represents the rate of penetration relative to the rotational axis of rotary drill bit  100   a  (ROP a ). Arrow  202  represents the rate of side penetration of rotary drill bit  200  (ROP s ) as steering unit  92   a  pushes or directs rotary drill bit  100   a  along a desired trajectory or path. 
     Tilt rate  174  and associated tilt angle may remain relatively constant for some portions of a directional wellbore such as a slant hole segment or a horizontal hole segment. For other portions of a directional wellbore tilt rate  174  may increase during formation of respective portions of the wellbore such as a kick off segment. Bend length  204   a  may be a function of the distance between arm  94   a  contacting adjacent portions of wellbore  60  and the end of rotary drill bit  100   a.    
     Bend length (L Bend ) may be used as one of the inputs to simulate forming portions of a wellbore in accordance with teachings of the present disclosure. Bend length or tilt length may be generally described as the distance from a fulcrum point of an associated bent subassembly to a furthest location on a “bit face” or “bit face profile” of an associated rotary drill bit. The furthest location may also be referred to as the extreme end of the associated rotary drill bit. 
     Some directional drilling techniques and systems may not include a bent subassembly. For such applications bend length may be taken as the distance from a first contact point between an associated bottom hole assembly with adjacent portions of the wellbore to an extreme end of a bit face on an associated rotary drill bit. 
     During formation of a kick off section or any other portion of a deviated wellbore, axial penetration of an associated drill bit will occur in response to weight on bit (WOB) and/or axial forces applied to the drill bit by a downhole drilling motor. Also, bit tilting motion relative to a bent sub, not side cutting or lateral penetration, will typically result from a side force or lateral force applied to the drill bit as a component of WOB and/or axial forces applied by a downhole drilling motor. A model such as shown in  FIGS. 17A-17G  may then be used to obtain the total bit lateral force (F lat ) as a function of time. Bit motion is often a combination of bit axial penetration and bit tilting while drilling a directional wellbore. 
     When bit axial penetration rate is very small (close to zero) and the distance from the bit to the bent sub or bend length is very large, side penetration or side cutting may be a dominated motion of the drill bit. The resulting bit motion may or may not be continuous when using a push-the-bit directional drilling system depending upon the weight on bit, revolutions per minute, applied side force and other parameters associated with rotary drill bit  100   a.    
       FIG. 4C  is a schematic drawing showing one example of a rotary drill bit which may be designed in accordance with teachings of the present disclosure for optimum performance in a push-the-bit directional drilling system. For example, a three dimensional model such as shown in  FIGS. 17A-17G  may be used to design a rotary drill bit with optimum active and/or passive gage length for use with a push-the-bit directional drilling system. Rotary drill bit  100   a  may be generally described as a fixed cutter drill bit. For some applications rotary drill bit  100   a  may also be described as a matrix drill bit, steel body drill bit and/or a PDC drill bit. 
     Rotary drill bit  100   a  may include bit body  120   a  with shank  122   a . The dimensions and configuration of bit body  120   a  and shank  122   a  may be substantially modified as appropriate for each rotary drill bit. See  FIGS. 5C and 5D . 
     Shank  122   a  may include bit breaker slots  124   a  formed on the exterior thereof. Pin  126   a  may be formed as an integral part of shank  122   a  extending from bit body  120   a . Various types of threaded connections, including but not limited to, API connections and premium threaded connections may be formed on the exterior of pin  126   a.    
     A longitudinal bore (not expressly shown) may extend from end  121   a  of pin  126   a  through shank  122   a  and into bit body  120   a . The longitudinal bore may be used to communicate drilling fluids from drilling string  32  to one or more nozzles (not expressly shown) disposed in bit body  120   a . Nozzle outlet  150   a  is shown in  FIG. 4C . 
     A plurality of cutter blades  128   a  may be disposed on the exterior of bit body  120   a . Respective junk slots or fluid flow slots  148   a  may be formed between adjacent blades  128   a . Each blade  128  may include a plurality of cutting elements  130  formed from very hard materials associated with forming a wellbore in a downhole formation. For some applications cutting elements  130  may also be described as “face cutters”. 
     Respective gage cutter  130   g  may be disposed on each blade  128   a . For embodiments such as shown in  FIG. 4C  rotary drill bit  100   a  may be described as having an active gage or active gage elements disposed on exterior portion of each blade  128   a . Gage surface  154  of each blade  128   a  may also include a plurality of active gage elements  156 . Active gage elements  156  may be formed from various types of hard abrasive materials sometimes referred to as “hardfacing”. Active elements  156  may also be described as “buttons” or “gage inserts”. As discussed later in more detail with respect to  FIGS. 7B ,  8 A and  8 B active gage elements may contact adjacent portions of a wellbore and remove some formation materials as a result of such contact. 
     Exterior portions of bit body  120   a  opposite from shank  122   a  may be generally described as a “bit face” or “bit face profile.” As discussed later in more detail with respect to rotary drill bit  100   e  as shown in  FIG. 7A , a bit face profile may include a generally cone-shaped recess or indentation having a plurality of inner cutters and a plurality of shoulder cutters disposed on exterior portions of each blade  128   a . One of the benefits of the present disclosure includes the ability to design a rotary drill bit having an optimum number of inner cutters, shoulder cutters and gage cutters to provide desired walk rate, bit steerability, and bit controllability. 
       FIG. 5A  shows portions of bottom hole assembly  90   b  disposed in a generally vertical section of wellbore  60   a  as rotary drill bit  100   b  begins to form kick off segment  60   b . Bottom hole assembly  90   b  includes rotary drill bit steering unit  92   b  which may provide one portion of a point-the-bit directional drilling system. 
     Point-the-bit directional drilling systems typically form a directional wellbore using a combination of axial bit penetration, bit rotation and bit tilting. Point-the-bit directional drilling systems may not produce side penetration such as described with respect to steering unit  92   b  in  FIG. 5A . Therefore, bit side penetration is generally not created by point-the-bit directional drilling systems to form a directional wellbore. It is particularly advantageous to simulate forming a wellbore using a point-the-bit directional drilling system using a three dimensional model operable to consider bit tilting motion in accordance with teachings of the present disclosure. One example of a point-the-bit directional drilling system is the Geo-Pilot® Rotary Steerable System available from Sperry Drilling Services at Halliburton Company. 
       FIG. 5B  is a graphical representation showing various parameters associated with a point-the-bit directional drilling system. Steering unit  92   b  will generally include bent subassembly  96   b . A wide variety of bent subassemblies may be satisfactorily used to allow drill string  32  to rotate drill bit  100   c  while bent subassembly  96   b  directs or points drill bit  100   c  at angle away from vertical axis  174 . Some bent subassemblies have a constant “bent angle”. Other bent subassemblies have a variable or adjustable “bent angle”. Bend length  204   b  is a function of the dimensions and configurations of associated bent subassembly  96   b.    
     As previously noted, side penetration of rotary drill bit will generally not occur in a point-the-bit directional drilling system. Arrow  200  represents the rate of penetration along rotational axis of rotary drill bit  100   c . Additional features of a model used to simulate drilling of directional wellbores for push-the-bit directional drilling systems and point-the-bit directional drilling systems will be discussed with respect to  FIGS. 9-13B . 
       FIG. 5C  is a schematic drawing showing one example of a rotary drill bit which may be designed in accordance with teachings of the present disclosure for optimum performance in a point-the-bit directional drilling system. For example, a three dimensional model such as shown in  FIGS. 17A-17F  may be used to design a rotary drill bit with an optimum ratio of inner cutters, shoulder cutters and gage cutters in forming a directional wellbore for use with a point-the-bit directional drilling system. Rotary drill bit  100   c  may be generally described as a fixed cutter drill bit. For some applications rotary drill bit  100   c  may also be described as a matrix drill bit steel body drill bit and/or a PDC drill bit. Rotary drill bit  100   c  may include bit body  120   c  with shank  122   c.    
     Shank  122   c  may include bit breaker slots  124   c  formed on the exterior thereof. Shank  122   c  may also include extensions of associated blades  128   c . As shown in  FIG. 5C  blades  128   c  may extend at an especially large spiral or angle relative to an associated bit rotational axis. 
     One of the characteristics of rotary drill bits used with point-the-bit directional drilling systems may be increased length of associated gage surfaces as compared with push-the-bit directional drilling systems. 
     Threaded connection pin (not expressly shown) may be formed as part of shank  122   c  extending from bit body  120   c . Various types of threaded connections, including but not limited to, API connections and premium threaded connections may be used to releasably engage rotary drill bit  100   c  with a drill string. 
     A longitudinal bore (not expressly shown) may extend through shank  122   c  and into bit body  120   c . The longitudinal bore may be used to communicate drilling fluids from an associated drilling string to one or more nozzles  152  disposed in bit body  120   c.    
     A plurality of cutter blades  128   c  may be disposed on the exterior of bit body  120   c . Respective junk slots or fluid flow slots  148   c  may be formed between adjacent blades  128   a . Each cutter blade  128   c  may include a plurality of cutters  130   d . For some applications cutters  130   d  may also be described as “cutting inserts”. Cutters  130   d  may be formed from very hard materials associated with forming a wellbore in a downhole formation. The exterior portions of bit body  120   c  opposite from shank  122   c  may be generally described as having a “bit face profile” as described with respect to rotary drill bit  100   a.    
       FIG. 5D  is a schematic drawing showing one example of a rotary drill bit which may be designed in accordance with teachings of the present disclosure for optimum performance in a point-the-bit directional drilling system. Rotary drill bit  100   d  may be generally described as a fixed cutter drill bit. For some applications rotary drill bit  100   d  may also be described as a matrix drill bit and/or a PDC drill bit. Rotary drill bit  100   d  may include bit body  120   d  with shank  122   d.    
     Shank  122   d  may include bit breaker slots  124   d  formed on the exterior thereof. Pin threaded connection  126   d  may be formed as an integral part of shank  122   d  extending from bit body  120   d . Various types of threaded connections, including but not limited to, API connections and premium threaded connections may be formed on the exterior of pin  126   d.    
     A longitudinal bore (not expressly shown) may extend from end  121   d  of pin  126   d  through shank  122   c  and into bit body  120   d . The longitudinal bore may be used to communicate drilling fluids from drilling string  32  to one or more nozzles  152  disposed in bit body  120   d.    
     A plurality of cutter blades  128   d  may be disposed on the exterior of bit body  120   d . Respective junk slots or fluid flow slots  148   d  may be formed between adjacent blades  128   d . Each cutter blade  128   d  may include a plurality of cutters  130   f . Respective gage cutters  130   g  may also be disposed on each blade  128   d . For some applications cutters  130   f  and  130   g  may also be described as “cutting inserts” formed from very hard materials associated with forming a wellbore in a downhole formation. The exterior portions of bit body  120   d  opposite from shank  122   d  may be generally described as having a “bit face profile” as described with respect to rotary drill bit  100   a.    
     Blades  128  and  128   d  may also spiral or extend at an angle relative to the associated bit rotational axis. One of the benefits of the present disclosure includes simulating drilling portions of a directional wellbore to determine optimum blade length, blade width and blade spiral for a rotary drill bit which may be used to form all or portions of the directional wellbore. For embodiments represented by rotary drill bits  100   a ,  100   c  and  100   d  associated gage surfaces may be formed proximate one end of blades  128   a ,  128   c  and  128   d  opposite an associated bit face profile. 
     For some applications bit bodies  120   a ,  120   c  and  120   d  may be formed in part from a matrix of very hard materials associated with rotary drill bits. For other applications bit body  120   a ,  120   c  and  120   d  may be machined from various metal alloys satisfactory for use in drilling wellbores in downhole formations. Examples of matrix type drill bits are shown in U.S. Pat. Nos. 4,696,354 and 5,099,929. 
       FIG. 6A  is a schematic drawing showing one example of a simulation of forming a directional wellbore using a directional drilling system such as shown in  FIGS. 4A and 4B  or  FIGS. 5A and 5B . The simulation shown in  FIG. 6A  may generally correspond with forming a transition from vertical segment  60   a  to kick off segment  60   b  of wellbore  60  such as shown in  FIGS. 4A and 5B . This simulation may be based on several parameters including, but not limited to, bit tilting motion applied to a rotary drill bit during formation of kick off segment  60   b . The resulting simulation provides a relatively smooth or uniform inside diameter as compared with the step hole simulation as shown in  FIG. 6C . 
     A rotary drill bit may be generally described as having three components or three portions for purposes of simulating forming a wellbore in accordance with teachings of the present disclosure. The first component or first portion may be described as “face cutters” or “face cutting elements” which may be primarily responsible for drilling action associated with removal of formation materials to form an associated wellbore. For some types of rotary drill bits the “face cutters” may be further divided into three segments such as “inner cutters,” “shoulder cutters” and/or “gage cutters”. See, for example,  FIGS. 6B and 7A . Penetration force (F p ) is often the principal or primary force acting upon face cutters. 
     The second portion of a rotary drill bit may include an active gage or gages responsible for protecting face cutters and maintaining a relatively uniform inside diameter of an associated wellbore by removing formation materials adjacent portions of the wellbore. Active gage cutting elements generally contact and remove partially the sidewall portions of a wellbore. 
     The third component of a rotary drill bit may be described as a passive gage or gages which may be responsible for maintaining uniformity of the adjacent portions of the wellbore (typically the sidewall or inside diameter) by deforming formation materials in adjacent portions of the wellbore. For active and passive gages the primary force is generally a normal force which extends generally perpendicular to the associated gage face either active or passive. 
     Gage cutters may be disposed adjacent to active and/or passive gage elements. Gage cutters are not considered as part of an active gage or passive gage for purposes of simulating forming a wellbore as described in this application. However, teachings of the present disclosure may be used to conduct simulations which include gage cutters as part of an adjacent active gage or passive gage. The present disclosure is not limited to the previously described three components or portions of a rotary drill bit. 
     For some applications a three dimensional (3D) model incorporating teachings of the present disclosure may be operable to evaluate respective contributions of various components of a rotary drill bit to forces acting on the rotary drill bit. The 3D model may be operable to separately calculate or estimate the effect of each component on bit walk rate, bit steerability and/or bit controllability for a given set of downhole drilling parameters. As a result, a model such as shown in  FIGS. 17A-17G  may be used to design various portions of a rotary drill bit and/or to select a rotary drill bit from existing bit designs for use in forming a wellbore based upon directional behavior characteristics associated with changing face cutter parameters, active gage parameters and/or passive gage parameters. Similar techniques may be used to design or select components of a bottom hole assembly or other portions of a directional drilling system in accordance with teachings of the present disclosure. 
       FIG. 6B  shows some of the parameters which would be applied to rotary drill bit  100  during formation of a wellbore. Rotary drill bit  100  is shown by solid lines in  FIG. 6B  during formation of a vertical segment or straight hole segment of a wellbore. Bit rotational axis  100   a  of rotary drill bit  100  will generally be aligned with the longitudinal axis of the associated wellbore, and a vertical axis associated with a corresponding bit hole coordinate system. 
     Rotary drill bit  100  is also shown in dotted lines in  FIG. 6B  to illustrate various parameters used to simulate drilling kick off segment  60   b  in accordance with teachings of the present disclosure. Instead of using bit side penetration or bit side cutting motion, the simulation shown in  FIG. 6A  is based upon tilting of rotary drill bit  100  as shown in dotted lines relative to vertical axis. 
       FIG. 6C  is a schematic drawing showing a typical prior simulation which used side cutting penetration as a step function to represent forming a directional wellbore. For the simulation shown in  FIG. 6C , the formation of wellbore  260  is shown as a series of step holes  260   a ,  260   b ,  260   c ,  260   d  and  260   e . As shown in  FIG. 6D  the assumption made during this simulation was that rotational axis  104   a  of rotary drill bit  100  remained generally aligned with a vertical axis during the formation of each step hole  260   a ,  260   b ,  260   c , etc. 
     Simulations of forming directional wellbores in accordance with teachings of the present disclosure have indicated the influence of gage length on bit walk rate, bit steerability and bit controllability. Rotary drill bit  100   e  as shown in  FIGS. 7A and 7B  may be described as having both an active gage and a passive gage disposed on each blade  128   e . Active gage portions of rotary drill bit  100   e  may include active elements formed from hardfacing or abrasive materials which remove formation material from adjacent portions of sidewall or inside diameter  63  of wellbore segment  60 . See for example active gage elements  156  shown in  FIG. 4C . 
     Rotary drill bit  100   e  as shown in  FIGS. 7A and 7B  may be described as having a plurality of blades  128   e  with a plurality of cutting elements  130  disposed on exterior portions of each blade  128   e . For some applications cutting elements  130  may have substantially the same configuration and design. For other applications various types of cutting elements and impact arrestors (not expressly shown) may also be disposed on exterior portions of blades  128   e . Exterior portions of rotary drill bit  100   e  may be described as forming a “bit face profile”. 
     The bit face profile for rotary drill bit  100   e  as shown in  FIGS. 7A and 7B  may include recessed portion or cone shaped section  132   e  formed on the end of rotary drill bit  100   e  opposite from shank  122   e . Each blade  128   e  may include respective nose  134   e  which defines in part an extreme end of rotary drill bit  100   e  opposite from shank  122   e . Cone section  132   e  may extend inward from respective noses  134   e  toward bit rotational axis  104   e . A plurality of cutting elements  130   i  may be disposed on portions of each blade  128   e  between respective nose  134   e  and rotational axis  104   e . Cutters  130   i  may be referred to as “inner cutters”. 
     Each blade  128   e  may also be described as having respective shoulder  136   e  extending outward from respective nose  134   e . A plurality of cutter elements  130   s  may be disposed on each shoulder  136   e . Cutting elements  130   s  may sometimes be referred to as “shoulder cutters.” Shoulder  136   e  and associated shoulder cutters  130   s  cooperate with each other to form portions of the bit face profile of rotary drill bit  100   e  extending outward from cone shaped section  132   e.    
     A plurality of gage cutters  130   g  may also be disposed on exterior portions of each blade  128   e . Gage cutters  130   g  may be used to trim or define inside diameter or sidewall  63  of wellbore segment  60 . Gage cutters  130   g  and associated portions of each blade  128   e  form portions of the bit face profile of rotary drill bit  100   e  extending from shoulder cutters  130   s.    
     For embodiments such as shown in  FIGS. 7A and 7B  each blade  128   e  may include active gage portion  138  and passive gage portion  139 . Various types of hardfacing and/or other hard materials (not expressly shown) may be disposed on each active gage portion  138 . Each active gage portion  138  may include a positive taper angle  158  as shown in  FIG. 7B . Each passive gage portion may include respective positive taper angle  159   a  as shown in  FIG. 7B . Active and passive gages on conventional rotary drill bits often have positive taper angles. 
     Simulations conducted in accordance with teachings of the present disclosure may be used to calculate side forces applied to rotary drill bit  100   e  by each segment or component of a bit face profile. For example inner cutters  130   i , shoulder cutters  130   s  and gage cutters  130   g  may apply respective side forces to rotary drill bit  100   e  during formation of a directional wellbore. Active gage portions  138  and passive gage portions  139  may also apply respective side forces to rotary drill bit  100   e  during formation of a directional wellbore. A steering difficulty index may be calculated for each segment or component of a bit face profile to determine if design changes should be made to the respective component. 
     Simulations conducted in accordance with teachings of the present disclosure have indicated that forming a passive gage with a negative taper angle such as angle  159   b  shown in  FIG. 7B  may provide improved or enhanced steerability when forming a directional wellbore. The size of negative taper angle  159   b  may be limited to prevent undesired contact between an associated passive gage and adjacent portions of a sidewall during drilling of a vertical wellbore or straight hole segments of a wellbore. 
     Since bend length associated with a push-the-bit directional drilling system is usually relatively large (greater than 20 times associated bit size), most of the cutting action associated with forming a directional wellbore may be a combination of axial bit penetration, bit rotation, bit side cutting and bit tilting. See  FIGS. 4A ,  4 B and  13 A. Simulations conducted in accordance with teachings of the present disclosure have indicated that an active gage with a gage gap such as gage gap  162  shown in  FIGS. 7A and 7B  may significantly reduce the amount of bit side force required to form a directional wellbore using a push-the-bit directional drilling system. A passive gage with a gage gap such as gage gap  164  shown in  FIGS. 7A and 7B  may also reduce required amounts of bit side force, but the effect is much less than that of an active gage with a gage gap. 
     Since bend length associated with a point-the-bit directional drilling system is usually relatively small (less than 12 times associated bit size), most of the cutting action associated with forming a directional wellbore may be a combination of axial bit penetration, bit rotation and bit tilting. See  FIGS. 5A ,  5 B and  13 B. Simulations conducted in accordance with teachings of the present disclosure have shown that rotary drill bits with positively tapered gages and/or gage gaps may be satisfactorily used with point-the-bit directional drilling systems. Simulations conducted in accordance with teachings of the present disclosure have further indicated that there is an optimum set of tapered gage angles and associated gage gaps depending upon respective bend length of each directional drilling system and required DLS for each segment of a directional wellbore. 
     Simulations conducted in accordance with teachings of the present disclosure have indicated that forming passive gage  139  with optimum negative taper angle  159   b  may result in contact between portions of passive gage  139  and adjacent portions of a wellbore to provide a fulcrum point to direct or guide rotary drill bit  100   e  during formation of a directional wellbore. The size of negative taper angle  159   b  may be limited to prevent undesired contact between passive gage  139  and adjacent portions of sidewall  63  during drilling of a vertical or straight hole segments of a wellbore. Such simulations have also indicated potential improvements in steerability and controllability by optimizing the length of passive gages with negative taper angles. For example, forming a passive gage with a negative taper angle on a rotary drill bit in accordance with teachings of the present disclosure may allow reducing the bend length of an associated rotary drill bit steering unit. The length of a bend subassembly included as part of the directional steering unit may be reduced as a result of having a rotary drill bit with an increased length in combination with a passive gage having a negative taper angle. 
     Simulations incorporating teachings of the present disclosure have indicated that a passive gage having a negative taper angle may facilitate tilting of an associated rotary drill bit during kick off drilling. Such simulations have also indicated benefits of installing one or more gage cutters at optimum locations on an active gage portion and/or passive gage portion of a rotary drill bit to remove formation materials from the inside diameter of an associated wellbore during a directional drilling phase. These gage cutters will typically not contact the sidewall or inside diameter of a wellbore while drilling a vertical segment or straight hole segment of the directional wellbore. 
     Passive gage  139  with an appropriate negative taper angle  159   b  and an optimum length may contact sidewall  63  during formation of an equilibrium portion and/or kick off portion of a wellbore. Such contact may substantially improve steerability and controllability of a rotary drill bit and associated steering difficulty index (SD index ). Such simulations have also indicated that multiple tapered gage portions and/or variable tapered gage portions may be satisfactorily used with both point-the-bit and push-the-bit directional drilling systems. 
       FIGS. 8A and 8B  show interaction between active gage element  156  and adjacent portions of sidewall  63  of wellbore segment  60   a .  FIGS. 8C and 8D  show interaction between passive gage element  157  and adjacent portions of sidewall  63  of wellbore segment  60   a . Active gage element  156  and passive gage element  157  may be relatively small segments or portions of respective active gage  138  and passive gage  139  which contacts adjacent portions of sidewall  63 . Active and passive gage elements may be used in simulations similar to previously described cutlets. 
     Arrow  180   a  represents an axial force (F a ) which may be applied to active gage element  156  as active gage element engages and removes formation materials from adjacent portions of sidewall  63  of wellbore segment  60   a . Arrow  180   p  as shown in  FIG. 8C  represents an axial force (F a ) applied to passive gage cutter  130   p  during contact with sidewall  63 . Axial forces applied to active gage  130   g  and passive gage  130   p  may be a function of the associated rate of penetration of rotary drill bit  100   e.    
     Arrow  182   a  associated with active gage element represents drag force (F d ) associated with active gage element  156  penetrating and removing formation materials from adjacent portions of sidewall  63 . A drag force (F d ) may sometimes be referred to as a tangent force (F t ) which generates torque on an associate gage element, cutlet, or mesh unit. The amount of penetration in inches is represented by Δ as shown in  FIG. 8B . 
     Arrow  182   p  represents the amount of drag force (F d ) applied to passive gage element  130   p  during plastic and/or elastic deformation of formation materials in sidewall  63  when contacted by passive gage  157 . The amount of drag force associated with active gage element  156  is generally a function of rate of penetration of associated rotary drill bit  100   e  and depth of penetration of respective gage element  156  into adjacent portions of sidewall  63 . The amount of drag force associated with passive gage element  157  is generally a function of the rate of penetration of associated rotary drill bit  100   e  and elastic and/or plastic deformation of formation materials in adjacent portions of sidewall  63 . 
     Arrow  184   a  as shown in  FIG. 8B  represents a normal force (F n ) applied to active gage element  156  as active gage element  156  penetrates and removes formation materials from sidewall  63  of wellbore segment  60   a . Arrow  184   p  as shown in  FIG. 8D  represents a normal force (F n ) applied to passive gage element  157  as passive gage element  157  plastically or elastically deforms formation material in adjacent portions of sidewall  63 . Normal force (F n ) is directly related to the cutting depth of an active gage element into adjacent portions of a wellbore or deformation of adjacent portions of a wellbore by a passive gage element. Normal force (F n ) is also directly related to the cutting depth of a cutter into adjacent portions of a wellbore. 
     The following algorithms may be used to estimate or calculate forces associated with contact between an active and passive gage and adjacent portions of a wellbore. The algorithms are based in part on the following assumptions:
         An active gage may remove some formation material from adjacent portions of a wellbore such as sidewall  63 . A passive gage may deform adjacent portions of a wellbore such as sidewall  63 . Formation materials immediately adjacent to portions of a wellbore such as sidewall  63  may be satisfactorily modeled as a plastic/elastic material.       

     For each cutlet or small element of an active gage which removes formation material: 
         F   n   =ka   1 *Δ 1   +ka   2 *Δ 2    
     
       
      
       F 
       a 
       =ka 
       3 
       *F 
       r  
      
     
     
       
      
       F 
       d 
       =ka 
       4 
       *F 
       r  
      
     
     Where Δ 1  is the cutting depth of a respective cutlet (gage element) extending into adjacent portions of a wellbore, and Δ 2  is the deformation depth of hole wall by a respective cutlet. 
     ka 1 , ka 2 , ka 3  and ka 4  are coefficients related to rock properties and fluid properties often determined by testing of anticipated downhole formation material. 
     For each cutlet or small element of a passive gage which deforms formation material: 
     
       
      
       F 
       n 
       =kp 
       1 
       *Δp  
      
     
     
       
      
       F 
       a 
       =kp 
       2 
       *F 
       r  
      
     
     
       
      
       F 
       d 
       =kp 
       3 
       *F 
       r  
      
     
     Where Δp is depth of deformation of formation material by a respective cutlet of adjacent portions of the wellbore. 
     kp 1 , kp 2 , kp 3  are coefficients related to rock properties and fluid properties and may be determined by testing of anticipated downhole formation material. 
     Many rotary drill bits have a tendency to “walk” or move laterally relative to a longitudinal axis of a wellbore while forming the wellbore. The tendency of a rotary drill bit to walk or move laterally may be particularly noticeable when forming directional wellbores and/or when the rotary drill bit penetrates adjacent layers of different formation material and/or inclined formation layers. An evaluation of bit walk rates requires consideration of all forces acting on rotary drill bit  100  which extend at an angle relative to tilt plane  170 . Such forces include interactions between bit face profile active and/or passive gages associated with rotary drill bit  100  and adjacent portions of the bottom hole may be evaluated. 
       FIG. 9  is a schematic drawing showing portions of rotary drill bit  100  in section in a two dimensional hole coordinate system represented by X axis  76  and Y axis  78 . Arrow  114  represents a side force applied to rotary drill bit  100  from directional drilling system  20  in tilt plane  170 . This side force generally acts normal to bit rotational axis  104   a  of rotary drill bit  100 . Arrow  176  represents side cutting or side displacement (D s ) of rotary drill bit  100  projected in the hole coordinate system in response to interactions between exterior portions of rotary drill bit  100  and adjacent portions of a downhole formation. Bit walk angle  186  is measured from F s  to D s . 
     When angle  186  is less than zero (opposite to bit rotation direction represented by arrow  178 ) rotary drill bit  100  will have a tendency to walk to the left of applied side force  114  and titling plane  170 . When angle  186  is greater than zero (the same as bit rotation direction represented by arrow  178 ) rotary drill bit  100  will have a tendency to walk right relative to applied side force  114  and tilt plane  170 . When bit walk angle  186  is approximately equal to zero (0), rotary drill bit  100  will have approximately a zero (0) walk rate or neutral walk tendency. 
       FIG. 10  is a schematic drawing showing an alternative definition of bit walk angle when a side displacement (D s ) or side cutting motion represented by arrow  176   a  is applied to bit  100  during simulation of forming a directional wellbore. An associated force represented by arrow  114   c  required to act on rotary drill bit  100  to produce the applied side displacement (D s ) may be calculated and projected in the same hole coordinate system. Applied side displacement (D s ) represented by arrow  176   a  and calculated force (F c ) represented by arrow  114   c  form bit walk angle  186 . Bit walk angle  186  is measured from F c  to D s . 
     When angle  186  is less than zero (opposite to bit rotation direction represented by arrow  178 ), rotary drill bit  100  will have a tendency to walk to the left of calculated side force  176  and titling plane  170 . When angle  186  is greater than zero (the same as bit rotation direction represented by arrow  178 ) rotary drill bit  100  will have a tendency to walk right relative to calculated side force  176  and tilt plane  170 . When bit walk angle  186  is approximately equal to zero (0), rotary drill bit  100  will have approximately a zero (0) walk rate or neutral walk tendency. 
     As discussed later in this application both walk force (F w ) and walk moment or bending moment (M w ) along with an associated bit steer rate and steer force may be used to calculate a resulting bit walk rate. However, the value of walk force and walk moment are generally small compared to an associated steer force and therefore need to be calculated accurately. Bit walk rate may be a function of bit geometry and downhole drilling conditions such as rate of penetration, revolutions per minute, lateral penetration rate, bit tilting rate or steer rate and downhole formation characteristics. 
     Simulations of forming a directional wellbore based on a 3D model incorporating teachings of the present disclosure indicate that for a given axial penetration rate and a given revolutions per minute and a given bottom hole assembly configuration that there is a critical tilt rate. When the tilt rate is greater than the critical tilt rate, the associated drill bit may begin to walk either right or left relative to the associated wellbore. Simulations incorporating teachings of the present disclosure indicate that transition drilling through an inclined formation such as shown in  FIGS. 14A ,  14 B and  14 C may change a bit walk tendencies from bit walk right to bit walk left. 
     For some applications the magnitude of bit side forces required to achieve desired DLS or tilt rates for a given set of drilling equipment parameters and downhole drilling conditions may be used as an indication of associated bit steerability or controllability. See  FIG. 11  for one example. Fluctuations in the amount of bit side force, torque on bit (TOB) and/or bit bending moment may also be used to provide an evaluation of bit controllability or bit stability during the formation of various portions of a directional wellbore. See  FIG. 12  for one example. 
       FIG. 11  is a schematic drawing showing rotary drill bit  100  in solid lines in a first position associated with forming a generally vertical section of a wellbore. Rotary drill bit  100  is also shown in dotted lines in  FIG. 11  showing a directional portion of a wellbore such as kick off segment  60   a . The graph shown in  FIG. 11  indicates that the amount of bit side force required to produce a tilt rate corresponding with the associated dogleg severity (DLS) will generally increase as the dogleg severity of the deviated wellbore increases. The shape of curve  194  as shown in  FIG. 11  may be a function of both rotary drill bit design parameters and associated downhole drilling conditions. 
     As previously noted fluctuations in drilling parameters such as bit side force, torque on bit and/or bit bending moment may also be used to provide an evaluation of bit controllability or bit stability. 
       FIG. 12  is a graphical representation showing variations in torque on bit with respect to revolutions per minute during the tilting of rotary drill bit  100  as shown in  FIG. 12 . The amount of variation or the ΔTOB as shown in  FIG. 12  may be used to evaluate the stability of various rotary drill bit designs for the same given set of downhole drilling conditions. The graph shown in  FIG. 11  is based on a given rate of penetration, a given RPM and a given set of downhole formation data. 
     For some applications steerability of a rotary drill bit may be evaluated using the following steps. Design data for the associated drilling equipment may be inputted into a three dimensional model incorporating teachings of the present disclosure. For example design parameters associated with a drill bit may be inputted into a computer system (see for example  FIG. 1C ) having a software application such as shown and described in  FIGS. 17A-17G . Alternatively, rotary drill bit design parameters may be read into a computer program from a bit design file or drill bit design parameters such as International Association of Drilling Contractors (IADC) data may be read into the computer program. 
     Drilling equipment operating data such as RPM, ROP, and tilt rate for an associated rotary drill bit may be selected or defined for each simulation. A tilt rate or DLS may be defined for one or more formation layers and an associated inclination angle for adjacent formation layers. Formation data such as rock compressive strength, transition layers and inclination angle of each transition layer may also be defined or selected. 
     Total run time, total number of bit rotations and/or respective time intervals per the simulation may also be defined or selected for each simulation. 3D simulations or modeling using a system such as shown in  FIG. 1C  and software or computer programs as outlined in  FIGS. 17A-17G  may then be conducted to calculate or estimate various forces including side forces acting on an associated rotary drill bit or other associated downhole drilling equipment. Average walk rate of an associated rotary drill bit may also be calculated over the simulated time interval. 
     The preceding steps may be conducted by changing DLS or tilt rate and repeated to develop a curve of bit side forces corresponding with each value of DLS. A curve of side force versus DLS may then be plotted (See  FIG. 11 ) and bit steerability calculated. Another set of rotary drill bit operating parameters may then be inputted into the computer and steps 3 through 7 repeated to provide additional curves of side force (F s ) versus dogleg severity (DLS). Bit steerability may then be defined by the set of curves showing side force versus DLS. 
       FIG. 13A  may be described as a graphical representation showing portions of a bottom hole assembly and rotary drill bit  100   a  associated with a push-the-bit directional drilling system. A push-the-bit directional drilling system may be sometimes have a bend length greater than 20 to 35 times an associated bit size or corresponding bit diameter in inches. Bend length  204   a  associated with a push-the-bit directional drilling system is generally much greater than length  206   a  of rotary drill bit  100   a . Bend length  204   a  may also be much greater than or equal to the diameter D B1  of rotary drill bit  100   a.    
       FIG. 13B  may be generally described as a graphical representation showing portions of a bottom hole assemble and rotary drill bit  100   c  associated with a point-the-bit directional drilling system. A point-the-bit directional drilling system may sometimes have a bend length less than or equal to 12 times the bit size. For the example shown in  FIG. 13B , bend length  204   c  associated with a point-the-bit directional drilling system may be approximately two or three times greater than length  206   c  of rotary drill bit  100   c . Length  206   c  of rotary drill bit  100   c  may be significantly greater than diameter D B2  of rotary drill bit  100   c . The length of a rotary drill bit used with a push-the-bit drilling system will generally be less than the length of a rotary drill bit used with a point-the-bit directional drilling system. 
     Due to the combination of tilting and axial penetration, rotary drill bits may have side cutting motion. This is particularly true during kick off drilling. However, the rate of side cutting is generally not a constant for a drill bit and is changed along drill bit axis. The rate of side penetration of rotary drill bits  100   a  and  100   c  is represented by arrow  202 . The rate of side penetration is generally a function of tilting rate and associated bend length  204   a  and  204   d . For rotary drill bits having a relatively long bit length and particularly a relatively long gage length such as shown in  FIG. 5C , the rate of side penetration at point  208  may be much less than the rate of side penetration at point  210 . As the length of a rotary drill bit increases the side penetration rate decreases from the shank as compared with the extreme end of the rotary drill bit. The difference in rate of side penetration between point  208  and  210  may be small, but the effects on bit steerability may be very large. 
     Simulations conducted in accordance with teachings of the present disclosure may be used to calculate bit walk rate for a given set of data (See for example Appendix A) Average walk force and/or average walk rate may also be calculated for one or more simulated time intervals. 
     Walk force (F w ) may be obtained by simulating forming a directional wellbore as a function of drilling time. Walk force (F w ) corresponds with the amount of force which is applied to a rotary drill bit in a plane extending generally perpendicular to an associated azimuth plane or tilt plane. 
       FIGS. 14A ,  14 B and  14 C are schematic drawings showing representations of various interactions between rotary drill bit  100  and adjacent portions of first formation  221  and second formation layer  222 . Software or computer programs such as outlined in  FIGS. 17A-17G  may be used to simulate or model interactions with multiple or laminated rock layers forming a wellbore. 
     For some applications first formation layer may have a rock compressibility strength which is substantially larger than the rock compressibility strength of second layer  222 . For embodiments such as shown in  FIGS. 14A ,  14 B and  14 C first layer  221  and second layer  222  may be inclined or disposed at inclination angle  224  (sometimes referred to as a “transition angle”) relative to each other and relative to vertical. Inclination angle  224  may be generally described as a positive angle relative associated vertical axis  74 . 
     Three dimensional simulations may be performed to evaluate forces required for rotary drilling bit  100  to form a substantially vertical wellbore extending through first layer  221  and second layer  222 . See  FIG. 14A . Three dimensional simulations may also be performed to evaluate forces which must be applied to rotary drill bit  100  to form a directional wellbore extending through first layer  221  and second layer  222  at various angles such as shown in  FIGS. 14B and 14C . A simulation using software or a computer program such as outlined in  FIG. 17A-17G  may be used calculate the side forces which must be applied to rotary drill bit  100  to form a wellbore to tilt rotary drill bit  100  at an angle relative to vertical axis  74 . 
       FIG. 14D  is a schematic drawing showing a three dimensional meshed representation of the bottom hole or end of wellbore segment  60   a  corresponding with rotary drill bit  100  forming a generally vertical or horizontal wellbore extending therethrough as shown in  FIG. 14A . Transition plane  226  as shown in  FIG. 14D  represents a dividing line or boundary between rock formation layer and rock formation layer  222 . Transition plane  226  may extend along inclination angle  224  relative to vertical. 
     The terms “meshed” and “mesh analysis” may describe analytical procedures used to evaluate and study complex structures such as cutters, active and passive gages, other portions of a rotary drill bit, other downhole tools associated with drilling a wellbore, bottom hole configurations of a wellbore and/or other portions of a wellbore. The interior surface of end  62  of wellbore  60   a  may be finely meshed into many small segments or “mesh units” to assist with determining interactions between cutters and other portions of a rotary drill bit and adjacent formation materials as the rotary drill bit removes formation materials from end  62  to form wellbore  60 . See  FIG. 14D . The use of mesh units may be particularly helpful to analyze distributed forces and variations in cutting depth of respective mesh units or cutlets as an associated cutter interacts with adjacent formation materials. 
     Three dimensional mesh representations of the bottom of a wellbore and/or various portions of a rotary drill bit and/or other downhole tools may be used to simulate interactions between the rotary drill bit and adjacent portions of the wellbore. For example cutting depth and cutting area of each cutting element or cutlet during one revolution of the associated rotary drill bit may be used to calculate forces acting on each cutting element. Simulation may then update the configuration or pattern of the associated bottom hole and forces acting on each cutter. For some applications the nominal configuration and size of a unit such as shown in  FIG. 14D  may be approximately 0.5 mm per side. However, the actual configuration size of each mesh unit may vary substantially due to complexities of associated bottom hole geometry and respective cutters used to remove formation materials. 
     Systems and methods incorporating teachings of the present disclosure may also be used to simulate or model forming a directional wellbore extending through various combinations of soft and medium strength formation with multiple hard stringers disposed within both soft and/or medium strength formations. Such formations may sometimes be referred to as “interbedded” formations. Simulations and associated calculations may be similar to simulations and calculations as described with respect to  FIGS. 14A-14D . 
     Spherical coordinate systems such as shown in  FIGS. 15A-15C  may be used to define the location of respective cutlets, gage elements and/or mesh units of a rotary drill bit and adjacent portions of a wellbore. The location of each mesh unit of a rotary drill bit and associated wellbore may be represented by a single valued function of angle phi (φ), angle theta (θ) and radius rho (ρ) in three dimensions (3D) relative to Z axis  74 . The same Z axis  74  may be used in a three dimensional Cartesian coordinate system or a three dimensional spherical coordinate system. 
     The location of a single point such as center  198  of cutter  130  may be defined in the three dimensional spherical coordinate system of  FIG. 15A  by angle φ and radius ρ. This same location may be converted to a Cartesian hole coordinate system of X h , Y h , Z h  using radius r and angle theta (θ) which corresponds with the angular orientation of radius r relative to X axis  76 . Radius r intersects Z axis  74  at the same point radius ρ intersects Z axis  74 . Radius r is disposed in the same plane as Z axis  74  and radius ρ. Various examples of algorithms and/or matrices which may be used to transform data in a Cartesian coordinate system to a spherical coordinate system and to transform data in a spherical coordinate system to a Cartesian coordinate system are discussed later in this application. 
     As previously noted, a rotary drill bit may generally be described as having a “bit face profile” which includes a plurality of cutters operable to interact with adjacent portions of a wellbore to remove formation materials therefrom. Examples of a bit face profile and associated cutters are shown in  FIGS. 2A ,  2 B,  4 C,  5 C,  5 D,  7 A and  7 B. The cutting edge of each cutter on a rotary drill bit may be represented in three dimensions using either a Cartesian coordinate system or a spherical coordinate system. 
       FIGS. 15B and 15C  show graphical representations of various forces associated with portions of cutter  130  interacting with adjacent portions of bottom hole  62  of wellbore  60 . For examples such as shown in  FIG. 15B  cutter  130  may be located on the shoulder of an associated rotary drill bit. 
       FIGS. 15B and 15C  also show one example of a local cutter coordinate system used at a respective time step or interval to evaluate or interpolate interaction between one cutter and adjacent portions of a wellbore. A local cutter coordinate system may more accurately interpolate complex bottom hole geometry and bit motion used to update a 3D simulation of a bottom hole geometry such as shown in  FIG. 14D  based on simulated interactions between a rotary drill bit and adjacent formation materials. Numerical algorithms and interpolations incorporating teachings of the present disclosure may more accurately calculate estimated cutting depth and cutting area of each cutter. 
     In a local cutter coordinate system there are two forces, drag force (F d ) and penetration force (F p ), acting on cutter  130  during interaction with adjacent portions of wellbore  60 . When forces acting on each cutter  130  are projected into a bit coordinate system there will be three forces, axial force (F a ), drag force (F d ) and penetration force (F p ). The previously described forces may also act upon impact arrestors and gage cutters. 
     For purposes of simulating cutting or removing formation materials adjacent to end  62  of wellbore  60  as shown in  FIG. 15B , cutter  130  may be divided into small elements or cutlets  131   a ,  131   b ,  131   c  and  131   d . Forces represented by arrows F e  may be simulated as acting on cutlet  131   a - 131   d  at respective points such as  191  and  200 . For example, respective drag forces may be calculated for each cutlet  131   a - 131   d  acting at respective points such as  191  and  200 . The respective drag forces may be summed or totaled to determine total drag force (F d ) acting on cutter  130 . In a similar manner, respective penetration forces may also be calculated for each cutlet  131   a - 131   d  acting at respective points such as  191  and  200 . The respective penetration forces may be summed or totaled to determine total penetration force (F p ) acting on cutter  130 . 
       FIG. 15C  shows cutter  130  in a local cutter coordinate system defined in part by cutter axis  198 . Drag force (F d ) represented by arrow  196  corresponds with the summation of respective drag forces calculated for each cutlet  131   a - 131   d . Penetration force (F p ) represented by arrow  192  corresponds with the summation of respective penetration forces calculated for each cutlet  131   a - 131   d.    
       FIG. 16  shows portions of bottom hole  62  in a spherical hole coordinate system defined in part by Z axis  74  and radius R h . The configuration of a bottom hole generally corresponds with the configuration of an associated bit face profile used to form the bottom hole. For example, portion  62   i  of bottom hole  62  may be formed by inner cutters  130   i . Portion  62   s  of bottom hole  62  may be formed by shoulder cutters  130   s . Side wall  63  may be formed by gage cutters  130   g.    
     Single point  200  as shown in  FIG. 16  is located on the exterior of cutter  130   s . In the hole coordinate system, the location of point  200  is a function of angle φ h  and radius ρ h .  FIG. 16  also shows the same single point  200  on the exterior of cutter  130   s  in a local cutter coordinate system defined by vertical axis Z c  and radius R c . In the local cutter coordinate system, the location of point  200  is a function of angle φ c  and radius ρ c . Cutting depth  212  associated with single point  200  and associated removal of formation material from bottom hole  62  corresponds with the shortest distance between point  200  and portion  62   s  of bottom hole  62 . 
     Simulating Straight Hole Drilling (Path B, Algorithm A) 
     The following algorithms may be used to simulate interaction between portions of a cutter and adjacent portions of a wellbore during removal of formation materials proximate the end of a straight hole segment. Respective portions of each cutter engaging adjacent formation materials may be referred to as cutting elements or cutlets. Note that in the following steps y axis represents the bit rotational axis. The x and z axes are determined using the right hand rule. Drill bit kinematics in straight hole drilling is fully defined by ROP and RPM. 
     Given ROP, RPM, current time t, dt, current cutlet position (x i , y i , z i ) or (θ i , φ i , ρ i ) 
     (1) Cutlet position due to penetration along bit axis Y may be obtained 
       x p =x i   ;y   p   =y   i +rop* d   t ;z p =z i    
     (2) Cutlet position due to bit rotation around the bit axis may be obtained as follows: 
       N_rot={010} 
     Accompany matrix: 
     
       
         
           
             
               M 
               rot 
             
             = 
             
               
                 
                   0 
                 
                 
                   
                     
                       - 
                       N_rot 
                     
                      
                     
                       ( 
                       3 
                       ) 
                     
                   
                 
                 
                   
                     N_rot 
                      
                     
                       ( 
                       2 
                       ) 
                     
                   
                 
               
               
                 
                   
                     N_rot 
                      
                     
                       ( 
                       3 
                       ) 
                     
                   
                 
                 
                   0 
                 
                 
                   
                     
                       - 
                       N_rot 
                     
                      
                     
                       ( 
                       1 
                       ) 
                     
                   
                 
               
               
                 
                   
                     
                       - 
                       N_rot 
                     
                      
                     
                       ( 
                       2 
                       ) 
                     
                   
                 
                 
                   
                     N_rot 
                      
                     
                       ( 
                       1 
                       ) 
                     
                   
                 
                 
                   0 
                 
               
             
           
         
       
     
     The transform matrix is: 
         R _rot=cos ω tI +(1−cos ω t ) N _rot N _rot′+sin ω tM _rot,         where I is 3×3 unit matrix and ω is bit rotation speed.       
     New cutlet position after bit rotation is: 
     
       
         
           
             
               
                 
                   x 
                   
                     i 
                     + 
                     1 
                   
                 
               
               
                 
                     
                 
               
               
                 
                   x 
                   p 
                 
               
             
             
               
                 
                   y 
                   
                     i 
                     + 
                     1 
                   
                 
               
               
                 
                   = 
                   
                     R 
                     rot 
                   
                 
               
               
                 
                   y 
                   p 
                 
               
             
             
               
                 
                   z 
                   
                     i 
                     + 
                     1 
                   
                 
               
               
                 
                     
                 
               
               
                 
                   z 
                   p 
                 
               
             
           
         
       
     
     (3) Calculate the cutting depth for each cutlet by comparing (x i+1 , y i+1 , z i+1 ) of this cutlet with hole coordinate (x h , y h , z h ) where X h =x i+1  &amp; z h =z i+1 , and d p =y i+1 −y h ; 
     (4) Calculate the cutting area of this cutlet 
       A cutlet= d   p   *d   r            where d r  is the width of this cutlet.       
     (5) Determine which formation layer is cut by this cutlet by comparing y i+1  with hole coordinate y h , if Y i+1 &lt;y h  then layer A is cut. y h  may be solved from the equation of the transition plane in Cartesian coordinate: 
         l ( x   h   −x   1 )+ m ( y   h   −y   1 )+ n ( z   h   −z   1 )=0 
     where (x 1 ,y 1 ,z 1 ) is any point on the plane and {l,m,n} is normal direction of the transition plane. 
     (6) Save layer information, cutting depth and cutting area into 3D matrix at each time step for each cutlet for force calculation. 
     (7) Update the associated bottom hole matrix removed by the respective cutlets or cutters. 
     Simulating Kick Off Drilling (Path C) 
     The following algorithms may be used to simulate interaction between portions of a cutter and adjacent portions of a wellbore during removal of formation materials proximate the end of a kick off segment. Respective portions of each cutter engaging adjacent formation materials may be referred to as cutting elements or cutlets. Note that in the following steps, y axis is the bit axis, x and z are determined using the right hand rule. Drill bit kinematics in kick-off drilling is defined by at least four parameters: ROP, RPM, DLS and bend length. 
     Given ROP, RPM, DLS and bend length, L bend , current time t, dt, current cutlet position (x i , y i , z i ) or (θ i , φ i , ρ i ) 
     (1) Transform the current cutlet position to bend center: 
       x i =x i ; 
     
       
      
       y 
       i 
       =y 
       i 
       −L 
       bend  
      
     
       z i =z i ; 
     (2) New cutlet position due to tilt may be obtained by tilting the bit around vector N_tilt an angle γ: 
       N_tilt={sin α0.0 cos α} 
     Accompany matrix: 
     
       
         
           
             
               M 
               tilt 
             
             = 
             
               
                 
                   0 
                 
                 
                   
                     
                       - 
                       N_tilt 
                     
                      
                     
                       ( 
                       3 
                       ) 
                     
                   
                 
                 
                   
                     N_tilt 
                      
                     
                       ( 
                       2 
                       ) 
                     
                   
                 
               
               
                 
                   
                     N_tilt 
                      
                     
                       ( 
                       3 
                       ) 
                     
                   
                 
                 
                   0 
                 
                 
                   
                     
                       - 
                       N_tilt 
                     
                      
                     
                       ( 
                       1 
                       ) 
                     
                   
                 
               
               
                 
                   
                     
                       - 
                       N_tilt 
                     
                      
                     
                       ( 
                       2 
                       ) 
                     
                   
                 
                 
                   
                     N_tilt 
                      
                     
                       ( 
                       1 
                       ) 
                     
                   
                 
                 
                   0 
                 
               
             
           
         
       
     
     The transform matrix is: 
         R _tilt=cos γ I +(1−cos γ) N _tilt N _tilt′+sin γ M _tilt         where I is the 3×3 unit matrix.       
     New cutlet position after tilting is: 
     
       
         
           
             
               
                 
                   x 
                   t 
                 
               
               
                 
                     
                 
               
               
                 
                   x 
                   i 
                 
               
             
             
               
                 
                   y 
                   t 
                 
               
               
                 
                   = 
                   
                     R 
                     Tilt 
                   
                 
               
               
                 
                   y 
                   i 
                 
               
             
             
               
                 
                   z 
                   t 
                 
               
               
                 
                     
                 
               
               
                 
                   z 
                   i 
                 
               
             
           
         
       
     
     (3) Cutlet position due to bit rotation around the new bit axis may be obtained as follows: 
       N_rot={sin γ cos θ cos γ sin γ sin θ} 
     Accompany matrix: 
     
       
         
           
             
               M 
               rot 
             
             = 
             
               
                 
                   0 
                 
                 
                   
                     
                       - 
                       N_rot 
                     
                      
                     
                       ( 
                       3 
                       ) 
                     
                   
                 
                 
                   
                     N_rot 
                      
                     
                       ( 
                       2 
                       ) 
                     
                   
                 
               
               
                 
                   
                     N_rot 
                      
                     
                       ( 
                       3 
                       ) 
                     
                   
                 
                 
                   0 
                 
                 
                   
                     
                       - 
                       N_rot 
                     
                      
                     
                       ( 
                       1 
                       ) 
                     
                   
                 
               
               
                 
                   
                     
                       - 
                       N_rot 
                     
                      
                     
                       ( 
                       2 
                       ) 
                     
                   
                 
                 
                   
                     N_rot 
                      
                     
                       ( 
                       1 
                       ) 
                     
                   
                 
                 
                   0 
                 
               
             
           
         
       
     
     The transform matrix is: 
         R _rot=cos ω tI +(1−cos ω t ) N _rot N _rot′+sin ω tM _rot,         I is 3×3 unit matrix and ω is bit rotation speed       
     New cutlet position after tilting is: 
     
       
         
           
             
               
                 
                   x 
                   r 
                 
               
               
                 
                     
                 
               
               
                 
                   x 
                   t 
                 
               
             
             
               
                 
                   y 
                   r 
                 
               
               
                 
                   = 
                   
                     R 
                     rot 
                   
                 
               
               
                 
                   y 
                   t 
                 
               
             
             
               
                 
                   z 
                   r 
                 
               
               
                 
                     
                 
               
               
                 
                   z 
                   t 
                 
               
             
           
         
       
     
     (4) Cutlet position due to penetration along new bit axis may be obtained 
     
       
      
       d 
       p 
       =rop×dt;  
      
     
     
       
      
       x 
       i+1 
       =x 
       r 
       +d 
       p 
       
         — 
       
       x  
      
     
     
       
      
       y 
       i+1 
       =y 
       r 
       +d 
       p 
       
         — 
       
       y  
      
     
     
       
      
       z 
       i+1 
       =z 
       r 
       +d 
       p 
       
         — 
       
       z  
      
     
     With d p     —   x, d p     —   y and d p     —   z being projection of d p  on X, Y, Z. 
     (5) Transfer the calculated cutlet position after tilting, rotation and penetration into spherical coordinate and get (θ i+1 , φ i+1 , ρ i+1 ) 
     (6) Determine which formation layer is cut by this cutlet by comparing Y i+1  with hole coordinate y h , if y i+1 &lt;y h  first layer is cut (this step is the same as Algorithm A). 
     (7) Calculate the cutting depth of each cutlet by comparing (θ i+1 , φ i+1 , ρ i+1 ) of the cutlet and (θ h , φ h , ρ h ) of the hole where θ h =θ i+1  &amp; φ h =φ i+1 . Therefore d ρ =ρ i+1 −ρ h . It is usually difficult to find point on hole (θ h , φ h , ρ h ), an interpretation is used to get an approximate ρ h : 
       ρ h =interp2(θ h ,φ h ,ρ h ,θ i+1 ,φ i+1 ) 
     where θ h , φ h , ρ h  is sub-matrices representing a zone of the hole around the cutlet. Function interp2 is a MATLAB function using linear or nonlinear interpolation method. 
     (8) Calculate the cutting area of each cutlet using dφ, dρ in the plane defined by ρ i , ρ i+1 . The cutlet cutting area is 
         A= 0.5 *dφ* (ρ i+1 ̂2−(ρ i+1   −d ρ)̂2) 
     (9) Save layer information, cutting depth and cutting area into 3D matrix at each time step for each cutlet for force calculation. 
     (10) Update the associated bottom hole matrix removed by the respective cutlets or cutters. 
     Simulating Equilibrium Drilling (Path D) 
     The following algorithms may be used to simulate interaction between portions of a cutter and adjacent portions of a wellbore during removal of formation materials in an equilibrium segment. Respective portions of each cutter engaging adjacent formation materials may be referred to as cutting elements or cutlets. Note that in the following steps, y represents the bit rotational axis. The x and z axes are determined using the right hand rule. Drill bit kinematics in equilibrium drilling is defined by at least three parameters: ROP, RPM and DLS. 
     Given ROP, RPM, DLS, current time t, selected time interval dt, current cutlet position (x i , y i , z i ) or (θ i , φ i , ρ i ), 
     (1) Bit as a whole is rotating around a fixed point O w , the radius of the well path is calculated by 
         R= 5730*12/DLS(inch)         and angle       
       γ=DLS*rop/100.0/3600(deg/sec) 
     (2) The new cutlet position due to rotation γ may be obtained as follows: 
       Axis:N — 1={00−1} 
     Accompany matrix: 
     
       
         
           
             
               M 
               1 
             
             = 
             
               
                 
                   0 
                 
                 
                   
                     
                       - 
                       N_ 
                     
                      
                     1 
                      
                     
                       ( 
                       3 
                       ) 
                     
                   
                 
                 
                   
                     N_ 
                      
                     1 
                      
                     
                       ( 
                       2 
                       ) 
                     
                   
                 
               
               
                 
                   
                     N_ 
                      
                     1 
                      
                     
                       ( 
                       3 
                       ) 
                     
                   
                 
                 
                   0 
                 
                 
                   
                     
                       - 
                       N_ 
                     
                      
                     1 
                      
                     
                       ( 
                       1 
                       ) 
                     
                   
                 
               
               
                 
                   
                     
                       - 
                       N_ 
                     
                      
                     1 
                      
                     
                       ( 
                       2 
                       ) 
                     
                   
                 
                 
                   
                     N_ 
                      
                     1 
                      
                     
                       ( 
                       1 
                       ) 
                     
                   
                 
                 
                   0 
                 
               
             
           
         
       
     
     The transform matrix is: 
         R   — 1=cos γ I +(1−cos γ) N   — 1 N   — 1′+sin γ M 1         where I is 3×3 unit matrix       
     New cutlet position after rotating around O w  is: 
     
       
         
           
             
               
                 
                   x 
                   t 
                 
               
               
                 
                     
                 
               
               
                 
                   x 
                   i 
                 
               
             
             
               
                 
                   y 
                   t 
                 
               
               
                 
                   = 
                   
                     R 
                     1 
                   
                 
               
               
                 
                   y 
                   i 
                 
               
             
             
               
                 
                   z 
                   t 
                 
               
               
                 
                     
                 
               
               
                 
                   z 
                   i 
                 
               
             
           
         
       
     
     (3) Cutlet position due to bit rotation around the new bit axis may be obtained as follows: 
       N_rot={sin γ cos α cos γ sin γ sin α}         where α is the azimuth angle of the well path       
     Accompany matrix: 
     
       
         
           
             
               M 
               rot 
             
             = 
             
               
                 
                   0 
                 
                 
                   
                     
                       - 
                       N_rot 
                     
                      
                     
                       ( 
                       3 
                       ) 
                     
                   
                 
                 
                   
                     N_rot 
                      
                     
                       ( 
                       2 
                       ) 
                     
                   
                 
               
               
                 
                   
                     N_rot 
                      
                     
                       ( 
                       3 
                       ) 
                     
                   
                 
                 
                   0 
                 
                 
                   
                     
                       - 
                       N_rot 
                     
                      
                     
                       ( 
                       1 
                       ) 
                     
                   
                 
               
               
                 
                   
                     
                       - 
                       N_rot 
                     
                      
                     
                       ( 
                       2 
                       ) 
                     
                   
                 
                 
                   
                     N_rot 
                      
                     
                       ( 
                       1 
                       ) 
                     
                   
                 
                 
                   0 
                 
               
             
           
         
       
     
     The transform matrix is: 
         R _rot=cos θ I +(1−cos θ) N _rot N _rot′+sin θ M _rot,         where I is 3×3 unit matrix       
     New cutlet position after bit rotation is: 
     
       
         
           
             
               
                 
                   x 
                   
                     i 
                     + 
                     1 
                   
                 
               
               
                 
                     
                 
               
               
                 
                   x 
                   t 
                 
               
             
             
               
                 
                   y 
                   
                     i 
                     + 
                     1 
                   
                 
               
               
                 
                   = 
                   
                     R 
                     rot 
                   
                 
               
               
                 
                   y 
                   t 
                 
               
             
             
               
                 
                   z 
                   
                     i 
                     + 
                     1 
                   
                 
               
               
                 
                     
                 
               
               
                 
                   z 
                   t 
                 
               
             
           
         
       
     
     (4) Transfer the calculated cutlet position into spherical coordinate and get (θ i+1 , φ i+1 , ρ 1+1 ). 
     (5) Determine which formation layer is cut by this cutlet by comparing y i+1  with hole coordinate y h , if y i+1 &lt;y h  first layer is cut (this step is the same as Algorithm A). 
     (6) Calculate the cutting depth of each cutlet by comparing (θ i+1 , φ i+1 , ρ i+1 ) of the cutlet and (θ h , φ h , ρ h ) of the hole where θ h =θ i+1  &amp; φ h =φ i+1 . Therefore d ρ =ρ i+1 −ρ h . It is usually difficult to find point on hole (θ h , φ h , ρ h ), an interpretation is used to get an approximate ρ h : 
       ρ h =interp2(θ h ,φ h ,ρ h ,θ i+1 ,φ i+1 ) 
     where θ h , φ h , ρ h  is sub-matrices representing a zone of the hole around the cutlet. Function interp2 is a MATLAB function using linear or nonlinear interpolation method. 
     (7) Calculate the cutting area of each cutlet using dφ, dρ in the plane defined by ρ i , ρ i+1 . The cutlet cutting area is: 
         A= 0.5 *d φ*(ρ i+1 ̂2−(ρ i+1   −d ρ)̂2) 
     (8) Save layer information, cutting depth and cutting area into 3D matrix at each time step for each cutlet for force calculation. 
     (9) Update the associated bottom hole matrix for portions removed by the respective cutlets or cutters. 
     An Alternative Algorithm to Calculate Cutting Area of a Cutter 
     The following steps may also be used to calculate or estimate the cutting area of the associated cutter. See  FIGS. 15C and 16 . 
     (1) Determine the location of cutter center O c  at current time in a spherical hole coordinate system, see  FIG. 16 . 
     (2) Transform three matrices φ H , θ H  and ρ H  to Cartesian coordinate in hole coordinate system and get X h , Y h  and Z h ; 
     (3) Move the origin of X h , Y h  and Z h  to the cutter center O c  located at (φ C , θ C  and ρ C ); 
     (4) Determine a possible cutting zone on portions of a bottom hole interacted by a respective cutlet for this cutter and subtract three sub-matrices from X h , Y h  and Z h  to get x h , y h  and z h ; 
     (5) Transform x h , y h  and z h  back to spherical coordinate and get φ h , θ h  and ρ h  for this respective subzone on bottom hole; 
     (6) Calculate spherical coordinate of cutlet B: φ B , θ B  and ρ B  in cutter local coordinate; 
     (7) Find the corresponding point C in matrices φ h , θ h  and ρ h  with condition φ C =φ B  and θ C =θ B ; 
     (8) If ρ B &gt;ρ C , replacing ρ C  with ρ B  and matrix ρ h  in cutter coordinate system is updated; 
     (9) Repeat the steps for all cutlets on this cutter; 
     (10) Calculate the cutting area of this cutter; 
     (11) Repeat steps 1-10 for all cutters; 
     (12) Transform hole matrices in local cutter coordinate back to hole coordinate system and repeat steps 1-12 for next time interval. 
     Force Calculations in Different Drilling Modes 
     The following algorithms may be used to estimate or calculate forces acting on all face cutters of a rotary drill bit. 
     (1) Summarize all cutlet cutting areas for each cutter and project the area to cutter face to get cutter cutting area, A c    
     (2) Calculate the penetration force (F p ) and drag force (F d ) for each cutter using, for example, AMOCO Model (other models such as SDBS model, Shell model, Sandia Model may be used). 
         F   p   =σ*A   c *(0.16 *abs (β e )−1.15)) 
         F   d   =F   d   *F   p   +σ*A   c *(0.04 *abs (β e )+0.8)) 
     where σ is rock strength, βe is effective back rake angle and F d  is drag coefficient (usually F d =0.3) 
     (3) The force acting point M for this cutter is determined either by where the cutlet has maximal cutting depth or the middle cutlet of all cutlets of this cutter which are in cutting with the formation. The direction of F p  is from point M to cutter face center O c . F d  is parallel to cutter axis. See for example  FIGS. 15B and 15C . 
     One example of a computer program or software and associated method steps which may be used to simulate forming various portions of a wellbore in accordance with teachings of the present disclosure is shown in  FIGS. 17A-17G . Three dimensional (3D) simulation or modeling of forming a wellbore may begin at step  800 . At step  802  the drilling mode, which will be used to simulate forming a respective segment of the simulated wellbore, may be selected from the group consisting of straight hole drilling, kick off drilling or equilibrium drilling. Additional drilling modes may also be used depending upon characteristics of associated downhole formations and capabilities of an associated drilling system. 
     At step  804   a  bit parameters such as rate of penetration and revolutions per minute may be inputted into the simulation if straight hole drilling was selected. If kickoff drilling was selected, data such as rate of penetration, revolutions per minute, dogleg severity, bend length and other characteristics of an associated bottom hole assembly may be inputted into the simulation at step  804   b . If equilibrium drilling was selected, parameters such as rate of penetration, revolutions per minute and dogleg severity may be inputted into the simulation at step  804   c.    
     At steps  806 ,  808  and  810  various parameters associated with configuration and dimensions of a first rotary drill bit design and downhole drilling conditions may be inputted into the simulation. Appendix A provides examples of such data. 
     At step  812  parameters associated with each simulation, such as total simulation time, step time, mesh size of cutters, gages, blades and mesh size of adjacent portions of the wellbore in a spherical coordinate system may be inputted into the model. At step  814  the model may simulate one revolution of the associated drill bit around an associated bit axis without penetration of the rotary drill bit into the adjacent portions of the wellbore to calculate the initial (corresponding to time zero) hole spherical coordinates of all points of interest during the simulation. The location of each point in a hole spherical coordinate system may be transferred to a corresponding Cartesian coordinate system for purposes of providing a visual representation on a monitor and/or print out. 
     At step  816  the same spherical coordinate system may be used to calculate initial spherical coordinates for each cutlet of each cutter and each gage portions which will be used during the simulation. 
     At step  818  the simulation will proceed along one of three paths based upon the previously selected drilling mode. At step  820   a  the simulation will proceed along path A for straight hole drilling. At step  820   b  the simulation will proceed along path B for kick off hole drilling. At step  820   c  the simulation will proceed along path C for equilibrium hole drilling. 
     Steps  822 ,  824 ,  828 ,  830 ,  832  and  834  are substantially similar for straight hole drilling (Path A), kick off hole drilling (Path B) and equilibrium hole drilling (Path C). Therefore, only steps  822   a ,  824   a ,  828   a ,  830   a ,  832   a  and  834   a  will be discussed in more detail. 
     At step  822   a  a determination will be made concerning the current run time, the ΔT for each run and the total maximum amount of run time or simulation which will be conducted. At step  824   a  a run will be made for each cutlet and a count will be made for the total number of cutlets used to carry out the simulation. 
     At step  826   a  calculations will be made for the respective cutlet being evaluated during the current run with respect to penetration along the associated bit axis as a result of bit rotation during the corresponding time interval. The location of the respective cutlet will be determined in the Cartesian coordinate system corresponding with the time the amount of penetration was calculated. The information will be transferred from a corresponding hole coordinate system into a spherical coordinate system. 
     At step  828   a  the model will determine which layer of formation material has been cut by the respective cutlet. A calculation will be made of the cutting depth, cutting area of the respective cutlet and saved into respective matrices for rock layer, depth and area for use in force calculations. 
     At step  830   a  the hole matrices in the hole spherical coordinate system will be updated based on the recently calculated cutlet position at the corresponding time. At step  832   a  a determination will be made to determine if the current cutter count is less than or equal to the total number of cutlets which will be simulated. If the number of the current cutter is less than the total number, the simulation will return to step  824   a  and repeat steps  824   a  through  832   a.    
     If the cutlet count at step  832   a  is equal to the total number of cutlets, the simulation will proceed to step  834   a . If the current time is less than the total maximum time selected, the simulation will return to step  822   a  and repeat steps  822   a  through  834   a . If the current time is equal to the previously selected total maximum amount of time, the simulation will proceed to steps  840  and  860 . 
     As previously noted, if a simulation proceeds along path C as shown in  FIG. 17D  corresponding with kick off hole drilling, the same steps will be performed as described with respect to path B for straight hole drilling except for step  826   b . As shown in  FIG. 17D , calculations will be made at step  826   b  corresponding with location and orientation of the new bit axis after tilting which occurred during respective time interval dt. 
     A calculation will be made for the new Cartesian coordinate system based upon bit tilting and due to bit rotation around the location of the new bit axis. A calculation will also be made for the new Cartesian coordinate system due to bit penetration along the new bit axis. After the new Cartesian coordinate systems have been calculated, the cutlet location in the Cartesian coordinate systems will be determined for the corresponding time interval. The information in the Cartesian coordinate time interval will then be transferred into the corresponding spherical coordinate system at the same time. Path C will then proceed through steps  828   b ,  830   b ,  832   b  and  834   b  as previously described with respect to path B. 
     If equilibrium drilling is being simulated, the same functions will occur at steps  822   c  and  824   c  as previously described with respect to path B. For path D as shown in  FIG. 17E , the simulation will proceed through steps  822   c  and  824   c  as previously described with respect to steps  822   a  and  824   a  of path B. At step  826   a  a calculation will be made for the respective cutlet during the respective time interval based upon the radius of the corresponding wellbore segment. A determination will be made based on the center of the path in a hole coordinate system. A new Cartesian coordinate system will be calculated after bit rotation has been entered based on the amount of DLS and rate of penetration along the Z axis passing through the hole coordinate system. A calculation of the new Cartesian coordinate system will be made due to bit rotation along the associated bit axis. After the above three calculations have been made, the location of a cutlet in the new Cartesian coordinate system will be determined for the appropriate time interval and transferred into the corresponding spherical coordinate system for the same time interval. Path D will continue to simulate equilibrium drilling using the same functions for steps  828   c ,  830   c ,  832   c  and  834   c  as previously described with respect to Path B straight hole drilling. 
     When selected path B, C or D has been completed at respective step  834   a ,  834   b  or  834   c  the simulation will then proceed to calculate cutter forces including impact arrestors for all step times at step  840  and will calculate associated gage forces for all step times at step  860 . At step  842  a respective calculation of forces for a respective cutter will be started. 
     At step  844  the cutting area of the respective cutter is calculated. The total forces acting on the respective cutter and the acting point will be calculated. 
     At step  846  the sum of all the cutting forces in a bit coordinate system is summarized for the inner cutters and the shoulder cutters. The cutting forces for all active gage cutters may be summarized. At step  848  the previously calculated forces are projected into a hole coordinate system for use in calculating associated bit walk rate and steerability of the associated rotary drill bit. 
     At step  850  the simulation will determine if all cutters have been calculated. If the answer is NO, the model will return to step  842 . If the answer is YES, the model will proceed to step  880 . 
     At step  880  all cutter forces and all gage blade forces are summarized in a three dimensional bit coordinate system. At step  882  all forces are summarized into a hole coordinate system. 
     At step  884  a determination will be made concerning using only bit walk calculations or only bit steerability calculations. If bit walk rate calculations will be used, the simulation will proceed to step  886   b  and calculate bit steer force, bit walk force and bit walk rate for the entire bit. At step  888   b  the calculated bit walk rate will be compared with a desired bit walk rate. If the bit walk rate is satisfactory at step  890   b , the simulation will end and the last inputted rotary drill bit design will be selected. If the calculated bit walk rate is not satisfactory, the simulation will return to step  806 . 
     If the answer to the question at step  884  is NO, the simulation will proceed to step  886   a  and calculate bit steerability using associated bit forces in the hole coordinate system. At step  888   a  a comparison will be made between calculated steerability and desired bit steerability. At step  890   a  a decision will be made to determine if the calculated bit steerability is satisfactory. If the answer is YES, the simulation will end and the last inputted rotary drill bit design at step  806  will be selected. If the bit steerability calculated is not satisfactory, the simulation will return to step  806 . 
       FIG. 18  is a schematic drawing showing one comparison of bit steerability versus tilt rate for a rotary drill bit when used with point-the-bit drilling system and push-the-bit drilling system, respectively. The curves shown in  FIG. 18  are based upon a constant rate of penetration of thirty feet per hour, a constant RPM of 120 revolutions per minute, and a uniform rock strength of 18000 PSI. The simulations used to form the graphs shown in  FIG. 18  along with other simulations conducted in accordance with teachings of the present disclosure indicates that bit steerability or required steer force is generally a nonlinear function of the DLS or tilt rate. The drilling bit when used in point-the-bit drilling system required much less steer force than with the push-the-bit drilling system. The graphs shown in  FIG. 18  provide a similar result with respect to evaluating steerability as calculations represented by bit steer force as a function of bit tilt rate. The effect of downhole drilling conditions on varying the steerability of a rotary drill bit have previously been generally unnoticed by the prior art. 
     Bit Steerability Evaluation 
     The steerability of a rotary drill may be evaluated using the following steps. 
     (1) Input bit geometry parameters or read bit file from bit design software such as UniGraphics or Pro-E; 
     (2) Define bit motion: a rotation speed (RPM) around bit axis, an axial penetration rate (ROP, ft/hr), DLS or tilting rate (deg/100 ft) at an azimuth angle (to define the bit tilt plane); 
     (3) Define formation properties: rock compressive strength, rock transition layer, inclination angle; 
     (4) Define simulation time or total number of bit rotations and time interval; 
     (5) Run 3D PDC bit drilling simulator and calculate bit forces including bit side force; 
     (6) Change DLS and repeat step 5 to get bit side force corresponding to the given DLS; 
     (7) Plot a curve using (DLS, F s ) and calculate bit steerability; The steerability may be represented by the slop of the curve if the curve is close to a line, or the steerability may be represented by the first derivative of the nonlinear curve. 
     (8) Giving another set of bit operational parameters (ROP, RPM) and repeat step 3 to 7 to get more curves; 
     (9) Bit steerability is defined by a set of curves or their first derivative or slop. 
     The steerability of various rotary drill bit designs may be compared and evaluated by calculating a steering difficulty for each rotary drill bit. 
     Steering Difficulty Index may be defined using steer force as follows: 
       SD index   =F   steer /Tilt Rate 
     Steering Difficulty Index may also be defined using steer moment as follows: 
       SD index   =M   steer /Steer Rate 
       Steer Rate=Tilt Rate 
     A steering difficulty index may also be calculated for any zone of part on the drill bit. For example, when the steer force, F steer , is contributed only from the shoulder cutters, then the associated SD index  represents the difficulty level of the shoulder cutters. In accordance with teachings of the present disclosure, the steering difficulty index for each zone of the drilling bit may be evaluated. By comparing the steering difficulty index of each zone, a bit designer may more easily identify which zone or zones are more difficult to steer and design modifications may be focused on the difficult zone or zones. 
     The calculation of steerability index for each zone may be repeated and design changes made until the calculation of steerability for each zone is satisfactory and/or the steerability index for the overall drill bit design is satisfactory. 
     Bit Walk Rate Evaluation 
     Bit walk rate may be calculated using bit steer force, tilt rate and walk force: 
       Walk Rate=(Steer Rate/ F   steer )* F   walk    
     Bit walk rate may also be calculated using bit steer moment, tilt rate and walk moment: 
       Walk Rate=(Steer Rate/ M   steer )* M   walk    
     The walk rate may be applied to any zone of part on the drill bit. For example, when the steer force, F steer  and walk force, F walk , are contributed only from the shoulder cutters, then the associated walk rate represents the walk rate of the shoulder cutters. In accordance with teachings of the present disclosure, the walk rate for each zone of the drilling bit can be evaluated. By comparing the walk rate of each zone, the bit designer can easily identify which zone is the easiest zone to walk and modifications may be focused on that zone. 
     Although the present disclosure 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 disclosure as defined by the following claims. 
     
       
         
           
               
               
               
             
               
                 APPENDIX A 
               
             
            
               
                   
               
               
                 EXAMPLES OF 
                 EXAMPLES OF 
                 EXAMPLES OF 
               
               
                 DRILLING EQUIPMENT DATA 
                 WELLBORE 
                 FORMATION 
               
            
           
           
               
               
               
               
            
               
                 Design Data 
                 Operating Data 
                 DATA 
                 DATA 
               
               
                   
               
               
                 active gage 
                 axial bit 
                 azimuth angle 
                 compressive 
               
               
                   
                 penetration 
                   
                 strength 
               
               
                   
                 rate 
               
               
                 bend (tilt) 
                 bit ROP 
                 bottom hole 
                 down dip 
               
               
                 length 
                   
                 configuration 
                 angle 
               
               
                 bit face 
                 bit rotational 
                 bottom hole 
                 first layer 
               
               
                 profile 
                 speed 
                 pressure 
               
               
                 bit geometry 
                 bit RPM 
                 bottom hole 
                 formation 
               
               
                   
                   
                 temperature 
                 plasticity 
               
               
                 blade 
                 bit tilt rate 
                 directional 
                 formation 
               
               
                 (length, number, 
                   
                 wellbore 
                 strength 
               
               
                 spiral, width) 
               
               
                 bottom hole 
                 equilibrium 
                 dogleg 
                 inclination 
               
               
                 assembly 
                 drilling 
                 severity 
               
               
                   
                   
                 (DLS) 
               
               
                 cutter 
                 kick off 
                 equilibrium 
                 lithology 
               
               
                 (type, size, 
                 drilling 
                 section 
               
               
                 number) 
               
               
                 cutter density 
                 lateral 
                 horizontal 
                 number of 
               
               
                   
                 penetration 
                 section 
                 layers 
               
               
                   
                 rate 
               
               
                 cutter location 
                 rate of 
                 inside 
                 porosity 
               
               
                 (inner, outer, 
                 penetration 
                 diameter 
               
               
                 shoulder) 
                 (ROP) 
               
               
                 cutter 
                 revolutions per 
                 kick off 
                 rock 
               
               
                 orientation 
                 minute (RPM) 
                 section 
                 pressure 
               
               
                 (back rake, side 
               
               
                 rake) 
               
               
                 cutting area 
                 side penetration 
                 profile 
                 rock 
               
               
                   
                 azimuth 
                   
                 strength 
               
               
                 cutting depth 
                 side penetration 
                 radius of 
                 second layer 
               
               
                   
                 rate 
                 curvature 
               
               
                 cutting 
                 steer force 
                 side 
                 shale 
               
               
                 structures 
                   
                 azimuth 
                 plasticity 
               
               
                 drill string 
                 steer rate 
                 side forces 
                 up dip angle 
               
               
                 fulcrum point 
                 straight hole 
                 slant hole 
               
               
                   
                 drilling 
               
               
                 gage gap 
                 tilt rate 
                 straight 
               
               
                   
                   
                 hole 
               
               
                 gage length 
                 tilt plane 
                 tilt rate 
               
               
                 gage radius 
                 tilt plane 
                 tilting 
               
               
                   
                 azimuth 
                 motion 
               
               
                 gage taper 
                 torque on bit 
                 tilt plane 
               
               
                   
                 (TOB) 
                 azimuth 
               
               
                   
                   
                 angle 
               
               
                 IADC Bit Model 
                 walk angle 
                 trajectory 
               
               
                 impact arrestor 
                 walk rate 
                 vertical 
               
               
                 (type, size, 
                   
                 section 
               
               
                 number) 
               
               
                 passive gage 
                 weight on 
               
               
                   
                 bit (WOB) 
               
               
                 worn (dull) bit 
               
               
                 data 
               
               
                   
               
               
                 EXAMPLES OF MODEL PARAMETERS FOR SIMULATING DRILLING A DIRECTIONAL WELLBORE 
               
               
                 Mesh size for portions of downhole equipment interacting with adjacent portions of a wellbore. 
               
               
                 Mesh size for portions of a wellbore. 
               
               
                 Run time for each simulation step. 
               
               
                 Total simulation run time. 
               
               
                 Total number of revolutions of a rotary drill bit per simulation.