Patent Publication Number: US-2023151697-A1

Title: Ridge shaped element

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of Provisional Application No. 62/983,883, filed on Mar. 2, 2020, which is hereby incorporated by reference in its entirety. 
    
    
     BACKGROUND 
     Drag bits, often referred to as “fixed cutter drill bits,” include bits that have cutting elements attached to the bit body, which may be a steel bit body or a matrix bit body formed from a matrix material such as tungsten carbide surrounded by a binder material. Drag bits may generally be defined as bits that have no moving parts. Drag bits having cutting elements made of an ultrahard cutting surface layer or “table” (generally made of polycrystalline diamond material or polycrystalline boron nitride material) deposited onto or otherwise bonded to a substrate are known in the art as polycrystalline diamond compact (“PDC”) bits. 
     An example of a drag bit having a plurality of cutting elements with ultrahard working surfaces is shown in  FIG.  1   . The drill bit  10  includes a bit body  11  having a threaded upper pin end  12  and a cutting end  13 . The cutting end  13  generally includes a plurality of ribs or blades  14  arranged about the rotational axis (also referred to as the longitudinal or central axis) of the drill bit and extending radially outward from the bit body  11 . Cutting elements, or cutters,  15  are embedded in the blades  14  at predetermined angular orientations and radial locations relative to a working surface and with a desired back rake angle and side rake angle against a formation to be drilled. 
     The cutters  15  are generally cylindrical in shape having an ultrahard material layer attached to a substrate, such as a cemented carbide substrate. The top surface of the ultrahard material layer may be referred to as a working surface, and the edge formed around the top surface may be referred to as the cutting edge, as the working surface and cutting edge of the cutting elements are typically the surfaces that contact and cut a formation. 
     SUMMARY 
     This summary is provided to introduce a selection of concepts that are further described below in the detailed description. This summary is not intended to identify key or essential features of the claimed subject matter, nor is it intended to be used as an aid in limiting the scope of the claimed subject matter. 
     Some embodiments of the present disclosure relate to cutting elements that include a substrate and an ultrahard layer on an upper surface of the substrate, a top surface of the ultrahard layer having a ridge extending along a major dimension of the top surface from an edge of the top surface, where the ridge may have a peak with at least two different roof radii of curvature, and at least two sidewalls sloping in opposite directions from the peak of the ridge at a roof angle, where a first roof angle of the ridge proximate the edge may be smaller than a second roof angle in a central portion of the ridge around a longitudinal axis of the cutting element. 
     Some embodiments of the present disclosure relate to cutting elements that include a top surface having a ridge extending from an edge of the top surface along a major dimension of the top surface, and a peak of the ridge having a width measured between opposite points of transition from the peak to a sidewall, wherein the width of the peak in a central portion of the ridge around a longitudinal axis of the cutting element may be greater than the width of the peak in the edge portion of the ridge, the edge portion extending a length of the ridge from the edge to the central portion, and wherein the peak may have a roof radius of curvature along an edge portion of the ridge less than 0.1 inches. 
     Some embodiments disclosed herein relate to cutting elements that include a substrate and an ultrahard layer on an upper surface of the substrate, a top surface of the ultrahard layer having a geometric surface axially extended from a plurality of recessed edge portions formed around an edge of the top surface, and at least one ridge extending radially outward from the geometric surface to the edge of the top surface, the at least one ridge having a peak with a roof radius of curvature. 
     Some embodiments disclosed herein relate to methods of forming a cutting element that includes providing a cutting element having a ridge formed at a top surface of the cutting element, the ridge extending along a major dimension of the top surface from an edge of the top surface, wherein the ridge has a peak with a first roof radius of curvature and sidewalls sloping away from the peak at a first roof angle, and removing an amount of ultrahard material from the top surface around an edge portion of the ridge to form a second peak having a second roof radius of curvature smaller than the first roof radius of curvature and recessed sidewalls sloping away from the second peak at a second roof angle smaller than the first roof angle, wherein the edge portion having the second roof radius of curvature and the second roof angle extends a partial length of the ridge from the edge toward a longitudinal axis of the cutting element. 
     Other aspects and advantages will be apparent from the following description and the appended claims. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG.  1    is a conventional drill bit. 
         FIGS.  2  and  3    show side views of a cutting element according to embodiments of the present disclosure. 
         FIG.  4    shows an ultrahard layer according to embodiments of the present disclosure. 
         FIG.  5    shows a side view of the ultrahard layer shown in  FIG.  4   . 
         FIG.  6    shows a top view of the ultrahard layer shown in  FIGS.  4  and  5   . 
         FIG.  7    shows another side view of the ultrahard layer shown in  FIGS.  4 - 6   . 
         FIG.  8    shows a cutting element according to embodiments of the present disclosure. 
         FIG.  9    shows an ultrahard layer according to embodiments of the present disclosure. 
         FIG.  10    shows a top view of the ultrahard layer shown in  FIG.  9   . 
         FIG.  11    shows a side view of the ultrahard layer shown in  FIGS.  9  and  10   . 
         FIG.  12    shows another side view of the ultrahard layer shown in  FIGS.  9 - 11   . 
         FIG.  13    shows an ultrahard layer according to embodiments of the present disclosure. 
         FIG.  14    shows a side view of the ultrahard layer shown in  FIG.  13   . 
         FIG.  15    shows a top view of the ultrahard layer shown in  FIGS.  13  and  14   . 
         FIG.  16    shows a cross-sectional view of the ultrahard layer of  FIGS.  13 - 15    along a plane intersecting the longitudinal axis of the ultrahard layer and extending through the length of the ridge on the ultrahard layer. 
         FIG.  17    shows another cross-sectional view of the ultrahard layer of  FIGS.  13 - 16    along a plane intersecting the longitudinal axis of the ultrahard layer and perpendicular to the length of the ridge on the ultrahard layer. 
         FIG.  18    shows another cross-sectional view of the ultrahard layer of  FIGS.  13 - 17    along a plane parallel to the longitudinal axis of the ultrahard layer and perpendicular to the length of the ridge on the ultrahard layer. 
         FIG.  19    shows a top view of a cutting element according to embodiments of the present disclosure. 
         FIG.  20    shows a top view of a cutting element according to embodiments of the present disclosure. 
         FIG.  21    shows a top view of a cutting element according to embodiments of the present disclosure. 
         FIG.  22    shows a top view of a cutting element according to embodiments of the present disclosure. 
         FIG.  23    shows a top view of a cutting element according to embodiments of the present disclosure. 
         FIG.  24    shows a top view of a cutting element according to embodiments of the present disclosure. 
         FIGS.  25 - 28    show different views of a cutting element according to embodiments of the present disclosure. 
         FIG.  29    shows a graph comparing forces and specific energy during testing of different cutting element types with a cutting element according to embodiments of the present disclosure. 
         FIGS.  30 - 34    show different views of a cutting element according to embodiments of the present disclosure. 
         FIG.  35    shows a comparison between the contacting area of a planar cutting element with ridge cutting elements at a depth of cut. 
         FIG.  36    shows a graph of the change in contacting area at different depths of cut for the cutting elements shown in  FIG.  35   . 
         FIG.  37    shows a schematic of forces acting on a ridge cutting element. 
         FIG.  38    shows a cross-sectional view of a ridge cutting element as it cuts a formation. 
     
    
    
     DETAILED DESCRIPTION 
     Embodiments of the present disclosure generally relate to shaped elements (e.g., shaped cutting elements), which may be mounted to drill bits for drilling earthen formations or other cutting tools. The shaped element geometry may include a non-planar top surface, also referred to as a working surface or cutting face, formed on an ultrahard layer of the shaped element. Further, the ultrahard layer of the shaped element may be on a substrate at a non-planar interface surface designed to improve the cutting performance of the non-planar top surface. Shaped elements of the present disclosure may be mounted to various types of downhole tools, including but not limited to, drill bits, such as drag bits, reamers, and other downhole milling tools. 
     The non-planar top surface may have a ridge geometry optimized to improve drilling efficiency and stability. Three parameters of the ridge geometry—roof angle, roof radius of curvature, and roof ridge angle—have been identified as factors in determining the cutting element engagement with a rock formation and torque resistance in the cutting tool. According to embodiments of the present disclosure, roof angle, roof radius of curvature, and roof ridge angle may be designed in combination to provide improved cutting efficiency.  FIGS.  2  and  3    show side views of a cutting element  100  according to embodiments of the present disclosure identifying the roof angle  102 , roof radius of curvature  104 , and roof ridge angle  106  of the cutting element ridge geometry. 
     The cutting element  100  includes an ultrahard layer  160  disposed on a substrate  162  at an interface  164 , where the non-planar top surface  110  geometry is formed on the ultrahard layer  160 . The non-planar top surface  110  geometry includes a ridge  120  extending along a major dimension  180  of the top surface between opposite sides of an edge  114  surrounding (and defining the bounds of) the top surface  110 . The presence of the ridge  120  results in an undulating edge  114  having raised and recessed portions relative to each other. In the embodiment shown, the ridge  120  may extend across the entire diameter of the ultrahard layer  160  between two opposite raised portions of the edge  114 . 
     A chamfer  140  may be formed around the edge  114 , or periphery, of the top surface  110 , where the chamfer  140  extends radially inward from the edge  114  of the top surface  110 . In some embodiments, the chamfer  140  may extend around the entire periphery of the top surface  110 . In some embodiments, the chamfer  140  may extend partially around the periphery of the top surface  110  (i.e., less than the entire periphery of the top surface  110 ). In one or more embodiments, the chamfer  140  may vary in angle and/or width around the edge  114 . In some embodiments, a cutting element  100  may have a radiused edge  114 . 
     As shown, the ridge  120  has a peak  122  with a convex cross-sectional shape when viewed along a plane perpendicular to the length of the ridge  120  along the major dimension  180 , where the peak  122  has a roof radius of curvature  104 . The peak  122  of the ridge  120  may have a width  124  measured between opposite points  126 ,  128  of transition from the peak  122  to a sidewall  130 . A roof radius of curvature  104  may be selected from a range of 0.02 inches to 0.2 inches, depending on, for example, the size of the cutting element  100  and the other ridge geometry factors of interest in this disclosure, including the roof angle  102  and roof ridge angle  106 . Further, according to embodiments of the present disclosure, a roof radius of curvature  104  may be varied along the length of the ridge  120 . For example, as discussed more below, a first portion of the ridge  120  may have a peak  122  with a first roof radius of curvature  104 , and a second portion of the ridge  120  may have a peak  122  with a second roof radius of curvature  104  that is greater than the first roof radius of curvature  104 . 
     While the embodiment shown in  FIGS.  2  and  3    has a ridge  120  with a convex peak  122 , it is also within the scope of the present disclosure that the peak  122  may have a plateau or substantially planar face along a portion of the ridge  120 . In such embodiment, the peak  122  may have a substantially infinite roof radius of curvature  104 . Further, planar peak  122  portions of a ridge  120  may have radiused transitions to the sidewalls  130  on either side of the ridge  120 . 
     The roof angle  102  is the angle defined between the sidewalls  130  along a longitudinal plane parallel with the longitudinal axis  101  of the cutting element  100  and perpendicular to a plane tangent to each sidewall  130 . According to embodiments of the present disclosure, a roof angle  102  may be selected from a range of about 110 degrees to about 165 degrees, depending on, for example, the size of the cutting element  100  and the other ridge geometry factors of interest in this disclosure, including the roof radius of curvature  104  and roof ridge angle  106 . Further, according to embodiments of the present disclosure, a roof angle  102  may be varied along the length of the ridge  120 . For example, as discussed more below, a first portion of the ridge  120  may have a peak  122  with a first roof angle  102 , and a second portion of the ridge  120  may have a peak  122  with a second roof angle  102  that is greater than the first roof radius of curvature  104 . 
     In embodiments having a chamfer  140  formed around the edge  114  of the top surface  110 , the peak  122  of the ridge  120  may intersect with an interior boundary  141  of the chamfer  140 , where the peak  122  of the ridge  120  may extend from proximate the edge  114  of the cutting element  100  in a direction toward the longitudinal axis  101 . In some embodiments, the peak  122  of the ridge  120  may extend from the edge  114  of the cutting element  100  without a chamfer between the edge  114  and the peak  122 . 
     The ridge  120  may be axially separated a height  125  from a recessed edge portion  132  formed around the edge  114  of the top surface  110 , where the recessed edge portion  132  may be the axially farthest region of the edge  114  from the peak  122  of the ridge  120 . In some embodiments, the height  125  of the ridge  120  may be uniform along its length, where the entire peak  122  extends along a plane  123  perpendicular to the longitudinal axis  101 . In some embodiments, such as shown in  FIG.  3   , the height  125  of the ridge  120  may vary. For example, as shown in  FIG.  3   , the height  125  of the ridge may increase in a direction from the edge  114  toward the longitudinal axis  101 , such that the peak  122  of the ridge  120  has a sloped portion proximate the edge  114  of the top surface  110 . 
     A roof ridge angle  106  is the angle defined between a line  121  tangent to the peak  122  of the ridge  120  proximate the edge  114  and a plane  123  perpendicular to the longitudinal axis  101 . According to embodiments of the present disclosure, a ridge  120  may have a roof ridge angle  106  selected from a range of zero to about  10  degrees on one or both edge portions of the ridge  120 , such that the axial height of the ridge  120  at the edge portion of the ridge  120  is less the axial height of the ridge  120  at the central portion of the ridge  120 . 
     According to embodiments of the present disclosure, an edge portion of a ridge may have a roof ridge angle greater than zero in combination with a reduced roof radius of curvature and a reduced roof angle when compared with a central portion of the ridge. Such combination of ridge geometry factors may increase cutting efficiency. 
     For example, as shown in the embodiment of  FIGS.  4 - 7   , an ultrahard layer  200  may have an edge portion  221  of a ridge  220  with a roof ridge angle  206  greater than zero in combination with a reduced roof radius of curvature  204   a  and a reduced roof angle  202   a  when compared with a central portion  223  and/or other portions of the ridge  220  along its length  280 . 
     In some embodiments, an edge portion  221  of a ridge  220  may refer to a length  281  of the ridge  220  measured from the edge  214  of the top surface  210  that corresponds with a predicted depth of cut of the cutting element during operation. For example, if a predicted depth of cut of a cutting element during operation ranges up to 0.2 inches, a cutting edge portion  221  of a ridge  220  formed on the top surface  210  of the cutting element may refer to the portion of the ridge within 0.2 inches from the edge  214  of the top surface  214 . In some embodiments, an edge portion  221  of a ridge  220  may refer to a percentage of the entire length  280  of the ridge proximate the edge  214  of the top surface  210 . For example, an edge portion  221  may extend a length  281  from the edge  214  of the top surface  210  that is between 5 and 30 percent of the entire length  280  of the ridge  220 . 
       FIGS.  4 - 7    show the ultrahard layer  200  portion of a cutting element, which may be attached to (or formed to) a substrate at a planar or non-planar interface to form the cutting element. For example, in the embodiment shown in  FIGS.  4 - 7   , the ultrahard layer  200  may have a bottom surface  207  that may be attached to an upper surface of a substrate having a geometry corresponding to the bottom surface  207  geometry, forming an interface between the ultrahard layer  200  and substrate. 
     The geometry of the top surface  210  of the cutting element  200  may be described with respect to an x-y-z coordinate system, as shown in  FIG.  4   . The ultrahard layer  200  has a longitudinal axis  201  coinciding with the z-axis extending there through. The non-planar top surface  210  formed on the ultrahard layer  200  has a geometry formed by varying heights  250  along the x-axis and y-axis, wherein the height  250  is measured along the z-axis from a common base plane  205  through a bottom surface  207  of the ultrahard layer  200 . As shown in  FIG.  5   , which is a side view in an x-z coordinate plane of the ultrahard layer  200 , the peak  222  of the ridge  220  has the greatest heights  252  formed in the top surface  210 . As shown in  FIGS.  6  and  7   , which show a top view in an x-y coordinate plane and a side view in an y-z coordinate plane, respectively, the ridge  220  extends a length  280  across the diameter of the ultrahard layer  200  along the y-axis between opposite sides of the edge  214  of the top surface  210 . From the sake of convenience, the y-axis is consistently defined based on the extension direction of the ridge  220 ; however, one skilled in the art would appreciate that if defined differently, the remaining description based on the x-, y-, z-coordinate system would similarly vary. 
     At least two sidewalls  230 ,  232  slope in opposite directions from the peak  222  of the ridge  220  at a roof angle  202   a,    202   b  (collectively referred to as  202 ). First sidewalls  232  extend a length along the y-axis from proximate an edge  214  of the ultrahard layer  200  and slope outwardly from the peak  222  in opposite directions along the x-axis. Second sidewalls  230  may be adjacent to the first sidewalls  232 , extending a length along the y-axis from the first sidewalls  232  and sloping outwardly from the peak  222  and from a transition  234  to the first sidewalls  232  in opposite directions along the x-axis. As shown in  FIG.  5   , the slope of the sidewalls  230 ,  232  may be measured along a line  231 ,  233  tangent to the sidewalls  230 ,  232 . In the embodiment shown, the sidewalls  230 ,  232  are substantially planar faces sloping from the peak  222  of the ridge  220  in a direction toward the edge  214  of the top surface  210 . The transition between the sidewalls  230 ,  232  and the peak  222  may be radiused or angled. 
     The roof angle  202  may vary along the length of the ridge  220 . For example, different portions along the ridge  220  may have different roof angles  202   a,    202   b.  The roof angle  202  may gradually transition (e.g., through radiused transitions) between different roof angles  202  by differently sloping sidewalls, for example, undulating sidewalls, sloping from the peak  222  of the ridge  220  at different slopes  231 ,  233 . In the embodiment shown, an edge portion  221  of the ridge  220  proximate the edge  214  may have first sidewalls  232  extending from the peak  222  of the ridge  220  at a first roof angle  202   a,  and a central portion  223  of the ridge  220  around the longitudinal axis  201  may have second sidewalls  230  extending from the peak  222  of the ridge  220  at a second roof angle  202   b,  where the first roof angle  202   a  is smaller than the second roof angle  202   b.    
     The first sidewalls  232  sloping from the ridge  220  at the first roof angle  202   a  may be recessed from the second sidewalls  230  sloping from the ridge  220  at the second roof angle  202   b,  where the first sidewalls  232  may have a lesser height  250  than the second sidewalls  230  along the y-dimension at a shared x-position. The first sidewalls  232  may transition to the second sidewalls  230  through a gradual transition proximate the peak  222 , where the first sidewalls  232  have a relatively steeper slope  233  proximate the edge portion  221  of the ridge  220  that gets shallower in the direction from the edge portion  221  toward the central portion  223  until the first sidewalls  232  transitions to the second sidewalls  232 . The first sidewalls  232  may also transition to the second sidewalls  230  via one or more transition surfaces, such as landing  234  and radiused transitions  236 ,  238  between planar portions of the sidewalls  230 ,  232 . 
     The peak  222  may further have a varying roof radius of curvature  204   a,    204   b  (collectively referred to as  204 ) corresponding to changes in the roof angle  202 . For example, in the embodiments shown, the edge portion  221  of the ridge  220  may have a first roof radius of curvature  204   a,  where first sidewalls  232  extend at the first roof angle  202   a  from the peak  222 , and the central portion  223  of the ridge  220  may have a second roof radius of curvature  204   b,  where the second sidewalls  230  extend at the second roof angle  202   b  from the peak. Both the first roof radius of curvature  204   a  and the first roof angle  202   a  may be smaller than the second roof radius of curvature  204   b  and the second roof angle  202   b.  For example, the edge portion  221  of the ridge  220  may have a peak  222  with a first roof radius of curvature  204   a  of less than 0.1 inches, e.g., ranging between 0.02 inches and 0.09 inches, and a first roof angle  202   a  ranging between 110 degrees and 130 degrees, while the central portion  223  of the ridge  220  may have a peak  222  with a second roof radius of curvature  204   b  ranging between 0.1 and 0.2 inches and a second roof angle  202   b  ranging between 135 degrees and 165 degrees. 
     According to embodiments of the present disclosure, an edge portion  221  of a ridge  220  may have a peak  222  with a first roof radius of curvature  204   a  that is, for example, less than 80 percent (e.g., ranging from 40 to 60 percent) of a second roof radius of curvature  204   b  in a central portion  223  of the ridge  220  and a first roof angle  202   a  that is, for example, less than 80 percent (e.g., ranging from 40 to 60 percent) of a second roof angle  202   b  in the central portion  223  of the ridge  220 . 
     A ridge  220  may have a peak  222  with at least two different roof radii of curvature  204 . For example, the peak  222  of a ridge  220  may have relatively smaller roof radii of curvature  204   a  proximate the edges  214  of the top surface  210  than the roof radius of curvature  204   b  in a central portion  223  of the ridge  220 . In some embodiments, a ridge  220  may have a peak  222  with more than three different roof radii of curvature  204 . In the embodiment shown, the peak  222  has a relatively smaller first roof radius of curvature  204   a  at one side of the ridge  220  and a relatively larger second roof radius of curvature  204   b  at the opposite side of the ridge  220 . 
     Further, the peak  222  of the ridge  220  may have a continuously increasing height  250  along an edge portion  221  of the ridge  220  in a direction from the edge  214  toward the longitudinal axis  201 . For example, a ridge  220  may have a peak  222  having a curvature along the y-axis. A roof ridge angle  206  may be defined between a line  228  tangent to the peak  222  of the ridge  220  proximate the edge  214  and a plane  229  perpendicular to the longitudinal axis  201 . The roof ridge angle  206  may range from greater than zero to 10 degrees, for example, between 2 and 8 degrees. According to embodiments of the present disclosure, a length of a ridge  220 , e.g., an edge portion  221  of the ridge  220 , having a first roof radius of curvature  204   a  and first roof angle  202   a  smaller relative to other portions of the ridge  220  may have a roof ridge angle  206  greater than zero degrees. 
     In embodiments having a chamfer  240  extending around the edge  214  of the ultrahard layer  200 , an edge portion  221  of the ridge  220  may include a chamfer  240 . In such embodiments, the ridge geometry parameters of the edge portion  221  including the roof angle  202 , roof ridge angle  206 , and roof radius of curvature  204 , may describe the geometry of the ridge peak  222  and sidewalls  232  within the edge portion  221 , exclusive of the chamfer  240 . In other words, description of ridge geometry parameters of an edge portion  221  having a chamfer  240  may include the roof angle  202 , roof ridge angle  206 , and roof radius of curvature  204  of the peak  222  and sidewalls  232  extending from an interior boundary  241  of the chamfer  240  in the edge portion  221 . 
     According to embodiments of the present disclosure, the ridge  220  may include one or more concave recesses  270  formed along the peak  222  of the ridge  220 . A concave recess  270  may form a concave discontinuous region along the profile of the ridge  220  along its length, e.g., as shown in  FIG.  7   . In the embodiment shown, the ridge  220  may have one concave recess  270 . In other embodiments, a ridge  220  may have more than one concave recess  270 . In some embodiments, a ridge  220  may have no concave recesses  270 . Further, the concave recess  270  may have a tear-drop shape when viewed from a top perspective (as shown in  FIG.  6   ), where the wider part of the tear-drop is proximate the edge portion  221  and the narrower/sharper part of the tear-drop is proximate the central portion  223  of the ridge  220 . 
     According to embodiments of the present disclosure, a ridge  220  may have a peak  222  with a varying width  225   a,    225   b  (collectively referred to as  225 ) measured between opposite points of transition from the peak  222  to the sidewall  230 ,  232 . For example, the peak  222  in the edge portion  221  of the ridge  220  may have a first width  225   a,  and the peak  222  in a remaining portion of the ridge  220 , e.g., the central portion  223  of the ridge  220 , may have a second width  225   b  that is greater than the first width  225   a.    
     In the embodiment shown in  FIGS.  4 - 7   , one edge portion  221  of a ridge  220  is modified to have, e.g., a relatively smaller roof angle  202   a  than a central portion  223 , a relatively smaller roof radius of curvature  204   a  than the central portion  223 , a relatively smaller width  225   a  than the central portion  223 , and a roof ridge angle  206 , and one concave recess  270  is formed in the ridge  220  radially from the edge portion  221 . In some embodiments, both ends of a ridge  220  may be modified to have at least one of a relatively smaller roof angle  202   a  than a central portion  223 , a relatively smaller roof radius of curvature  204   a  than the central portion  223 , a relatively smaller width  225   a  than the central portion  223 , and a roof ridge angle  206 . Further, in some embodiments, more than one concave recess  270  may be formed along the ridge  220 . 
     For example,  FIG.  8    shows an embodiment of a cutting element  290  having a modified edge portion  221  on each end of the ridge  220 . Each edge portion  221  may have at least one of a relatively smaller roof angle  202   a  than a central portion  223  of the ridge  220 , a relatively smaller roof radius of curvature  204   a  than the central portion  223  of the ridge  220 , a relatively smaller width  225   a  than the central portion  223  of the ridge  220 , and a roof ridge angle  206 . Further, two concave recesses  270  are formed along the ridge  220 , each concave recess  270  located radially between an edge portion  221  and the central portion  223 . The ridge geometry may be symmetrical about a line  285  extending along a major dimension of the top surface  210  and through the longitudinal axis  201  of the cutting element  290 . By providing symmetrical edge portions  221  of a ridge  220 , the cutting element  290  may be used in two cutting positions. For example, the cutting element  290  may be positioned in a cutting tool in a first orientation where a first edge portion  221  is oriented to contact and cut a formation during operation. The cutting element  290  may further be positioned in a cutting tool in a second orientation (e.g., if the first edge portion  221  wears or fails from use) where the second edge portion  221  is oriented to contact and cut a formation during operation. 
     In some embodiments, a ridge  220  may have two different edge portion  221  geometries, which may allow for a single cutting element  200  to have two cutting efficiency options. For example, a cutting element  200  may have a first edge portion  221  extending a first length from an edge  214  of the cutting element  200  and a second edge portion  221  extending a second length from an opposite side of the edge  214 , where both the first and second edge portions  221  may have at least two of a relatively smaller roof angle  202   a  than a central portion  223  of the ridge  220 , a relatively smaller roof radius of curvature  204   a  than the central portion  223  of the ridge  220 , a relatively smaller width  225   a  than the central portion  223  of the ridge  220 , and a roof ridge angle  206 . At least one geometry parameter in the first edge portion  221  may be different than the second edge portion  221 . For example, the first length of the first edge portion  221  may be different from the second length of the second edge portion  221 , which may be selected, for example, based on different expected depths of cut. 
       FIGS.  9 - 12    show another example of an ultrahard layer  300  according to embodiments of the present disclosure.  FIG.  9    is a perspective view,  FIG.  10    is a top view, and  FIGS.  11  and  12    are side views of the ultrahard layer  300 . The ultrahard layer  300  has a top surface  310  and a bottom surface  307  opposite the top surface, where a thickness  350  of the ultrahard layer  300  is measured axially between the top surface  310  and bottom surface  307  of the ultrahard layer  300 . 
     The top surface  310  of the ultrahard layer  300  has a ridge  320  geometry, which includes a ridge  320  extending a length  380  across the major dimension (e.g., diameter) of the top surface  310 . The top surface  310  may also include a chamfer  340  formed around the edge  314  of the top surface  310 , where the chamfer  340  extends radially inward from the edge  314  to an interior boundary  341  of the chamfer  340 . The ridge  320  includes a peak  322  extending linearly between opposite sides of the edge  314  and sidewalls  330 ,  332  extending from the peak  322  toward the edge  314 . In embodiments having a chamfer  340  formed around the entire edge  314 , the peak  322  may extend to and meet with opposite sides of the interior boundary  341  of the chamfer  340 . 
     The ridge  320  may further include an edge portion  321  extending a length  381  from the edge  314  of the top surface  310  that has at least one of a roof ridge angle  306 , reduced roof angle  302 , and a reduced roof radius of curvature  304  when compared with a remaining portion  323  of the ridge  320 . In the embodiment shown in  FIGS.  9 - 12   , the edge portion  321  of the ridge may extend a length  381  that is between 25 and 45 percent of the entire length  380  of the ridge  320 . 
     The edge portion  321  of the ridge  320  may have a first roof angle  302   a  measured between oppositely sloping first sidewalls  332  from the peak  322 , and the remaining portion  323  of the ridge  320  may have a second roof angle  302   b  measured between oppositely sloping second sidewalls  330 . The first sidewalls  332  may appear to be scooped or recessed from the adjacent second sidewalls  330 , such that the first sidewalls  332  extend from the peak  322  at a steeper slope relative the longitudinal axis  301  of the ultrahard layer than the second sidewalls  330 , and thus, the first roof angle  302   a  is smaller than the second roof angle  302   b.  In embodiments having a convex and/or concave cross sectional profile of a sidewall, such as shown in  FIG.  12   , where first sidewalls  332  have a concave cross-sectional profile and second sidewalls  330  have a convex cross-sectional profile, the slope of the sidewalls  330 ,  332  may be measured along a line  331   a,    331   b  tangent to the portion of the sidewalls  330 ,  332  extending from a point  326  of transition from the peak  322  to the sidewalls  330 ,  332 . 
     The edge portion  321  of the ridge  320  may also have a first roof radius of curvature  304   a  that is smaller than a second roof radius of curvature  304   b  of the remaining portion  323  of the ridge  320 . For example, the first roof radius of curvature  304   a  may be less than 80 percent, less than 60 percent, less than 50 percent, or less than 40 percent of the second roof radius of curvature  304   b.    
     As shown in  FIG.  11   , the ridge  320  geometry may further include a first roof ridge angle  306   a  formed along the peak  322  in the edge portion  321  of the ridge  320 . The first roof ridge angle  306   a  may be formed between a plane  329  perpendicular to the longitudinal axis  301  and a line  328   a  extending along the peak  322  from a point where the peak  322  meets the interior boundary  341  of the chamfer  340  (or the edge  314  in embodiments without a chamfer  340 ) to a point  327  where the peak  322  transitions to being parallel with the plane  329 . The peak  322  in the edge portion  321  of the ridge  320  may have a concave cross-sectional profile when viewed along a profile intersecting the longitudinal axis  301  and extending through the length  312  of the ridge  320 . According to embodiments of the present disclosure, the first roof ridge angle  306   a  may be selected from a range of zero to about 10 degrees. 
     A second roof ridge angle  306   b  may be formed along the peak  322  at the edge  314  opposite the edge portion  321 . The portion of the peak  322  proximate the edge  314  and defining a second roof ridge angle  306   b  may have substantially planar cross-sectional profile when viewed along the profile intersecting the longitudinal axis  301  and extending through the length  312  of the ridge  320 . In such case, the second roof ridge angle  306   b  may be measured between a line  328   b  tangent to the peak  322  of the ridge  320  proximate the edge  314  and the plane  329  perpendicular to the longitudinal axis  301 . According to embodiments of the present disclosure, the second roof ridge angle  306   b  may be selected from a range of zero to about 10 degrees. The second roof ridge angle  306   b  may be less than, greater than, or equal to the first roof ridge angle  306   a.    
       FIGS.  13 - 18    show another example of an ultrahard layer  400  according to embodiments of the present disclosure.  FIG.  13    is a perspective view of the ultrahard layer  400 .  FIG.  14    is a side view of the ultrahard layer  400 .  FIG.  15    is a schematic of a top view of the ultrahard layer  400  (looking at the top surface  410  of the ultrahard layer  400 ).  FIGS.  16 - 18    are cross-sectional views of the ultrahard layer  400  taken along cross-sections A-A, B-B, and C-C, respectively, shown in  FIG.  15   . 
     The ultrahard layer  400  has a non-planar top surface  410  with ridge  420  geometry and a non-planar bottom surface  407  opposite the top surface  410 . The ridge  420  geometry includes a ridge  420  extending linearly along a major dimension  480  of the ultrahard layer  400  between opposite sides of an edge  414  of the ultrahard layer  400 . The ultrahard layer  400  may have a cylindrical side surface  403 , where the major dimension  480  of the ultrahard layer  400  is a diameter  480  of the cylindrical side surface  403 . In other embodiments, the side surface(s)  403  of an ultrahard layer may define non-circular cross-sectional shapes along a cross-section perpendicular to the longitudinal axis  401 , such as oblong, elliptical, or polygonal cross-sectional shapes. The non-planar bottom surface  407  of the ultrahard layer  400  may be attached to (or formed to) an upper surface of a substrate having a geometry corresponding to the bottom surface  407  geometry, forming a non-planar interface between the ultrahard layer  400  and substrate. 
     In the embodiment shown, the geometry of the bottom surface  407  includes one or more protrusions  408  formed at circumferential positions around the perimeter of the bottom surface  407 , for example, at opposite sides of a diameter  480  of the ultrahard layer  400 . In some embodiments, one or more protrusions  408  may be formed at a circumferential position around the ultrahard layer  400  that corresponds with an edge portion  421  of a ridge  420  formed on the top surface  410 . For example, as shown in  FIGS.  13  and  14   , protrusions  408  may be formed on the bottom surface  407  at circumferential positions opposite edge portions  421  of the ridge  420 . The ultrahard layer  400  may be attached or formed to a substrate having an upper surface with a corresponding geometry to the bottom surface  407  of the ultrahard layer  400 , e.g., one or more recessed portions having a corresponding shape to one or more protrusions  408  formed on the bottom surface  407  of the ultrahard layer  400 . 
     A thickness  450  of the ultrahard layer  400  is measured axially between the top surface  410  and bottom surface  407  of the ultrahard layer  400 . According to embodiments of the present disclosure, the ultrahard layer  400  may have a combination top surface  410  geometry and bottom surface  407  geometry to provide the ultrahard layer  400  with the greatest thickness  456  at the edge portions  421  of the ridge  420  relative to the remaining areas of the ultrahard layer  400 . 
     The ridge  420  geometry of the top surface  410  includes a peak  422  of the ridge  420  and sidewalls  430  extending outwardly from the peak  422  to an interior boundary  441  of a chamfer  440  formed around the edge  414  of the top surface  410  (or in embodiments without a chamfer  440 , extending to the edge  414  of the top surface  410 ). The peak  422  may have a width  425  measured between opposite points  426  of transition from the peak  422  to a sidewall  430 . The transition from the peak  422  to a sidewall  430  may be angled or radiused. The width  425  of the peak  422  may vary along the length  480  of the ridge  420 . For example, the peak  422  in the edge portions  421  of the ridge  420  may have a first width  425   a,  and the portion of the peak  422  extending between the opposite edge portions  421  (e.g., including a central portion  423  around the longitudinal axis  401  of the ultrahard layer) may have a second width  425   b  greater than the first width  425   a.    
     According to embodiments of the present disclosure, the first width  425   a  of a peak  422  in an edge portion  421  of a ridge  420  may be, for example, between 20 percent to 80 percent less than the second width  425   b  of the peak  422  in the central portion  423  of the ridge  420 . For example, in the embodiment shown, the peak  422  in the edge portions  421  of the ridge  420  may have a first width  425   a  ranging between 20 percent to 50 percent less than the second width  425   b  of the portion of the peak  422  extending between the edge portions  421 . The width values may vary depending on the overall size of the ultrahard layer  400  and the other dimensions of the ridge geometry, such as the ridge height  460 , roof angle  402 , roof radius of curvature  404 , and roof ridge angle (e.g.,  206 ). In some embodiments, a first width  425   a  of the peak  422  in the edge portions  421  may range, for example, between 0.02 inches to 0.05 inches, or between 0.03 inches to 0.06 inches, and the second width  425   b  of the portion of the peak  422  extending between the edge portions  421  may range, for example, between 0.04 inches to 0.08 inches, or between 0.05 inches to 0.1 inch. 
     The roof radius of curvature  404  is a measurement of the curvature of the peak  422  and may vary along the length  480  of the ridge  420 . For example, in the embodiment shown in  FIGS.  13 - 18   , the peak  422  may have a first roof radius of curvature  404   a  in the edge portions  421  of the ridge  420  and a second roof radius of curvature  404   b  in the central portion  423  of the ridge  420 , where the second roof radius of curvature  404   b  is greater than the first roof radius of curvature  404   a.  According to embodiments of the present disclosure, the roof radius of curvature  404  of a peak  422  in edge portion(s)  421  of a ridge  420  may be smaller than the roof radius of curvature  404  of the peak  422  in portions of the ridge  420  interior to and adjacent to the edge portion(s)  421 . For example, the first roof radius of curvature  404   a  of the peak  422  in the edge portions  421  of a ridge  420  may range from about 20 percent to 60 percent less than the roof radius of curvature  404  of a portion of the ridge  420  adjacent to and interior to the edge portions  421 . 
     In some embodiments, the first roof radius of curvature  404   a  of the peak  422  in an edge portion  421  of a ridge  420  may vary along the length of the edge portion  421  and/or the roof radius of curvature  404  may vary along the remaining portion of the ridge  420 , where the greatest value of the first roof radius of curvature  404   a  may be less than the roof radius or radii of curvature  404  along the remaining portion of the ridge  420 . For example, the first roof radius of curvature  404   a  of the peak  422  in the edge portions  421  of a ridge  420  may be less than 0.1 inches, e.g., ranging from a lower limit of 0.02 inches, 0.04 inches, or 0.06 inches to an upper limit of 0.05 inches, 0.08 inches, or 0.09 inches, and a portion of the ridge  420  adjacent to and interior to the edge portions  421  may have a roof radius of curvature that is 0.1 inches or greater, e.g., ranging from a lower limit of 0.1 inches, 0.14 inches, or 0.15 inches to an upper limit of 0.15 inches, 0.2 inches, or 0.25 inches. 
     In some embodiments, at least a portion of the ridge  420  extending between the edge portions  421  may have a peak  422  with a planar surface, in which case the radius of curvature  404  of the planar surface portion of the peak  422  would be infinity. 
     The ridge  420  may further have a roof angle  402  measured between oppositely sloping sidewalls  430  from the peak  422 . The slope of a sidewall  430  may be measured along a line  431  extending from an interior boundary  441  of a chamfer  440  (or from the edge  414  in embodiments without a chamfer  440 ) to a point  426  of transition from the peak  422  to the sidewall  430 . In embodiments having planar sidewalls  430 , the line  431  may be tangent to the sidewall  430  surface. According to embodiments of the present disclosure, the roof angle  402  may vary along the length  480  of the ridge  420 . For example, in the embodiment shown in  FIGS.  13 - 18   , the edge portions  421  of the ridge  420  may have a first roof angle  402   a  smaller than a second roof angle  402   b  along a central portion  423  of the ridge  420 . As shown in  FIG.  17    which is a cross-sectional view of the ultrahard layer  400  taken at plane B-B from  FIG.  15    through the central portion  423  of the ridge  420 , the second roof angle  402   b  is measured between the lines  431   b  tangent to the sidewalls  430  extending laterally from the peak  422  toward the edge  414  of the ultrahard layer  400 . As shown in  FIG.  18   , which is a cross-sectional view of the ultrahard layer  400  taken at plane C-C from  FIG.  15    through an edge portion  421  of the ridge  420 , the first roof angle  402   a  is measured between the lines  431   a  tangent to the sidewalls  430  extending laterally from the peak  422  toward the edge  414  of the ultrahard layer  400 . 
     According to embodiments of the present disclosure, a first roof angle  402   a  of an edge portion  421  of a ridge  420  may be less than 145 degrees, for example, ranging between 100 degrees and 145 degrees. The sidewalls  430  on opposite sides of the peak  422  in the portion of the ridge  420  between the edge portions  421  (including central portion  423 ) may extend from the peak  422  at a second roof angle greater than 135 degrees, for example, ranging between 140 degrees and 170 degrees. The sidewalls  430  may transition from sloping at the first roof angle  402   a  from the peak  422  to sloping at the second roof angle  402   b  from the peak  422  along a radiused or curved transition  424  along the peak  422 . Further, the transition between the sidewalls sloping at the first roof angle  402   a  (represented by tangent line slope  431   a ) and the sidewalls sloping at the second roof angle  402   b  (represented by tangent line slope  431   b ) may be gradual, such that there is a continuously changing slope between the first slope  431   a  and the second slope  431   b.    
     The ridge  420  may have a ridge height  460  measured axially from a lowest portion  462  of the edge  414  to the ridge peak  422 . According to embodiments of the present disclosure, the ridge height  460  may range, for example, from a lower limit of 0.05 inches, 0.08 inches, or 0.1 inch to an upper limit of 0.07 inches, 0.1 inch, 0.15 inches, or 0.2 inches. In some embodiments, the ridge height  460  may vary along the length  480  of the ridge  420 . For example, in embodiments where the peak  422  of the ridge  420  slopes at a roof ridge angle (e.g.,  206  in  FIG.  7   ), the ridge height  460  may continuously change along the sloping portion of the ridge  420 . In embodiments having one or more concave recesses (e.g.,  270  in  FIG.  6   ), the ridge height  460  may vary between the peak  422  and the concave recess(es). In embodiments such as shown in  FIGS.  13 - 18    having a ridge  420  with a roof ridge angle of zero and no concave recesses, the peak  422  may be at a uniform ridge height  460  along the entire length  480  of the ridge  420 . 
     According to embodiments of the present disclosure, the length  481  of an edge portion  421 , as measured by a radial distance from the edge  414  of the top surface  410  toward the longitudinal axis  401 , may be designed to be greater than or equal to a predicted depth of cut when the cutting element is cutting. For example, in some embodiments, the length  481  of the edge portion  421  may range from about 0.07 inches to 0.3 inches. In embodiments having a chamfer  440  formed around the edge  414 , the peak  422  of the ridge  420  within the edge portion  421  may extend radially inward from an interior boundary  441  of the chamfer  440 . A chamfer may extend a radial distance  442  between the edge  414  of the ultrahard layer  400  to the interior boundary  441  of the chamfer  440  ranging, for example, from about 0.01 inches to about 0.03 inches. Further, a chamfer may have a slope  443  with respect to the longitudinal axis  401  ranging from, for example, about 40 degrees to about 50 degrees or 15 degrees to 70 degrees. 
     The geometry of the ridge  420  in an edge portion  421  may include a peak  422  having a reduced roof angle  402  and a reduced roof radius of curvature  404  relative to a central portion  423  of the ridge. Further, ridge  420  geometry may include opposite ends of the ridge  420  (two edge portions  421 ) having a peak width  425   a  that is less than the peak width  425   b  in a central portion  423  of the ridge  420 . Such ridge geometry may provide edge portion(s)  421  having a relatively reduced contacting area (i.e., the area of the top surface  410  and side surface  403  of the edge portion  421  that contacts a formation during operation), which may reduce the workload of the cutting element when cutting. 
     Ridge geometry may vary while still providing edge portion(s) of the ridge having at least one of a reduced roof angle, a reduced roof radius of curvature, and a reduced peak width relative to a central portion of the ridge. For example,  FIGS.  19 - 24    show additional examples of cutting elements having a ridge geometry according embodiments to the present disclosure, where the edge portion(s) of the ridge have at least one of a reduced roof angle, a reduced roof radius of curvature, and a reduced peak width relative to a central portion of the ridge. 
       FIGS.  19 - 21    show top views of cutting elements  500 ,  510 ,  520  having ridge  501 ,  511 ,  521  geometries that include edge portions  502 ,  512 ,  522  having a reduced peak width  505 ,  515 ,  525  relative to a central portion  503 ,  513 ,  523  of the ridge  501 ,  511 ,  521 . As shown in  FIG.  19   , the ridge  501  extends linearly across a major dimension of the top surface  504 , where edge portions  502  of the ridge  501  are at opposite ends of the ridge  501 . A central portion  503  of the ridge  501  extending between the two edge portions  502  has a peak width  505  that is greater than the peak width  505  along the edge portions  502 . The peak width  505  is measured between opposite points  507  of transition from the peak  506  of the ridge  501  to the sidewalls  508  extending outwardly from the peak  506  toward an edge  509  of the top surface  504 . 
     The central portion  503  of the ridge  501  may have a peak  506  with a planar surface having a polygonal shape, which is a diamond-shaped in the embodiment shown in  FIG.  19   . The planar surface portion of the peak  506  (in the central portion  503  of the ridge  501 ) may have its planar surface extending along a plane (e.g., plane  329  in  FIG.  11   ) perpendicular to the longitudinal axis (e.g.,  301  in  FIG.  11   ) of the cutting element. The transitions  507  from the planar surface of the peak  506  in the central portion  503  to the sidewalls  508  of the ridge  501  may be curved or radiused. Further, the peak  506  may be a curved surface along the edge portions  502  of the ridge  501 , where the curved surface peak  506  portions may have a roof radius of curvature (e.g.,  404   a,    404   b  in  FIG.  13   ) ranging from, for example, less than 0.1 inches. 
       FIG.  20    shows another example of a cutting element  510  with a ridge  511  geometry having a central portion  513  of the ridge  511  with a peak  516  having a polygonal shape. The ridge  511  extends linearly across a major dimension of the top surface  514 , where edge portions  512  of the ridge  511  extend inwardly from opposite sides of the edge  519  of the top surface  514  to a central portion  513  of the ridge  511 . The width  515  of the peak  516  in the central portion  513  is greater than the width  515  of the peak  516  along the edge portions  512 . The peak width  515  is measured between opposite points  517  of transition from the peak  516  of the ridge  511  to the sidewalls  518  extending outwardly from the peak  516  toward the edge  519  of the top surface  514 . 
       FIG.  21    shows an example of a cutting element  520  with a ridge  521  geometry having a central portion  523  of the ridge  521  with an oval-shaped peak  526  surface. The ridge  521  extends linearly across a major dimension of the top surface  524 , where edge portions  522  of the ridge  521  extend inwardly from opposite sides of the edge  529  of the top surface  524  to the central portion  523  of the ridge  521 . The width  525  of the peak  526  in the central portion  523  is greater than the width  525  of the peak  526  along the edge portions  522 . The oval-shaped portion of the peak  526  may have a planar surface, while the peak  526  in the edge portions  522  may have a curved surface with a radius of curvature (e.g.,  404   a,    404   b  in  FIG.  13   ) ranging from, for example, less than 0.1 inches. 
     According to some embodiments of the present disclosure, the width  525  of the peak  526  in the central portion  523  of the ridge  521  may be up to 2 times greater than the width  525  at the edge portion  522 , up to 3 times greater than the width  525  at the edge portion  522 , or more. In some embodiments, the width of a peak in the central portion of the ridge may extend greater than 20 percent of the major dimension, greater than 50 percent of the major dimension, or up to the entire major dimension. 
       FIGS.  22 - 24    show top views of cutting elements  600 ,  610 ,  620  having ridge geometry that includes a central portion  603 ,  613 ,  623  of the ridge  601 ,  611 ,  621  that extends to opposite sides of the edge  609 ,  619 ,  629  of the top surface  604 ,  614 ,  624 , across a major dimension of the top surface  604 ,  614 ,  624 . The edge portions  602 ,  612 ,  622  of the ridge  601 ,  611 ,  621  have a reduced peak width  605 ,  615 ,  625  relative to the central portion  603 ,  613 ,  623  of the ridge  601 ,  611 ,  621 . 
     Described another way, the cutting element  600 ,  610 ,  620  ridge geometry may include a geometric surface  606 ,  616 ,  626  axially extended from a plurality of recessed edge portions  607 ,  617 ,  627  formed around the edge  609 ,  619 ,  629  of the top surface  604 ,  614 ,  624 . At least one ridge  601 ,  611 ,  621  extends radially outward from the geometric surface  606 ,  616 ,  626  to the edge  609 ,  619 ,  629  of the top surface  604 ,  614 ,  624 . Sidewalls may slope downwardly from the geometric surface  606 ,  616 ,  626  and ridge  601 ,  611 ,  621  to the recessed edge portions  607 ,  617 ,  627 . 
     As shown in  FIG.  22   , the ridge geometry includes a geometric surface  606  axially extended from multiple recessed edge portions  607  formed around the edge  609  of the top surface  604 , where the geometric surface  606  has a polygonal shape. The ridges  601  may have a curved peak  608  with a roof radius of curvature, and the geometric surface  606  may be a planar surface. Further, the peak  608  of the ridges  601  and the geometric surface  606  may lie on a shared plane (e.g., plane  329  in  FIG.  11   ) perpendicular to the longitudinal axis of the cutting element  600 . In some embodiments, the peak  608  of one or more ridges  601  may slope at a roof ridge angle from the geometric surface  606  (e.g., where a line tangent to the ridge peak  608  may slope at a roof ridge angle from the plane perpendicular to the longitudinal axis, such as shown in  FIG.  11   ). 
       FIG.  23    shows another example of ridge geometry according to embodiments of the present disclosure, where a geometric surface  616  is axially extended from multiple recessed edge portions  617  formed around the edge  619  of the top surface  614 . The geometric surface  616  may have an oval shape or other elongated curved shape. Further, the geometric surface  616  may extend across an entire major dimension  618  between opposite sides of the edge  619  of the top surface  614 . 
     In some embodiments, a geometric surface may have an irregular shape, e.g., including both straight and curved boundary lines. For example,  FIG.  24    shows a cutting element  620  with a ridge geometry including a geometric surface  626  axially extended from multiple recessed edge portions  627  formed around the edge  629  of the top surface  624 , where the geometric surface  626  has an irregular shape. The geometric surface  626  may extend across an entire major dimension  628   a  between opposite sides of the edge  629  of the top surface  624 . Further, the geometric surface  626  may have a shape that is symmetrical across both a line  628   b  bisecting the length of the ridges  621  and across the major dimension  628   a  of the geometric surface  626 . 
     As shown in the embodiments shown in  FIGS.  19 - 24   , at least a portion of a ridge peak may be formed of a planar surface lying along a plane perpendicular to the longitudinal axis of the cutting element. For example, as described above, the portion of the peaks forming a geometric surface may be a planar surface, while the portions of the peaks in the edge portions may be formed of a curved surface having a radius of curvature. In some embodiments, such as described below, a ridge peak may be entirely formed of a planar surface (along the entire length of the peak). 
     For example,  FIGS.  25 - 28    show another example of a cutting element  700  according embodiments to the present disclosure having a ridge geometry formed on a top surface  710  of an ultrahard layer, where the edge portion(s)  721  of the ridge  720  have at least one of a reduced roof angle, a reduced roof radius of curvature, and a reduced peak width relative to a central portion of the ridge. The ridge geometry of the top surface  710  includes a peak  722  of the ridge  720  and sidewalls  730  extending outwardly from the peak  722  to an interior boundary  741  of a chamfer  740  formed around the edge  714  of the top surface  710  (or in embodiments without a chamfer  740 , extending to the edge  714  of the top surface  710 ). The ridge  720  extends a length  780  linearly across a major dimension of the top surface  710 , where edge portions  721  of the ridge  720  extend a length  781  inwardly from opposite sides of the edge  714  of the top surface  710  to a central portion  723  of the top surface  710 . 
     The sidewalls  730  may extend downwardly and outwardly from the peak  722  to the interior boundary  741  of the chamfer  740  at a roof angle  702 . The roof angle  702  may be measured between the lines  731  tangent to the sidewalls  730  proximate to the peak  722 . The roof angle  702  may be substantially constant along the length  780  of the peak  722 . The roof angle  702  may range, for example, between about 140 degrees to about 155 degrees. 
     The peak  722  may be formed of a planar surface extending substantially perpendicular to the longitudinal axis  701  along the length  780  of the ridge  720 . The peak  722  planar surface may form a geometric surface (e.g., as described in  FIGS.  22 - 24   ) having a geometry defined between opposite points  726  of transition from the peak  722  to a sidewall  730  and between opposite sides of the chamfer  740 . 
     A width  725  of the peak  722  may be measured between opposite points  726  of transition from the peak  722  to a sidewall  730 . The transition from the peak  722  to a sidewall  730  may be angled or radiused. The width  725  of the peak  722  may vary along the length  780  of the ridge  720 . For example, the peak  722  in the edge portions  721  of the ridge  720  may have a first width  725   a  proximate the edge  714  of the top surface  710 , and the portion of the peak  722  in a central portion of the top surface  710  around the longitudinal axis  701  may have a second width  725   b  greater than the first width  725   a.  Further, as shown in  FIG.  25   , the width  725  of the peak  722  may gradually and continuously increase from the first width  725   a  proximate the edge  714  toward the central portion of the top surface  710 . 
     According to embodiments of the present disclosure, the first width  725   a  proximate the edge  714  of the peak  722  may range, for example, between about 0.05 to about 0.15 inches. By providing a first width  725   a  of about 0.05 inches or more proximate the edge  714  of the cutting element, the peak  722  may form two cutting tips  790  that may act as pinch points to build stress concentrations on a working surface, e.g., a rock formation being drilled, and to reduce forces required for the rock fracturing. Three cutting edges  792  alternatingly formed around the cutting tips  790  may also help with rock fracturing. 
     Further, cutting elements according to embodiments of the present disclosure having a peak  722  with a first width  725   a  proximate the edge  714  of the cutting element of about 0.1 inch, a second width  725   b  greater than the first width  725   a,  and a roof angle  702  of about 140 degrees have been shown experimentally to have a lower cutter specific energy (i.e., the energy required to remove a unit volume of rock for a single cutting element) when compared with cutting elements having different cutting face geometry. For example,  FIG.  29    shows a graph of test results comparing cutting performance of a conventional planar top cutting element  771 , a cutting element  772  having a ridge with a uniform curved peak along its length, a cutting element  773  having a ridge geometry such as shown in  FIGS.  13 - 18   , and a cutting element  700  having a ridge geometry such as shown in  FIGS.  25 - 28    with a peak first width  725   a  of about 0.1 inches and a roof angle of about 140 degrees. The graph shows the measured normalized forces (cutting force and vertical force) and specific energy of the cutting elements  771 ,  772 ,  773 , and  700  as they cut a rock sample at a depth of cut (DOC) of 0.1 inches at a 20 degrees back rake angle. As shown, the cutting element  700  having a ridge geometry with a peak first width  725   a  of about 0.1 inches and a roof angle of about 140 degrees has the lowest cutting force, the lowest vertical force, and the lowest specific energy when compared with the other cutting elements  771 ,  772 , and  773  in the same rock-cutting movement. Such results indicate that the ridge geometry shown in  FIGS.  25 - 28    may use less drilling effort and provide better cutting efficiency when compared with other cutting element geometries. 
     In addition to cutting element geometry that provides improved cutting efficiency by lowering forces during rock fracturing, embodiments of the present disclosure may also include cutting element geometry that aids in rock chip removal. For example,  FIGS.  30 - 34    show a cutting element  800  having a top surface  810  ridge geometry according to embodiments of the present disclosure that includes at least one scooped feature for directing rock chips or other cutting debris away from the cutting tips of the ridge  820 . 
     The ridge geometry of the top surface  810  includes a ridge  820  extending a length  880  across an entire major dimension (e.g., diameter) of the cutting element between opposite edges  814  of the top surface  810 , where the ridge geometry varies along its length  880 . For example, edge portions  821  of the ridge  820  (e.g., portions of the ridge  820  extending a length  881  radially from the opposite edges  814  of the cutting element) may have a different geometry than the central portion  823  of the ridge  820  (the portion surrounding the longitudinal axis  801  of the cutting element). In the embodiment shown, the width  825  of the ridge  820  may be smaller in the edge portions  821  of the ridge  820  than in the central portion  823  of the top surface  810 . 
     Similar to the embodiment shown in  FIGS.  25 - 28   , the ridge peak  822  may have a planar surface lying along a plane perpendicular to the longitudinal axis  801  of the cutting element, where the planar surface peak  822  may form a raised geometric surface relative to recessed edge portions  815 . The geometric surface of the peak  822  may have a geometry defined between opposite lateral sides of the peak  822  and between opposite sides of the edge  814 . 
     The width  825  of the peak  822  may measured between opposite sides of the peak  822  planar surface. The width  825  of the peak  822  may increase from a first width  825   a  proximate the edge  814  of the cutting element to a second width  825   b  in the central portion  823  of the top surface  810 . As shown in  FIG.  34   , two cutting tips  890  may be formed at the cutting element edge  814  on opposite sides of the peak  822  at the first width  825   a,  and three cutting edges  892  may be alternatingly formed around the cutting tips  890 . The alternating cutting tips  890  and cutting edges  892  may contact and fracture rock during cutting. 
     Further, the top surface geometry may include undulating sidewalls  830  formed on opposite sides of the ridge  820 . The undulating sidewalls  830  may include scooped regions  831  positioned proximate to and on opposite sides of the peak  822  in the edge portions  821 . The scooped regions  831  may have a generally concave geometry and extend between the transition region  835   a,  cutting edges  892 , and a recessed edge portion  815  of the edge  814 . The scooped regions  831  may provide a path for the flow of rock debris around the peak  820  and away from the cutting element. The undulating sidewalls  830  may further include raised regions  832  positioned between the scooped regions  831  on opposite sides of the peak  822  and extending from the transition region  835   b  to a raised edge portion  816  of the edge  814 . In such manner, the edge  814  formed around the cutting element may undulate in height between the ridge peak  822 , the recessed edge portions  815 , and the raised edge portions  816 . 
     The ridge geometry may further include a transition region  835   a,    835   b  (collectively referred to as  835 ) providing a curved transition between the ridge peak  822  and undulating sidewalls  830  positioned on opposite sides of the peak  822 . The transition region  835  may have a varying geometry along the length  880  of the ridge  820  and corresponding with at least one of the geometry of the undulating sidewalls  830  and the varying ridge width  825 . In the embodiment shown in  FIGS.  30 - 34   , first transition regions  835   a  on opposite sides of the peak  822  in the edge portions  821  of the ridge  820  may have a smaller size than a second transition region  835   b  in the central portion  823  of the top surface  810 . For example, the first transition regions  835   a  may have a relatively tighter curvature from the peak  822  to the scooped regions  831  in the undulating sidewalls  830  compared to the second transition region  835   b  having a relatively larger curvature from the peak  822  to the raised regions  832  of the undulating sidewalls  830 . Additionally, the first transition regions  835   a  may have a relatively smaller width, as measured laterally from the peak  822 , compared to the second transition region  835   b  having a relatively larger width, as measured laterally from the peak  822 . 
     Cutting elements according to embodiments of the present disclosure may be formed, for example, by forming an ultrahard layer having ridge geometry disclosed herein using a mold with a negative of the ridge geometry. The ultrahard layer having ridge geometry according to embodiments of the preset disclosure may be formed on a substrate (e.g., placing ultrahard material such as diamond powder adjacent to a preformed substrate or substrate material in a high pressure high temperature press and sintering the material together) or may be pre-formed and attached to a substrate. 
     In some embodiments, a method of forming a cutting element with ridge geometry according to embodiments disclosed herein may include providing a cutting element having a ridge formed at a top surface of the cutting element, where the ridge may extend along a major dimension of the top surface from an edge of the top surface and has a peak with a first roof radius of curvature and sidewalls sloping away from the peak at a first roof angle. An amount of ultrahard material from the top surface around an edge portion of the ridge may then be removed to form a second peak having a second roof radius of curvature smaller than the first roof radius of curvature and recessed sidewalls sloping away from the second peak at a second roof angle smaller than the first roof angle. The edge portion having the second roof radius of curvature and the second roof angle may extend a partial length of the ridge from the edge toward a longitudinal axis of the cutting element. For example, in some embodiments, an amount of ultrahard material may be removed from the top surface using a laser to form an edge portion of a ridge having a reduced roof radius of curvature and reduced roof angle (and in some embodiments also a roof ridge angle). 
     Substrates according to embodiments of the present disclosure may be formed of cemented carbides, such as tungsten carbide, titanium carbide, chromium carbide, niobium carbide, tantalum carbide, vanadium carbide, or combinations thereof cemented with iron, nickel, cobalt, or alloys thereof. For example, a substrate may be formed of cobalt-cemented tungsten carbide. Ultrahard layers according to embodiments of the present disclosure may be formed of, for example, polycrystalline diamond, such as formed of diamond crystals bonded together by a metal catalyst such as cobalt or other Group VIII metals under sufficiently high pressure and high temperatures (sintering under HPHT conditions), thermally stable polycrystalline diamond (polycrystalline diamond having at least some or substantially all of the catalyst material removed), or cubic boron nitride. Further, it is also within the scope of the present disclosure that the ultrahard layer may be formed from one or more layers, which may have a gradient or stepped transition of diamond content therein. In such embodiments, one or more transition layers (as well as the other layer) may include metal carbide particles therein. Further, when such transition layers are used, the combined transition layers and outer layer may collectively be referred to as the ultrahard layer, as that term has been used in the present application. That is, the interface surface on which the ultrahard layer (or plurality of layers including an ultrahard material) may be formed is that of the cemented carbide substrate. 
     Cutting elements having a ridge geometry according to embodiments of the present disclosure may have improved cutting efficiency. For example, cutting efficiency may be improved due to decreased contacting area between an edge portion of a ridge and a working surface. The inventors of this application have found that cutting element workload grows with the expanding engagement with the rock formation. This engagement is a function of the contacting area as well as the depth of cutting (DOC). 
     Referring to  FIGS.  35  and  36   , a study on the performance of ridge geometry according to embodiments of the present disclosure is shown. In the study, three models of cutting elements, including a conventional cutting element  900  having a planar top surface, a first ridge cutting element  910  having a roof angle of 159 degrees, and a second ridge cutting element  920  having a roof angle of 135 degrees, were built for the geometric study and contacted to a working surface at different DOCs. The highlighted portions of the cutting elements  900 ,  910 ,  920  show the contacting area  902 ,  912 ,  922 .  FIG.  36    shows a graph of the growth of the contacting area of each sample cutting element  902 ,  912 ,  922  with the increasing DOC at a constant back-rake angle of 15 degrees. From the study, it was evident that the growth rate varied with the roof angle, where the greater the roof angle, the faster the contacting area  902 ,  912 ,  922  enlarges with the increasing DOC. 
     Further, contacting area correlates to the penetrating resistance when a cutting element cuts into a rock formation. Therefore, combining various roof angles, e.g., forming an edge portion of a ridge with a smaller roof angle compared with a central portion of the ridge, as described herein, may be used to control the contacting area of the cutting element. When a relatively larger roof angle is formed in the central portion of the ridge, the cutting element may be limited on the amount of penetration at the transition between the smaller roof angle portion (in the edge portion of the ridge) and the larger roof angle portion (in the central portion of the ridge). In such manner, the contacting area of a cutting element may be controlled (and thus reduce effects of overloading the cutting element) by designing a selected edge portion of a ridge to have a reduced roof angle relative to a larger roof angle in a central portion of the ridge. 
     Further, embodiments of the present disclosure may have an edge portion of a ridge having a reduced peak width relative to the peak width of an adjacent central portion of the ridge. By increasing the width of the ridge peak in a central portion of the ridge relative to an edge portion of the ridge, crack propagation may be reduced. For example, if a crack initiates from an edge of a ridge cutting element according to embodiments of the present disclosure, the crack may propagate until meeting an increased amount of ultrahard material at the relatively wider central portion of the ridge, at which point, the relatively wider central portion of the ridge may inhibit further crack growth. 
     While ridge cutting elements having a generally uniform ridge geometry along the entire length of the ridge may have better drilling efficiency when compared with, for example, a conventional planar cutting element, such ridge geometry may suffer from increased loads in operation, and thus experience premature failures (most commonly ultrahard material layer fracturing. By modifying an edge portion of the ridge in accordance with embodiments disclosed herein, the loading may be controlled, and thus improve the life of the cutting element. 
     In another study, cutting elements having a generally uniform ridge geometry along the entire length of the ridge with a roof angle of 175 degrees in a blunter ridge cutting element and with a roof angle of 135 degrees in a sharper ridge cutting element were compared using rock cutting tests on a vertical turret lathe.  FIG.  37    shows a representation of the ridge cutting elements  930  moving in direction  932  on a rock sample  934  in the vertical turret lathe test. Three forces acting on the ridge cutting elements  930 , including vertical force  940 , cutting force  942 , and side force  944 , were recorded during testing. From the test results, it was found that the sharper ridge cutting element with the roof angle of 135 degrees required only half of the vertical force applied on the blunter ridge cutting element with 175 degrees to reach the same depth of cutting  936 . It was also found that the sharper ridge cutting element (with 135-degree roof angle) took about 60% of the cutting force applied on the blunter ridge cutting element (with 175-degree roof angle) to drag the ridge cutting element forward. 
     The ridge cutting elements  930  were further equipped on bits with back-rake angles  950  between 12 and 20 degrees, as shown in  FIG.  38   . In addition, the ridge cutting elements  930  included a roof ridge angle of around 5 degrees, which increased the effective back-rake angle  950 . In drilling, this back-rake angle  950  resulted in compression  960  on the ahead rock  970  (i.e., the rock directly ahead of the cutting element when cutting) from the vertical force  940  and the cutting force  942 . Such compression  960  may restrict the rock  970  fracturing and removal. Thus, a lower back-rake angle  950  may reduce such resistance to rock fracturing. Ridge cutting elements according to embodiments of the present disclosure having a reduced roof angle and reduced roof radius of curvature (either with or without a roof ridge angle) were shown to have noticeably reduced compression in the ahead rock  970  during testing. In addition, the ridge cutting element having a modified edge portion tended to break the fractured rocks into smaller pieces. 
     According to embodiments of the present disclosure, an edge portion of a ridge cutting element may be modified to have a reduced roof angle (e.g., 125 degrees or less) and a reduced roof radius of curvature (e.g., less than 0.11 inches). The smaller roof radius of curvature may smooth the sharper angle from the reduced roof angle. 
     Further, one or more concave recesses (e.g., a tear-drop shaped dimple) may be introduced on the peak of the ridge for reduced compression on ahead rock and the ease of rock chip breakdown. A concave recess may be employed on the ridge peak between an edge portion and central portion of the ridge (e.g., on a portion of the ridge sloping at a roof ridge angle) to bridge the modified edge portion and the remaining portion of the ridge. 
     The cutting efficiency of a ridge cutting element having a modified edge portion with a reduced roof angle and reduced roof radius of curvature according to embodiments of the present disclosure was estimated by finite element analysis (FEA) modeling. In comparison to a ridge cutting element having a generally uniform ridge geometry, a ridge cutting element having a modified edge portion with a roof angle of 120 degrees and roof radius of curvature of less than 0.11 inches required 10 percent less cutting force. By reducing the cutting force, the bit-turning resistance may also be reduced, thereby improving bit responses to drive changes. 
     Embodiments of a shaped element have been primarily described with reference to wellbore drilling operations; the shaped elements described herein may be used in applications other than the drilling of a wellbore. In other embodiments, shaped elements according to the present disclosure may be used outside a wellbore or other downhole environment used for the exploration or production of natural resources. For instance, shaped elements of the present disclosure may be used in a borehole used for placement of utility lines. Accordingly, the terms “wellbore,” “borehole” and the like should not be interpreted to limit tools, systems, assemblies, or methods of the present disclosure to any particular industry, field, or environment. 
     One or more specific embodiments of the present disclosure are described herein. These described embodiments are examples of the presently disclosed techniques. Additionally, in an effort to provide a concise description of these embodiments, not all features of an actual embodiment may be described in the specification. It should be appreciated that in the development of any such actual implementation, as in any engineering or design project, numerous embodiment-specific decisions will be made to achieve the developers&#39; specific goals, such as compliance with system-related and business-related constraints, which may vary from one embodiment to another. Moreover, it should be appreciated that such a development effort might be complex and time consuming, but would nevertheless be a routine undertaking of design, fabrication, and manufacture for those of ordinary skill having the benefit of this disclosure. 
     Additionally, it should be understood that references to “one embodiment” or “an embodiment” of the present disclosure are not intended to be interpreted as excluding the existence of additional embodiments that also incorporate the recited features. For example, any element described in relation to an embodiment herein may be combinable with any element of any other embodiment described herein. Numbers, percentages, ratios, or other values stated herein are intended to include that value, and also other values that are “about” or “approximately” the stated value, as would be appreciated by one of ordinary skill in the art encompassed by embodiments of the present disclosure. A stated value should therefore be interpreted broadly enough to encompass values that are at least close enough to the stated value to perform a desired function or achieve a desired result. The stated values include at least the variation to be expected in a suitable manufacturing or production process, and may include values that are within 5%, within 1%, within 0.1%, or within 0.01% of a stated value. 
     A person having ordinary skill in the art should realize in view of the present disclosure that equivalent constructions do not depart from the spirit and scope of the present disclosure, and that various changes, substitutions, and alterations may be made to embodiments disclosed herein without departing from the spirit and scope of the present disclosure. Equivalent constructions, including functional “means-plus-function” clauses are intended to cover the structures described herein as performing the recited function, including both structural equivalents that operate in the same manner, and equivalent structures that provide the same function. It is the express intention of the applicant not to invoke means-plus-function or other functional claiming for any claim except for those in which the words ‘means for’ appear together with an associated function. Each addition, deletion, and modification to the embodiments that falls within the meaning and scope of the claims is to be embraced by the claims. 
     The terms “approximately,” “about,” and “substantially” as used herein represent an amount close to the stated amount that is within standard manufacturing or process tolerances, or which still performs a desired function or achieves a desired result. For example, the terms “approximately,” “about,” and “substantially” may refer to an amount that is within less than 5% of, within less than 1% of, within less than 0.1% of, and within less than 0.01% of a stated amount. Further, it should be understood that any directions or reference frames in the preceding description are merely relative directions or movements. For example, any references to “up” and “down” or “above” or “below” are merely descriptive of the relative position or movement of the related elements. 
     The present disclosure may be embodied in other specific forms without departing from its spirit or characteristics. The described embodiments are to be considered as illustrative and not restrictive. Changes that come within the meaning and range of equivalency of the claims are to be embraced within their scope.