Patent Publication Number: US-10323465-B2

Title: Optimization of rolling elements on drill bits

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a U.S. national stage patent application of International Patent Application No. PCT/US2017/037799, filed on Jun. 15, 2017, the benefit of which is claimed and the disclosure of which is incorporated herein by reference in its entirety. 
     BACKGROUND 
     In wellbore drilling for the oil and gas industry, a drill bit may be mounted on the end of a drill string and rotated to break up a geologic formation. The drill bit may be rotated by turning the entire drill string, e.g., with a top drive at surface location, and/or the drill bit may be rotated using downhole equipment, such as a mud motor mounted within the drill string. When drilling, a drilling fluid is pumped through the drill string and discharged from the drill bit to remove cuttings and debris. The mud motor, if present in the drill string, may be selectively powered using the circulating drilling fluid. 
     One common type of drill bit is a “fixed cutter” bit, wherein cutters (also referred to as cutter elements, cutting elements, or inserts) are secured to a bit body at fixed positions. The bit body may be formed from a high strength material, such as tungsten carbide, steel, or a composite/matrix material, and the cutters may include a substrate or support stud made of a carbide (e.g., tungsten carbide), and an ultra-hard cutting surface layer or “table” made of a polycrystalline diamond material or a polycrystalline boron nitride material deposited onto or otherwise bonded to the substrate. Such cutters are commonly referred to as polycrystalline diamond compact (“PDC”) cutters. 
     Some cutters are strategically positioned along leading edges of blades defined on the bit body such that the cutters engage the formation during drilling. In use, high forces are exerted on the cutters, and over time, a working surface or cutting edge of each cutter eventually wears down or fails. The cutting edge of a fixed cutter may be continuously exposed to the formation, while an exposed surface of a rolling element may be successively exposed to the formation and withdrawn from the formation as it rotates on the drill bit. In some instances, rolling elements may provide depth of cut control to the fixed cutters. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The following figures are included to illustrate certain aspects of the present disclosure, and should not be viewed as exclusive embodiments. The subject matter disclosed is capable of considerable modifications, alterations, combinations, and equivalents in form and function, without departing from the scope of this disclosure. 
         FIG. 1  is a perspective view of a rotary drill bit that illustrates both fixed cutters and rolling element assemblies secured on a bit body thereof. 
         FIG. 2A  is a schematic side view of a rolling element assembly having a rolling element that defines a generally cylindrical body, wherein the cylindrical body is in a generally unbalanced operational loading. 
         FIG. 2B  is a schematic view side of the rolling element assembly of  FIG. 2A  illustrating a generally balanced loading of the cylindrical body. 
         FIG. 3  is a schematic top view of a drill bit illustrating the locations of fixed cutters and three rolling elements on a bit face of the drill bit, which may be arranged to provide an improved operational life of the rolling elements and depth of cut control for the drill bit. 
         FIG. 3  illustrates a circle intersecting a top surface of one of the rolling elements at a particular radial coordinate as well as a plurality of fixed cutters at the radial coordinate. 
         FIG. 4  is a schematic profile view of the bit face of  FIG. 3  illustrating the axial and radial positions of the rolling element and the fixed cutters that intersect the circle. 
         FIG. 4A  is a schematic graphical view of the relative axial positions of the top surface of the rolling element of  FIG. 4  at the radial coordinate and the intersection points defined where the fixed cutters intersect the circle, illustrating an axial underexposure of each of the fixed cutters at the radial coordinate. 
         FIG. 4B  is a schematic graphical view of the relative axial positions of one of the intersection points of  FIG. 4A  and the top surface of each of the rolling elements of  FIG. 4 . 
         FIG. 5  is a schematic profile view of the bit face of  FIG. 3  illustrating the orientation, axial position and radial position of the fixed cutters and rolling elements. 
         FIG. 6  is a schematic top view of one of the rolling elements of  FIG. 3  illustrating a rotational orientation of the rolling element. 
         FIG. 7A  is flowchart illustrating a procedure for selecting the location and orientation of the rolling elements on a drill bit face to balance operational forces on the rolling elements. 
         FIG. 7B  is a flowchart illustrating a procedure for calculating a critical depth of cut as specified in the procedure of  FIG. 7A . 
         FIG. 7C  is a flow chart illustrating a procedure for calculating the forces and moment acting on a rolling element as specified in the procedure of  FIG. 7A . 
         FIGS. 8A and 8B  are side and end views, respectively, of a rolling element of  FIG. 3  illustrating the operational loads acting on the rolling element as specified in the procedure of  FIG. 7C . 
         FIG. 9A  is a schematic view of the three rolling elements of  FIG. 3  illustrating an example operational loading prior to performing the procedure of  FIG. 7A  wherein operational loads are balanced on a first of the three rolling elements and unbalanced on second and third rolling elements. 
         FIG. 9B  is a schematic view of the second and third rolling elements of  FIG. 9A  illustrating an example operational loading subsequent to performing the procedure of  7 A wherein the operational loads are balanced on the second and third rolling elements. 
         FIG. 10  is a diagrammatic view of the critical depth of cut calculated in the procedure of  FIG. 7A  for all three of the rolling elements of  FIG. 3 , wherein the critical depth of cut is charted against a bit radius for a radial portion of the drill bit. 
     
    
    
     DETAILED DESCRIPTION 
     The present disclosure relates to earth-penetrating drill bits and, more particularly, to rolling-type cutting or depth of cut control (DOCC) elements that can be used in drill bits. A rolling DOCC element may include a generally cylindrical body strategically positioned and secured to the drill bit so that the rolling element is able to engage the formation during drilling. In response to drill bit rotation, and depending on the selected orientation of the rolling element with respect to the body of the drill bit, the rolling element may roll against the underlying formation, cut against the formation, or may both roll against and cut the formation. Embodiments of the disclosure are directed methods for selecting the location and orientation of the rolling elements on the drill bits such that an outer surface of the rolling element maintains multiple zones of contact with the geologic formation to balance the forces acting the cylindrical body. Damage to the drill bit may thereby be prevented. In some embodiments, the methods include calculating a critical depth of cut for each point along a rolling element length of the rolling elements, changing at least one design variable, and recalculating the depth of cut until at least three contact points exist along the rolling element. 
       FIG. 1  is a perspective view of an example drill bit  100  that illustrates both fixed cutters and rolling elements on a bit body  102 . The drill bit  100  the present teachings may be applied to any fixed cutter drill bit category, including polycrystalline diamond compact (PDC) drill bits, drag bits, matrix drill bits, and/or steel body drill bits. While the drill bit  100  is depicted in  FIG. 1  as a fixed cutter drill bit, the principles of the present disclosure are equally applicable to other types of drill bits operable to form a wellbore including, but not limited to, fixed cutter core bits, impregnated diamond bits and roller cone drill bits. 
     The bit body  102  of the drill bit includes radially and longitudinally extending blades  104  having leading faces  106 . The bit body  102  may be made of steel or a matrix of a harder material, such as tungsten carbide. The bit body  102  rotates about a longitudinal drill bit axis  107  to drill into underlying subterranean formation under an applied weight-on-bit. Corresponding junk slots  112  are defined between circumferentially adjacent blades  104 , and a plurality of nozzles or ports  114  can be arranged within the junk slots  112  for ejecting drilling fluid that cools the drill bit  100  and otherwise flushes away cuttings and debris generated while drilling. 
     The bit body  102  further includes a plurality of fixed cutters  116  secured within a corresponding plurality of cutter pockets sized and shaped to receive the cutters  116 . Each cutter  116  in this example comprises a fixed cutter secured within its corresponding cutter pocket via brazing, threading, shrink-fitting, press-fitting, snap rings, or any combination thereof. The fixed cutters  116  are held in the blades  104  and respective cutter pockets at predetermined angular orientations and radial locations to present the fixed cutters  116  with a desired angle against the formation being penetrated. As the drill bit  100  is rotated, the fixed cutters  116  are driven through the rock by the combined forces of the weight-on-bit and the torque experienced at the drill bit  100 . During drilling, the fixed cutters  116  may experience a variety of forces, such as drag forces, axial forces, reactive moment forces, or the like, due to the interaction with the underlying formation being drilled as the drill bit  100  rotates. 
     Each fixed cutter  116  may include a generally cylindrical substrate made of an extremely hard material, such as tungsten carbide, and a cutting face secured to the substrate. The cutting face may include one or more layers of an ultra-hard material, such as polycrystalline diamond, polycrystalline cubic boron nitride, impregnated diamond, etc., which generally forms a cutting edge and the working surface for each fixed cutter  116 . The working surface is typically flat or planar, but may also exhibit a curved exposed surface that meets the side surface at a cutting edge. 
     Generally, each fixed cutter  116  may be manufactured using tungsten carbide as the substrate. While a cylindrical tungsten carbide “blank” can be used as the substrate, which is sufficiently long to act as a mounting stud for the cutting face, the substrate may equally comprise an intermediate layer bonded at another interface to another metallic mounting stud. To form the cutting face, the substrate may be placed adjacent a layer of ultra-hard material particles, such as diamond or cubic boron nitride particles, and the combination is subjected to high temperature at a pressure where the ultra-hard material particles are thermodynamically stable. This results in recrystallization and formation of a polycrystalline ultra-hard material layer, such as a polycrystalline diamond or polycrystalline cubic boron nitride layer, directly onto the upper surface of the substrate. When using polycrystalline diamond as the ultra-hard material, the fixed cutter  116  may be referred to as a polycrystalline diamond compact cutter or a “PDC cutter,” and drill bits made using such PDC fixed cutters  116  are generally known as PDC bits. 
     As illustrated, the drill bit  100  may further include a plurality of rolling element assemblies  118 , each including a rolling element  120 . The rolling element  120  may include a generally cylindrical body strategically positioned in a predetermined position and orientation on the bit body  102  so that the rolling element  120  is able to engage the formation during drilling. The orientation of a rotational axis A 0  ( FIG. 2A ) of each rolling element  120  with respect to a tangent to an outer surface of the blade  104  may dictate whether the particular rolling element  120  operates purely as a rolling DOCC element, purely a rolling cutting element, or a hybrid of both. The terms “rolling element” and “rolling DOCC element” are used herein to describe a rolling element  120  in any orientation, whether it acts purely as a DOCC element, a purely as cutting element or as a hybrid of both. Rolling elements  120  may prove advantageous in allowing for additional weight-on-bit (WOB) to enhance directional drilling applications without over engagement of the fixed cutters  116 . Effective DOCC also limits fluctuations in torque and minimizes stick-slip, which can cause damage to the fixed cutters  116 . An optimized three-dimensional position and three-dimensional orientation of the rolling element  120  may be selected to extend the life of the rolling element assemblies, and thereby improve the efficiency of the drill bit  100  over its operational life. As described herein, the three-dimensional position and orientation may be expressed in terms of a Cartesian coordinate system with the Y-axis positioned along the longitudinal axis  107 , and a polar coordinate system with a polar axis along the X-axis of Cartesian coordinate system. 
       FIG. 2A  is a schematic side view of a rolling element assembly  118  having a rolling element  120  experiencing a generally unbalanced operational loading. As illustrated, the rolling element  120  defines a generally cylindrical body arranged to rotate about the rotational axis A 0  within a frame  124 . In other embodiments, a rolling element (not shown) having an alternate profile, e.g., a convex, concave or irregular profile, may be provided to rotate within the frame  124  without departing from aspects of the disclosure. The frame  124  supports the rolling element  120  therein such that an entire rolling element length Lr of the rolling element  120  protrudes from the frame  124 . In operation, the rolling element  120  may thus contact a geologic formation along the entire rolling element length Lr thereof. A portion of a rolling element diameter Dr of the rolling element is disposed generally within the frame  124  such that the frame  124  retains the rolling element  120  therein. 
     In operation, the rolling element  120  may contact the geologic formation over a single contact area E 0  along the rolling element length Lr. The rolling element  120  may thereby experience a resultant operational load P 0  at a top surface  128  of the rolling element  120 . Where the resultant force P 0  is laterally offset from a centerline C L  of the rolling element, the force P 0  generates a moment M 0 . The moment M 0  may deform or damage the frame  124 , and potentially lead to the loss of the rolling element  120  from the frame  124 . 
       FIG. 2B  is a schematic side view of the rolling element assembly  118  with the rolling element  120  experiencing a generally balanced operational loading. As illustrated in  FIG. 2B , where the rolling element  120  is arranged to contact the formation over at least three contact areas E 1 , E 2 , E 3 , along the rolling element length Lr, the moment induced by an applied force P 3  may be at least partially balanced by forces P 1  and P 2  applied on an opposite side of the centerline C L . In this manner, the resultant moment M 1  may be reduced, wear on an outer rolling surface of the rolling element  120  may be relatively even across the rolling element length Lr, and durability of the rolling element assembly  118  will be improved. In ideal conditions, the entire rolling element length Lr of the rolling element  120  is maintained in contact with the formation for a critical depth of cut, and the moment M 1  acting on the rolling element assembly  118  is very close to zero. 
       FIG. 3  is a schematic top view of an example drill bit  200  illustrating design locations of fixed cutters  116  and rolling element assemblies  118  on a bit face  202  of the drill bit  200 . The bit face  202  may be defined at the leading end of a bit body  102  ( FIG. 1 ), and in the example embodiment illustrated includes twelve fixed cutters  116  numbered 1 through 12 and three rolling elements  120   a ,  120   b  and  120   c  (collectively or generally  120 ) having control points thereon respectively numbered F 1  through F 3 . The drill bit  200  represents one example arrangement of the cutters  116  and rolling elements  120  that may be considered in determining an optimized position and orientation of the rolling elements  120  in accordance with principles of the disclosure. Aspects of the disclosure may be practiced with more or fewer cutters  116  and or rolling elements  120  arranged in various other configurations. 
     Once the locations of the fixed cutters  116  are determined, and an initial location and orientation of the rolling elements  120  is selected, the design variables associated with the position and orientation of the rolling elements  120  may be defined. As illustrated in  FIG. 3 , an angular position θ of a component on the bit face  202  may generally be defined between the X-axis and the plane extending through the Y-axis and the component. For example, an angular position of the control point F 1  on the rolling element  120   a  is generally represented by the coordinate θ f1 . A radial spacing from the Y-axis may be generally represented by a radius “R.” For example, the radial offset of the control point F 1  (and the rolling center “O”) of the rolling element  120   a  may be represented by the radius Rf. 
     A circle  204  having the radius Rf intersects cutting edges  206  at the leading faces of the fixed cutters  116  numbered 6, 7, 8 and 9 at intersection points P 6 , P 7 , P 8  and P 9  respectively. The intersection points P 6 , P 7 , P 8  and P 9  may have the same rotational path as control points F 1 , F 2  and F 3  and, thus, may have a depth of cut that may be affected by the control points of the rolling elements  120 . The angular position of a point “P” intersecting the circle  204  is generally represented by the coordinate θ p . For example, θ p8  represents the angle defined between the X-axis and the line extending from the Y-axis to the intersection point “P 8 .” Since the radial positions of the rolling elements  120   a ,  120   b  and  120   c  are not necessarily the same, the rolling centers “O” of the rolling elements  120   a ,  120   b  and  120   c  may not all fall on the same circle. 
       FIG. 4  is a schematic view of the bit face  202  of  FIG. 3  illustrating the axial and radial positions of a rolling element  120   a  having a control point F 1  arranged to control the depth of cut of fixed cutters  116  (6, 7, 8 and 9). The rolling element  120   a  and the fixed cutter  116  (8) are both secured on the same blade  104  ( FIG. 1 ) having a profile  208  in the Y-R plane, while the rolling element  120   a  and fixed cutters (6, 7 and 9) are secured on different blades  104 . An axial underexposure δ ( FIG. 4A ) generally defines an axial distance that the control point F 1  on the rolling element  120   a  is disposed below each of the fixed cutters  116  on the profile  208 . For a particular radial coordinate dr, e.g., Rf, an axial underexposure δ is defined as the axial distance between an axial coordinate Yf of top surface  128  of the rolling element  120  and the axial coordinate of each of the intersection points “P.” For example, δ 8  represents the axial distance between the top surface  128  of the rolling element  120  (F 1 ) at the radial coordinate Rf and the top of the fixed cutter  116  (8) at the radial coordinate Rf, e.g., at point P 8 . The axial underexposure δ 6  is illustrated as being generally negative since the intersection point P 6  is disposed axially below the top surface  128  at the radial coordinate Rf. 
     As illustrated in  FIG. 4B , an axial underexposure δ (measured along Y axis) is defined between each intersection point “P” and each of the rolling elements  120 . Since each of the rolling elements  120  (F 1 , F 2 , F 3 ) may be disposed at different axial coordinates Yf (Yf F1 , Yf F2 , Yf F3 ), an axial underexposure δ 8  (e.g., δ 8   F1 , δ 8   F2 , δ 8   F3 ) may be defined with respect to each of the rolling elements  120  (F 1 , F 2 , F 3 ). 
       FIG. 5  is a schematic profile view of the bit face of  FIG. 3  generally illustrating orientations of a rolling element  120  in a Y-R plane defined by axial (Y) and radial (R) axes. The rolling element  120  defines a rolling center “O” as described above, and at least three control points A, B and C along the top surface  128 . The control points A and B are generally located at ends of the top surface  128  and define a radial interval of the rolling element  120  on the bit face  202 . The control point C is located between the control points A and B. The control points A, B, C generally represent locations along the top surface  128  that may be evaluated for contact with the formation during drilling operations. More or fewer control points may be evaluated, and in some embodiments, tens or hundreds of control points may be evaluated in practice. 
     An axis A, extends normally to the rolling axis A 0  in the Y-R plane and extends through control point C and the rolling center O. A profile angle φ is defined between the axis A 1  and the vertical or Y-axis. In the optimization and/or selection processes described below, a rolling element  120  may or may not ever initially be disposed at the profile angle φ, but the profile angle φ provides a basis for an adjusted profile angle Δφ, which defines an orientation of the rolling element  120  in the Y-R plane. The adjusted profile angle Δφ is defined between the axis A 1  of the rolling element  120  in the “initial” orientation and the axis A 2  of the rolling element  120  in the adjusted orientation. 
       FIG. 6  is a schematic profile view of the bit face of  FIG. 3  generally illustrating an orientation of the rolling element  120  in a Z-X plane defined by the horizontal axes Z and X. An angular position of θ of the rolling element  120  may be defined between the X axis and a radial plane RP passing through the Y-axis and the rolling center “O” of the rolling element  120 . An adjusted angular position dθ of the rolling element  120  is defined as the angle subtended between the radial plane RP and rolling axis A 0  of the rolling element  120 . The adjusted angular position dθ of the rolling element  120  thereby defines the orientation of the rolling element  120  in the Z-X plane. 
     The control points A and B are defined along the radial plane RP. The rolling element  120  may control a depth of cut within the radial interval defined between the control points A and B. 
       FIG. 7A  is flowchart illustrating a procedure for selecting the location and orientation of the rolling elements  120  on a drill bit face to balance operational forces on the rolling elements  120 . The steps of method  300  may be performed by various computer programs, models or any combination thereof, configured to simulate and design drilling systems, apparatuses and devices. The programs and models may include instructions stored on a computer readable medium and operable to perform, when executed, one or more of the steps described below. The computer readable media may include any system, apparatus or device configured to store and retrieve programs or instructions such as a hard disk drive, a compact disc, flash memory or any other suitable device. The programs and models may be configured to direct a processor or other suitable unit to retrieve and execute the instructions from the computer readable media. Collectively, the computer programs and models used to simulate and design drilling systems may be referred to as a “drilling engineering tool” or “engineering tool.” 
     Initially at step  302 , a plurality of fixed cutters  116  and rolling element assemblies  118  may be laid out on a bit body to achieve a desired set of design objectives. The initial position and orientation of the rolling element assemblies  118  may be selected such that the rolling elements  120  provide particular depth of cut control characteristics and cutting characteristics. Once the initial position and orientation of the rolling elements  120  are selected, an initial set of design variables is established. Each rolling element  120  will be defined by at least the following variables: 
     1) δ=under exposure, i.e., the distance from top of a fixed cutter  116   
     2) φ=profile angle 
     3) Δφ=adjusted profile angle 
     4) θ=angular position from an X-axis 
     5) dθ=adjusted angular position 
     6) dr=radial offset from a Y-axis (e.g., a bit rotational axis) 
     7) Dr=rolling element diameter 
     8) Lr=rolling element length 
     Generally, the rolling element diameter Dr and the rolling element length Lr of a cylindrical rolling element defines the shape of the rolling element  120 , the radial offset dr, angular position θ and the under exposure δ define a position of the rolling element  120 , and the profile angle φ, adjusted profile angle Δφ and adjusted angular position dθ define an orientation of the rolling element  120  on the bit body  102 . Once an initial set of design variables is determined for each of the rolling element  120 , the procedure may proceed to step  304 . 
     At step  304 , using the set of design variables, a critical depth of cut is determined for each control point, e.g., A, B. C ( FIG. 5 ), along the top surface  128  rolling element  120 . The critical depth of cut may be expressed in the units of inches per revolution, and generally indicates the degree to which each point along the rolling element length Lr of the rolling element  120  will penetrate the geologic formation in operation. The critical depth of cut calculation may be used to assess how even the depth of cut is across the rolling element length Lr. Any number of control points may be selected, and in some embodiments, at least three control points are selected. In some embodiments, an engineering tool may calculate a critical depth of cut at hundreds of control points evenly spaced along the top surface  128  of the rolling element  120 . Specific steps for calculating the critical depth of cut are described herein with respect to  FIG. 7B  below. 
     At decision  306 , it is determined whether the critical depth of cut calculation revealed at least a predetermined number of contact zones where the rolling element is in contact with the geologic formation. The contact zones represent the radial distance dr where the rolling element  120  is controlling the depth of cut achieved by the fixed cutters  116  at the same radial distance dr. In some embodiments, the predetermined number of contact zones is two distinct contact zones located on opposite lateral sides of the rolling center O. This arrangement permits the forces acting on the rolling element  120  to balance one another to some extent. In some embodiments, the predetermined number of contact zones is at least three contact zones. If the number of contact zones identified along the rolling element length Lr of the rolling element from  120  is fewer than the predetermined number of contact zones, then the procedure may proceed to step  308 . 
     At step  308 , at least one design variable may be changed to establish an adjusted set of design variables. For example, the under exposure δ may be increased or decreased. The adjusted profile angle Δφ and/or the adjusted angular position dθ of rolling element may be changed to rotate the rolling element  120  to an orientation expected to increase contact between the rolling element and the geologic formation. In some embodiments, the shape and/or position of the rolling element  120  may also be changed to increase contact with the geologic formation. In some embodiments, the engineering tool may be configured to systematically change the at least one design variable and in other embodiments, a bit designer may input the change to the at least one design variable into the engineering tool. 
     Once the at least one design variable has been changed, the critical depth of cut Δ for each control point along the rolling element length of the rolling element  120  may be recalculated using the adjusted set of design variables. The procedure  300  may then return to decision  306  where it is again determined whether the predetermined number of contact zones exists for a given critical depth of cut. The decision  306  and step  308  may be repeated iteratively until the predetermined number of contact zones is found to exist for a given critical depth of cut. Then, the procedure  300  may progress to step  310 . 
     At step  310 , an engagement area Af is calculated along with the resultant operational loads P acting on the rolling element  120 . The engagement area Af represents the cross sectional area of the rolling element  120  that penetrates into the geologic formation in operation (see  FIG. 8B ). The resultant operational loads P may each include a tangential component P tan  and a radial component P rad  with respect to the rolling element  120  (see  FIGS. 8A and 8B ). From the magnitude and position of the radial component P rad  of the operational loads P, the moment M acting on the rolling element  120  may be calculated. Specific steps for calculating the engagement area Af, resultant operational loads P, and moment M are specified in the procedure of  FIG. 7C . 
     At decision  312 , the engineering tool may determine whether the operational loads P and the moment M are within an acceptable predetermined range. If the rolling element  120  does not meet all design requirements, at least one of the operational loads P and the moment M may fall outside the acceptable predetermined range. The procedure  300  may then return to step  308  where at least one design variable is changed. Steps and decisions  308 ,  306 ,  310  and  312  may be repeated and iterated, at least until the operational loads P and the moment M fall within the acceptable predetermined range. The procedure  300  may then progress to step  314 . 
     At step  314 , the adjusted set of design variables that yielded the operational loads P and the moment M falling within the predetermined range may be recorded as a final set of design variables. In some embodiments, a drill bit may be constructed with rolling element(s)  120  located and oriented according to the final set of design variables. 
       FIG. 7B  is a flowchart illustrating a procedure  400  for calculating a critical depth of cut Δ specified in steps  304  and  308  the procedure  300  of  FIG. 7A . The procedure  400  permits a critical depth of cut Δ to be calculated for each radial position Rf on the drill bit that includes an RODCC. Since the procedure  400  may be performed after step  302  ( FIG. 7A ) of procedure  300  where the initial set of design variables is established, the coordinates Xf, Yf, and Zf of the points F located on a radial plane passing through the bit axis  107  (y-axis) and the center of a rolling element  120  may have been established prior to step  402  of the procedure  400 . 
     At step  402 , at a given radial location Rf i , the coordinates of the points P j  where the circle  204  ( FIG. 3 ) having a radius Rf i  intersects the edges of cutters  116  and the points F k  where the circle intersects the center of the rolling elements  120 . Here, the index “i” represents the specific control point, e.g., A to C, along the radial plane RP passing through one of the rolling elements  120  (see  FIG. 6 ), the index “j” represents the number of the fixed cutter  116  intersecting the circle  204 , e.g., 6, 7, 8, and 9, and the index “k” represents the number of the rolling element  120 , e.g., 1, 2 and 3. Once the Cartesian coordinates of the points P j , F k  for a given radial location are compiled, the procedure  400  may proceed to step  404 . 
     At step  404 , the angular positions θf k  of the points F k  (e.g., points F 1 -F 3 ) are calculated. Where the angular position θf k  is defined within 0°-360°, the angular position θf k  may be given by the equation:
 
θ f   k =arctan 2( Zf   k   ,Xf   k )·180.0/π.
 
     At step  406 , the angular positions θp j  of the points P j  (e.g., points P 6 -P 9 ) are similarly calculated. Where the angular position θp j  is defined within 0°-360°, the angular position θp j  may be given by the equation:
 
θ p   j =arctan 2( Zp   j   ,Xp   j )·180.0/π.
 
     At step  408  the critical depth of cut Δj provided to each point P j  by each F k  may be calculated. The critical depth of cut Δj may be given by the equation:
 
Δ j=δj· 360/(360−Δθ j ).
 
     To calculate the critical depth of cut Δj using the equation above, an angular offset Δθ j  between the points P j  and the points F k  must be determined as well as the axial underexposure δj of the points P j  with respect to the points F k . The angular offset Δθ j  (defined within 0°-360°), and the axial underexposure δj may be given by the equations below.
 
Δθ j   =θf   k   −θp   j  
 
δ j=Yp   j   −Yf   k  
 
     At step  410 , once the critical depths of cut Δj are calculated, the critical depths of cut provided by each of the points F k  may be calculated as the maximal of the critical depths of cut Δj by the equation given below.
 
Δ f   k =max [Δ j ].
 
     For example, the critical depth of cut provided by the point F 1  is the maximal of Δ 6 , Δ 7 , Δ 8 , and Δ 9  for the bit face  202  illustrated in  FIG. 3 . Once the critical depths of cut provided by each of the points F k  are calculated, the procedure  400  may progress to step  412 . 
     At step  412 , a bit critical depth of cut Δi at the given radius Rf i  is determined. The bit critical depth of cut at the radius Rf, is given by the equation below.
 
Δ i =min [Δ f   k ].
 
     Once the bit critical depth of cut Δi at the given radius Rf i  is determined, the procedure may progress to step  414  where the index “i” is updated, and the steps of the procedure  400  are repeated for another different radius Rf i . The index “i” may range from zero (0) to the radius of the bit face  202  such that the critical depth of cut Δi may be plotted as a function of radial position of the bit face (see, e.g.,  FIG. 10 ). 
       FIG. 7C  is a flow chart illustrating a procedure  500  for calculating the operational loads P and moment “m” acting on a rolling element  120  as specified in step  310  of the procedure  300  of  FIG. 7A .  FIGS. 8A and 8B  are side and end views, respectively, of the rolling element  120  illustrating the operational loads acting on the rolling element as  120  specified in the procedure  500  of  FIG. 7C . 
     At step  502 , the depth of cut ΔF l  for any point F l  on the top surface  128  of the rolling element  120  is determined in cross sectional plane (see  FIG. 8A ). The depth of cut Δ may be determined or based on the results of step  412  of the procedure  400  described above where the bit critical depth of cut Δi at the given radius Rf i  has been determined. At step  504 , the associated engagement area Af defined between the rolling element  120  and the geologic formation determined from the depth of cut ΔF l  and the particular geometry of the rolling element  120 . 
     At step  504 , a force model may be applied to determine the radial component p rad  of the point operational loads p applied at the particular point F l  of the rolling element  120 . The force model may define the radial component p rad  as a function of the depth of cut ΔF l  and the engagement area Af determined above, as well as other known, ascertainable or estimable variables such as the rock strength. The tangential component p tan  of the operational loads P tan  at the particular point F l  may then be determined from the radial component p rad  and a rolling coefficient of friction μ the by the following equation.
 
 p   tan   =μp   rad .
 
     At step  506 , a moment “m” about the rolling center “O” due to the radial component p rad  at the particular point F l  may be determined. At step  508 , the index l may be updated, and the operational loads p rad , p tan  and “m” may be determined for another point F l  on the top surface  128  of the rolling element  120 . Steps  502  through  508  may be repeated and iterated until all the points along the top surface  128  of the rolling element  120  are considered. 
     The procedure  500  may then proceed to step  510 . All the point loads p rad , p tan  and moments “m” may be summarized and simplified to the center “O’ to obtain combined loads P rad , P tan  and moment M for the rolling element  120 . At step  512  the combined forces P rad , P tan  may be projected into a bit coordinate system to obtain bit forces contributed from the rolling element  120 . At step  514  the combined forces P rad , P tan  may be projected into a hole or wellbore coordinate system to obtain steer force and a walk force for the rolling element  120 . At step  516 , the index k may be updated, and each of the rolling elements  120  on the bit face  202  may be considered. 
       FIG. 9A  is a schematic view of the three rolling elements  120  of  FIG. 3  illustrating an example operational loading prior to performing any optimization in the procedure  300  of  FIG. 7A . Generally, the operational loads P are balanced on a first rolling element  120   a  of the three rolling elements  120  and unbalanced on second  120   b  and third  120   c  of the rolling elements  120 . The operational loads P illustrated in  FIG. 9A  were determined based on a bit rotation rate of 120 RPM and rate of penetration (ROP) of 120 ft/hr. The location and orientation of the rolling elements  120  were selected based an initial layout of the bit face  202  (see step  302  of procedure  300  on  FIG. 7A ) before any optimization or change to any of the design variables defining the rolling elements  120  was implemented. 
     The operational loads P 4  through P 9  on the rolling elements  120   a ,  120   b  and  120 C represent the combined radial components p rad  of the point loads spanning a specific contact zone Z 4  through Z 9  existing along the rolling element length of the rolling elements  120 . The contact zones Z 4  through Z 9  may be identified from a plot of the critical depth of cut control curves for each of the rolling elements  120  plotted against the bit radius. 
     For example, three distinct contact zones Z 4 , Z 5  and Z 6  were identified along the first rolling element  120   a  where the critical depth of cut curve for the first rolling element  120   a  indicated a critical depth of cut control beneath a threshold “T.” The threshold “T” may represent a desired minimum depth of cut to be maintained in operation for the radial interval on the bit face  202  containing the rolling element  120   a . The upper shaded region of the critical depth of curves illustrated in  FIG. 9A  represent the critical depths of cut Δ where the rolling elements  120  will be in engagement with the geologic formation, and the lower un-shaded region critical depth of cut curves represent an un-controlled region. Thus, the portions of the shaded regions extending below the threshold “T” represent the zones of contact between the upper surfaces  128  of the rolling elements  120  and the geologic formation when the minimum depth of cut is maintained in operation. In some embodiments, the threshold “T” may be predetermined by a bit designer. For example, if a bit designer desired that the rolling element would be in contact with the formation only if ROP was over 120 ft/hr with an RPM equal to 120, the threshold “T” could be 0.2 in/rev. 
     Since three distinct contact zones Z 4 , Z 5  and Z 6  were identified along the first rolling element  120   a , this rolling element  120   a  may be found to provide the predetermined number of contact points described in above with reference to decision  306  of the procedure  300  (see  FIG. 7A ). Since only two distinct contact zones Z 7 , Z 8  were identified along the second rolling element  120   b , operational loads P 7  and P 8  may only partially balance one another on the second rolling element  120   b . For example, where the magnitude of the operational loads P 7  and P 8  are substantially different, the loads P 7  and P 8  may produce a moment on the second rolling element  120   b . Since only one distinct contact zone Z 9  was identified along the third rolling element  120   c , no operational load is available to balance the operational load P 9  on the third rolling element  120   c . Thus, if no optimization is performed, the third rolling element  120   c  may be estimated to be the first to fail in operation. 
       FIG. 9B  is a schematic view of the second and third rolling elements  120   b ,  120   c  of  FIG. 9A  illustrating an example operational loading subsequent to performing an optimization in the procedure of  7 A. By changing at least one design variable as specified in step  308  of the procedure  300  ( FIG. 7A ) to redesign, reposition and/or reorient the second and third rolling elements on the bit face  202  ( FIG. 3 ), at least three contact zones Z 10 , Z 11  and Z 12  may be identified on the second rolling element  120   b , and at least three contact zones Z 13 , Z 14  and Z 15  may be identified on the third rolling element  120   c . The operational loads P 10  and P 11  may be balanced by operational load P 12  on the second rolling element  120   b  and operational loads P 13  and P 14  may be balanced by operational load P 15  on the third rolling element. By balancing the operational loads on the all three rolling elements, the operational life of the rolling elements  120 , and thus the drill bit  200  ( FIG. 3 ) on which the rolling elements  120  are placed, may be extended. 
       FIG. 10  is a diagrammatic view of the optimized critical depth of cut curves calculated in the procedure of  FIG. 7A  for all three of the rolling elements  120  of  FIG. 3 . Where the critical depth of cut is charted against a bit radius for the radial portion of the drill bit including the rolling elements  120 , the controlled depth of cut provided by the rolling elements be evaluated. 
     The aspects of the disclosure described below are provided to describe a selection of concepts in a simplified form that are described in greater detail above. This section is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter. 
     In one aspect, the disclosure is directed to a method of configuring a rolling depth of cut controller (RDOCC) of a drill bit. The method includes (a) selecting a position and an orientation for a first rolling element of the RDOCC on a bit face of a drill bit, the first rolling element defining a top surface along a generally cylindrical body thereof, (b) establishing a set of design variables associated with the position, the orientation and a shape of the first rolling element, (c) calculating a critical depth of cut for a plurality of control points along the top surface of the first rolling element using the design variables, (d) identifying a number of contact zones existing along the top surface of the rolling element from the critical depth of cut calculated, (e) determining an engagement area and associated force magnitudes of operational forces acting on the rolling element for each of the contact zones identified, (f) ascertaining a moment acting on the rolling element from the force magnitudes determined, and (g) comparing the force magnitudes and the moment to predetermined limits. 
     In one or more example embodiments, identifying the number of contact zones existing along the rolling element length of the rolling element includes identifying at least two distinct contact zones on opposite lateral sides of a rolling center of the first rolling element. Identifying the number of contact zones may include identifying at least three distinct contact zones. 
     In some embodiments, the method further includes identifying a number of contact zones existing along the top surface of the first rolling element that includes less than three distinct contact zones, changing at least one of the design variables to establish an adjusted set of design variables, and recalculating the critical depth of cut for the plurality of control points along the top surface of the first rolling element using the adjusted set of design variables. Changing at least one of the design variables may include changing at least one of an adjusted profile angle and an adjusted angular position of the first rolling element defining an orientation of the first rolling element on the drill bit. 
     In example embodiments, the method further includes deciding that at least one of the force magnitudes and the moment are outside the predetermined limits, and changing at least one of the design variables to establish an adjusted set of design variables. The method may also further include determining a plurality of intersection points associated with cutting edges of fixed cutters on the bit face, each of the plurality of intersection points having substantially the same radial location as one of the control points, and calculating a critical depth of cut provided by each of the control points to each of the intersection points based on differences in position defined between the control points and the intersection points. The method may further include determining a critical depth of cut for each of the control points as a maximal of the depths of cut provided to each of the intersection points by each of the control points. In some embodiments, the method also includes determining a critical depth of cut for each of a plurality of control points defined on at least a second rolling element having substantially the same radial location as one of the intersection points, and determining a bit critical depth of at the radial locations of the intersection points as the minimum of critical depths of cut for each of the control points on each of the first and second rolling elements provided to each of the intersection points. The plurality of intersection points may include an intersection point defined on all of the fixed cutters located on the bit face that each include at least a portion of their cutting edges at the same radial location as a corresponding control point. In some embodiments, the method further includes projecting the operational forces into at least one of a bit coordinate system and a hole coordinate system. 
     According to another aspect, the disclosure is directed to a drill bit including a bit body defining a rotational axis about which the bit body rotates. A bit face is defined at a leading end of the bit body and a first rolling element is disposed on the bit face. The first rolling element defines a top surface along a generally cylindrical body thereof, and the top surface defines a first radial interval of the bit face. A first cutting element is defined on the bit face, and the first cutting element has a cutting edge extending at least partially into the first radial interval on the bit face. A position and orientation of the first rolling element on the bit face is configured to maintain at least three distinct contact zones between the top surface and a geologic formation to control a depth of cut associated with the first cutting element. 
     In some example embodiments, the drill bit further includes at least a second cutter having a cutting edge extending at least partially into the first radial interval, and the depth of cut may be controlled by the first rolling element is based on at least the first and second cutters. The drill bit may further include a second rolling element on the bit face, and the second rolling element may define a second radial interval overlapping a portion of the first radial interval into which the cutting edge of the first cutting element extends. In one or more example embodiments, the first cutting element may be a fixed cutting element on the bit face. 
     In another aspect, the disclosure is directed to a method of configuring a rolling depth of cut controller (RDOC) of a drill bit. The method includes (a) determining a desired minimum depth of cut for a radial interval defined on a bit face of the drill bit, (b) identifying all cutting elements located on the bit face that each include a cutting edge defined at least a partially within the radial interval, (c) determining a radial position of a rolling element of the depth of cut controller within the radial interval, the rolling element defining a cylindrical body, (d) identifying a number of contact zones existing along a top surface of the rolling element in an initial position and orientation based on the desired minimum depth of cut and each of the cutting edges defined at least a partially within the radial interval, and (e) determining a final an axial position, an angular position and an orientation of the rolling element based on the number of contact zones identified. 
     In some example embodiments, the number of contact zones existing along the top surface of the rolling element is at least three contact zones. In one or more example embodiments, the method further includes determining the final axial position, angular position and orientation of the rolling element based on a moment acting on the rolling element due to operational forces applied to the rolling element at the contact zones. 
     Therefore, the disclosed systems and methods are well adapted to attain the advantages mentioned as well as those that are inherent therein. The particular embodiments disclosed above are illustrative only, as the teachings of the present disclosure may be modified and practiced in different but equivalent manners apparent to those skilled in the art having the benefit of the teachings herein. Furthermore, no limitations are intended to the details of construction or design herein shown, other than as described in the claims below. It is therefore evident that the particular illustrative embodiments disclosed above may be altered, combined, or modified and all such variations are considered within the scope of the present disclosure. The systems and methods illustratively disclosed herein may suitably be practiced in the absence of any element that is not specifically disclosed herein and/or any optional element disclosed herein. While compositions and methods are described in terms of “comprising,” “containing,” or “including” various components or steps, the compositions and methods can also “consist essentially of” or “consist of” the various components and steps. All numbers and ranges disclosed above may vary by some amount. Whenever a numerical range with a lower limit and an upper limit is disclosed, any number and any included range falling within the range is specifically disclosed. In particular, every range of values (of the form, “from about a to about b,” or, equivalently, “from approximately a to b,” or, equivalently, “from approximately a-b”) disclosed herein is to be understood to set forth every number and range encompassed within the broader range of values. Also, the terms in the claims have their plain, ordinary meaning unless otherwise explicitly and clearly defined by the patentee. Moreover, the indefinite articles “a” or “an,” as used in the claims, are defined herein to mean one or more than one of the elements that it introduces. If there is any conflict in the usages of a word or term in this specification and one or more patent or other documents that may be incorporated herein by reference, the definitions that are consistent with this specification should be adopted. 
     As used herein, the phrase “at least one of” preceding a series of items, with the terms “and” or “or” to separate any of the items, modifies the list as a whole, rather than each member of the list (i.e., each item). The phrase “at least one of” allows a meaning that includes at least one of any one of the items, and/or at least one of any combination of the items, and/or at least one of each of the items. By way of example, the phrases “at least one of A, B, and C” or “at least one of A, B, or C” each refer to only A, only B, or only C; any combination of A, B, and C; and/or at least one of each of A, B, and C. 
     The Abstract of the disclosure is solely for providing the United States Patent and Trademark Office and the public at large with a way by which to determine quickly from a cursory reading the nature and gist of technical disclosure, and it represents solely one or more examples. 
     While various examples have been illustrated in detail, the disclosure is not limited to the examples shown. Modifications and adaptations of the above examples may occur to those skilled in the art. Such modifications and adaptations are in the scope of the disclosure.