Patent Publication Number: US-11378363-B2

Title: Contoured liner for a rectangular slotted shaped charge

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a national phase of and claims priority to Patent Cooperation Treaty (PCT) Application No. PCT/EP2019/063773 filed May 28, 2019, which claims the benefit of United States Provisional Patent Application No. 62/683,474 filed Jun. 11, 2018. The entire contents of each application listed above are incorporated herein by reference. 
    
    
     FIELD 
     The disclosure relates generally to a slotted shaped charge for use in oilfield and gas well perforating operations. More specifically, the disclosure relates to a contoured, curvilinear liner for a slotted shaped charge. 
     BACKGROUND 
     Slotted shaped charges are commercially available and used as part of, for example, perforating gun assemblies in oilfield and gas well completions. The slotted shaped charges are explosive components and are typically arranged in a helical pattern around at least one substantially cylindrical charge carrier in a perforating gun assembly. The charges may be used for various purposes, for example to generate holes in, e.g., steel casing such as piping or tubing, and cement lining a well, to generate flow paths for fluids that may be used to clean and/or seal the well, and to perforate surrounding geological formations to access oil and/or gas deposits within the formations. A slotted shaped charge is usually rectangular and is referred to as “slotted” because the perforations caused by the slotted shaped charge are rectangularly-shaped slots. As such, slotted shaped charges arranged in a helical fashion around a cylindrical charge carrier may be overlapped to provide 360° access to the structures and formations within a wellbore. 
     A slotted shaped charge typically includes a casing housing an explosive material and a liner enclosing the casing above the explosive material. The charge also includes a detonation initiator, such as a detonating cord, that is configured within the perforating gun assembly to electrically or mechanically initiate an explosion of the explosive material. The explosion collapses the liner above the explosive material and thereby releases a jet of thermal energy and liner particulate from the slotted shaped charge. Thus, the jet provides a focused ballistic energy that may be used to perforate the well casings, geological formations, and other targets in the path of the jet. The slotted shaped charge may be designed with, among other things, a particular size, explosive load, and liner for a particular application. The liner, too, may be designed from particular materials and may have a particular shape depending on the application for the slotted shaped charge. The various design considerations may affect, for example, the jet geometry, perforation geometry, depth of penetration, and other properties of a slotted shaped charge and associated ballistics. 
     Specifically, the explosion of a rectangular, slotted shaped charge produces ballistic energy that creates a detonation wave that moves toward the open end of the casing that houses the explosive charge. The wave is shaped by the opening to create a linear perforating jet upon initiation which, in turn, creates a rectangular perforation in the target surface. Thus, the jet pierces the casing and/or cement liner and forms a rectangular tunnel in the surrounding target formation. Larger perforating jets create larger perforations in the target formation and increase the potential oil and/or gas flow. The overall size of the liner in the slotted shaped charge may contribute to the size/span of the perforating jet that is formed upon detonation of the slotted shaped charge and provide for a larger perforation in the target formation. 
     In addition to providing for oil and/or gas flow in the wellbore, another objective of slotted shaped charges is to assist in abandoning wells and/or oilfields. Well abandonment typically involves complicated procedures wherein the wellbore must be shut in and permanently sealed using cement. It is essential that elements of the geo-formation such as layers of sedimentary rock, and in particular freshwater aquifers, are pressure isolated. Unwanted vertical channels or voids in a previously cemented wellbore annulus such as the space between an inner well casing and an outer well casing may produce migration pathways for fluids or gas. Thus, an objective behind perforating with a slotted shaped charge may be to produce a longitudinal slot or linear-shaped slit or hole on the target piping/tubing that is particularly useful in closing/abandonment procedures. 
     Based on the above considerations, various liners for slotted shaped charges have been developed to, among other things, increase/optimize the size of the perforating jet and perforations in wellbore casings and target formations. However, with ever-evolving economic and environmental considerations in oil and gas completions, liners that further improve the perforating performance of slotted shaped charges are needed to increase the potential oil and/or gas flow in wellbores and effectively close the wellbores for abandonment. 
     BRIEF DESCRIPTION 
     According to one aspect of the disclosure, the disclosure relates generally to a contoured, curvilinear liner for use with a slotted shaped charge. An exemplary curvilinear liner may include a first wing and a second wing, wherein each of the first wing and the second wing includes a curvilinear exterior surface that extends from a curvilinear internal central edge to a curvilinear exterior peripheral edge, each of the first wing and the second wing includes a curvilinear interior surface that extends from a curvilinear interior central edge to a curvilinear interior peripheral edge, the first wing and the second wing converge at an apex of the curvilinear exterior central edge, and the curvilinear interior surface is separated from the curvilinear exterior surface by a thickness of the wing; a face surface including a first end and a second end that extends away from and is opposite the first end, wherein the face surface spans between the exterior peripheral edge and the interior peripheral edge; and, a curvilinear bottom edge including a first end and a second end that extends away from and is opposite of the first end, wherein the curvilinear bottom edge is defined by the curvilinear interior central edge. Each of the curvilinear bottom edge, curvilinear exterior central edge, face surface, and first wing and second wing may define a contour of the curvilinear liner. For purposes of this disclosure, “curvilinear” is defined as contained by, or including, at least one curved line and/or a shape contained by or including at least one curved line. “Contour” is defined without limitation as a profile, shape, or the like. 
     In an exemplary embodiment, one or each of the first wing and the second wing may have a curvilinear contour defined by at least one of the exterior surface or the interior surface of the wing. In the same or different exemplary embodiments, one or each of the first wing and the second wing may be substantially straight. 
     According to another aspect of the exemplary disclosed embodiments, each of the first wing and the second wing may have a thickness that varies or remains substantially constant. 
     The disclosure also relates to a shaped charge including a liner according to the exemplary disclosed embodiments, a system including a perforating gun containing at least one shaped charge with a liner according to the exemplary disclosed embodiments, and a method of perforating structures and formations in a wellbore using a perforating gun containing at least one shaped charge having a liner according to the exemplary disclosed embodiments. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
       A more particular description will be rendered by reference to specific embodiments thereof that are illustrated in the appended drawings. Understanding that these drawings depict only typical embodiments thereof and are not therefore to be considered to be limiting of its scope, exemplary embodiments will be described and explained with additional specificity and detail through the use of the accompanying drawings in which: 
         FIG. 1A  shows a cross-sectional view of a non-axisymmetric shaped charge according to the prior art; 
         FIG. 1B  shows a perforating gun containing slotted shaped charges, according to the prior art; 
         FIG. 2A  shows a side perspective view of an exemplary trumpet concave contoured liner; 
         FIG. 2B  shows a side plan view of the exemplary trumpet concave contoured liner; 
         FIG. 2C  shows a cross-sectional view of the exemplary trumpet concave contoured liner along a line A-A in  FIG. 2B ; 
         FIG. 3A  shows a side perspective view of an exemplary trumpet convex contoured liner; 
         FIG. 3B  shows a side plan view of the exemplary trumpet convex contoured liner; 
         FIG. 3C  shows a cross-sectional view of the exemplary trumpet convex contoured liner along a line B-B in  FIG. 3B ; 
         FIG. 4A  shows a side perspective view of an exemplary tulip concave contoured liner; 
         FIG. 4B  shows a side plan view of the exemplary tulip concave contoured liner; 
         FIG. 4C  shows a cross-sectional view of the exemplary tulip concave contoured liner along a line C-C in  FIG. 4B ; 
         FIG. 5A  shows a side perspective view of an exemplary tulip convex contoured liner; 
         FIG. 5B  shows a side plan view of the exemplary tulip convex contoured liner; 
         FIG. 5C  shows a cross-sectional view of the exemplary tulip convex contoured liner along a line D-D in  FIG. 5B ; 
         FIG. 6A  shows a side perspective view of an exemplary V-shape concave contoured liner; 
         FIG. 6B  shows a side plan view of the exemplary V-shape concave contoured liner; 
         FIG. 6C  shows a cross-sectional view of the exemplary V-shape concave contoured liner along a line E-E in  FIG. 6B ; 
         FIG. 7A  shows a side perspective view of an exemplary V-shape convex contoured liner; 
         FIG. 7B  shows a side plan view of the exemplary V-shape contoured liner; and, 
         FIG. 7C  shows a cross-sectional view of the exemplary V-shape contoured liner along a line F-F in  FIG. 7B . 
     
    
    
     Various features, aspects, and advantages of the embodiments will become more apparent from the following detailed description, along with the accompanying figures in which like numerals represent like components throughout the figures and text. The various described features are not necessarily drawn to scale, but are drawn to emphasize specific features relevant to some embodiments. 
     The headings used herein are for organizational purposes only and are not meant to limit the scope of the description or the claims. To facilitate understanding, reference numerals have been used, where possible, to designate like elements common to the figures. 
     DETAILED DESCRIPTION 
     Reference will now be made in detail to various exemplary embodiments. Each example is provided by way of explanation, and is not meant as a limitation and does not constitute a definition of all possible embodiments. 
       FIG. 1A  illustrates a cross-section of one typical embodiment of a non-axisymmetric shaped charge  100  having a liner  130  with a linear, angular (i.e., “v-shaped”) profile, according to the prior art. In the illustrated embodiment, the non-axisymmetric shaped charge  100  is a slotted shaped charge. The non-axisymmetric shaped charge  100  is illustrated having a casing  120  and the liner  130  is housed within the casing  120 . According to an aspect, the casing  120  is a non-axisymmetric shaped casing. The casing  120  is shown including two sidewalls  123  (the boundary of a third sidewall  123  is shown by a dashed line and a fourth sidewall is not visible in the cross-sectional view of  FIG. 1A ), a back wall  124 , and an open front portion  122  opposite the back wall  124 . The casing  120  includes a hollow interior  121  bounded by the back wall  124  and sidewalls  123  within which the liner  130  is housed. The liner  130  may be arranged within the hollow interior  121  in a manner configured to close the open front portion  122  relative to the back wall  124  at any suitable position within the hollow interior  121  as applications dictate. In the embodiment shown in  FIG. 1A , the liner  130  has an apex  135  which is a substantially central portion of the liner  130  positioned below all other portions of the liner  130  in the hollow interior  121 . Opposing linear wings  136  extend away from the apex  135 . The liner  130  is made of a material selected based on the target to be penetrated, and may be made of powdered metal and/or metal alloys held together by a percentage of binder materials. The powdered metal and/or metal alloy forming the liner  130  may include at least one of copper, tin, tungsten, lead, nickel, bronze, molybdenum or combinations thereof. In some embodiments, the liner  130  may be made of a formed solid metal sheet, rather than compressed powdered metal and/or metal alloys. In another embodiment, the liner  130  may be made of a non-metal material, such as glass, cement, high-density composite or plastic. 
     With further reference to  FIG. 1A , an explosive load  140  may be disposed within the hollow interior  121  and the liner  130  may be positioned to enclose, encase or otherwise cover the explosive load  140  between the liner  130  and the back wall  124 . In other words, the explosive load  140  may be enclosed, encased or positioned between the liner  130  and the back wall  124  in such a manner that it is secured within the casing  120 . In some embodiments, the liner  130  may be pressed into and/or positioned on or over the explosive load  140 . In various embodiments, the liner  130  may extend, e.g., via the wings  136 , to the open front portion  122  or any portion of the sidewalls  123  suitable for a particular application of a slotted shaped charge. 
     Continuing with reference to the typical embodiment of a slotted shaped charge as shown in  FIG. 1A , a detonating cord  160  is received by the casing  120  via an aperture or gap  150  in the back wall  124 . The detonating cord  160  in the embodiment of  FIG. 1A  contacts or otherwise abuts or is positioned in a manner to initiate detonation of the explosive load  140  upon firing. The type, configuration, and function of the detonating cord  160  may be according to any known detonating cord techniques consistent with this disclosure. 
     With reference now to  FIG. 1B , one or more shaped charges  100  may be used in a perforating gun assembly  800  for downhole perforating operations. Perforating gun  800  includes a carrier tube  810  that houses a charge carrier  820 . Shaped charges  100  may be contained by, and arranged in a helical fashion around the charge carrier  820 . In operation, the perforating gun  800  may be lowered into a desired position within a wellbore, and the shaped charges  100  initiated at the desired position. The explosive jets that are generated by the shaped charge explosions may then perforate the carrier tube  810 , well casings (not shown), cement lining (not shown), and hydrocarbon formations (not shown), for example. A detonating cord (not shown) and other internal components of the perforating gun assembly  800  may also be contained within the charge carrier  820 . 
     With reference now to  FIGS. 2A-2C , an exemplary embodiment of a contoured, curvilinear liner  200  for use with a slotted shaped charge according to the disclosure is shown. The exemplary curvilinear liner  200  is configured to be inserted in, e.g., a slotted shaped charge, in a manner such as the liner  130  is inserted in the non-axisymmetric shaped charge  100  shown in  FIG. 1A . However, the various disclosed exemplary embodiments of a curvilinear liner  200  ( FIGS. 2A-2C ),  300  ( FIGS. 3A-3C ),  400  ( FIGS. 4A-4C ),  500  ( FIGS. 5A-5C ),  600  ( FIGS. 6A-6C ),  700  ( FIGS. 7A-7C ) may include one or more curvilinear portions that define contours such as  210 ,  220 ,  230 ,  240  (as described below), that define in part an overall contoured, curvilinear shape of the exemplary curvilinear liners. 
     For example, the exemplary curvilinear liner  200  shown in  FIGS. 2A-2C  has a “trumpet concave” configuration with at least four contours  210 ,  220 ,  230 ,  240 ,  250  as explained below. The “trumpet” designation is indicative of the profile shown, for example, in  FIG. 2C , wherein an edge contour  250  of a first wing  204  and a second wing  204  arcs outward, resembling the horn of a trumpet.  FIG. 2A  shows the trumpet concave curvilinear liner  200  from a side perspective view.  FIG. 2B  is a side plan view of the trumpet concave curvilinear liner  200  and  FIG. 2C  is a cross-sectional view of the trumpet concave curvilinear liner  200  along a line A-A in  FIG. 2B . As shown in those figures, the exemplary trumpet concave curvilinear liner  200  includes, among other things, a curvilinear exterior central edge  205   a , a curvilinear interior central edge  205   b , and a curvilinear bottom edge  203  that is defined by the curvilinear interior central edge  205   b . The first wing  204  and the second wing  204  converge towards an apex  205  of the curvilinear exterior central edge  205   a . Each of the first wing  204  and the second wing  204  includes a curvilinear exterior surface  254  that extends between the curvilinear exterior central edge  205   a  and a curvilinear exterior peripheral edge  206 , and a curvilinear interior surface  264  that extends between the curvilinear interior central edge  205   b  and a curvilinear interior peripheral edge  208 . For purposes of this disclosure, the direction generally from the exterior/interior central edge  205   a / 205   b  toward the exterior/interior peripheral edge  206 / 208  is the “upward” direction. The “downward” direction is opposite the upward direction. Further, for purposes of this disclosure, a first point that is nearer to the exterior/interior peripheral edge  206 / 208  is “above” a second point that is nearer to the exterior/interior central edge  205   a / 205   b , and the second point is “below” the first point. 
     The trumpet concave curvilinear liner  200  and other exemplary disclosed embodiments of a curvilinear liner ( 300 ,  400 ,  500 ,  600 ,  700 ) may be formed, without limitation, from the materials and/or techniques discussed with respect to the liner  130  that is shown in  FIG. 1A . In the exemplary embodiment shown in  FIGS. 2A and 2C , the first wing  204  and the second wing  204  have a varying thickness t 1 , t 2 . In the same or alternative embodiments, one or each of the first wing  204  and the second wing  204  may have a constant thickness. In certain exemplary embodiments, a maximum thickness of the first wing  204  and the second wing  204  may be from approximately 1 millimeter (mm) to approximately 8 mm. In other embodiments, the maximum thickness may be any value required for a particular use and consistent with this disclosure. Further, in the exemplary embodiment shown in  FIGS. 2A-2C , the trumpet concave curvilinear liner  200  is symmetrical about at least the curvilinear exterior central edge  205   a  and line A-A in  FIG. 2B . In other embodiments, a contoured liner may be symmetrical or asymmetrical about any boundary. 
     The general aspects of a contoured liner that are discussed above with respect to the trumpet concave curvilinear liner  200  are applicable to other exemplary disclosed embodiments  300 ,  400 ,  500 ,  600 ,  700  and further embodiments consistent with this disclosure, except where otherwise indicated, and will not be repeated. 
     Within continuing reference to  FIGS. 2A-2C , the trumpet concave curvilinear liner  200  further includes a face surface  207  that spans between the exterior peripheral edge  206  and interior peripheral edge  208  of each of the first wing  204  and the second wing  204 . Each of the curvilinear exterior central edge  205   a , the curvilinear bottom edge  203 , the face surface  207  (vis-à-vis the exterior peripheral edge  206  and/or the interior peripheral edge  208 ), and the curvilinear exterior surface  254  and/or the curvilinear interior surface  264  of each of the first wing  204  and the second wing  204  may define one or more contours  210 ,  220 ,  230 ,  240  of the exemplary trumpet concave curvilinear liner  200 . For example, the curvilinear bottom edge  203  defines a bottom edge contour  210  of the trumpet concave curvilinear liner  200 . The bottom edge contour  210  is substantially arc-shaped, is bounded by a first end  203   a  of the curvilinear bottom edge  203  and a second end  203   b  of the curvilinear bottom edge  203 , and includes an apex  203   c . The bottom edge contour  210  is concave with respect to a line or plane ‘i’ that includes the first end  203   a  of the curvilinear bottom edge  203  and the second end  203   b  of the curvilinear bottom edge  203 —i.e., the apex  203   c  of the bottom edge contour  210  is above the line i that includes the first end  203   a  and the second end  203   b  of the curvilinear bottom edge  203 . Thus, the arc represented by the bottom edge contour  210  extends in an upward direction from each boundary at the first end  203   a  and the second end  203   b  of the curvilinear bottom edge  203  to the apex  203   c.    
     With continuing reference to the trumpet concave curvilinear liner  200  shown in  FIGS. 2A-2C , the curvilinear exterior central edge  205   a  defines a central edge contour  220 . The central edge contour  220  is substantially arc-shaped and may be defined by, or span between, the convergence of the curvilinear exterior surface  254  of the first wing  204  and the curvilinear exterior surface  254  of the second wing  204  at the apex  205  of the curvilinear exterior central edge  205   a.    
     The curvilinear exterior surface  254  and/or the curvilinear interior surface  264  of each of the first wing  204  and the second wing  204  may also define a wing contour  230 . For example, in the exemplary trumpet concave curvilinear liner  200  shown in  FIGS. 2A-2C , the curvilinear exterior surface  254  and the curvilinear interior surface  264  are concave with respect to a corresponding line or plane ‘c’ that includes the apex  205  of the curvilinear exterior central edge  205   a  and respectively the exterior peripheral edge  206  and the interior peripheral edge  208 . In other words, an apex  204   c  of, e.g., the curvilinear exterior surface  254  is above the line c that includes the apex  205  of the curvilinear exterior central edge  205   a  and the exterior peripheral edge  206 . 
     With further reference to  FIGS. 2A-2C , the exterior peripheral edge  206  and the interior peripheral edge  208  define a face surface contour  240 . The face surface contour  240  may extend between the first end  207   a  of the face surface  207  and the second end  207   b  of the face surface  207 . In the exemplary trumpet concave curvilinear liner  200  shown in  FIGS. 2A-2C , the face surface contour  240  is concave with respect to a line or plane ‘e’ that includes each of the first end  207   a  of the face surface  207  and the second end  207   b  of the face surface  207 ; that is, an apex  207   c  of, e.g., the exterior peripheral edge  206  is above the line e. 
     With reference now to  FIGS. 3A-3C , an exemplary embodiment of a “trumpet convex” curvilinear liner  300  is shown.  FIG. 3A  shows the trumpet convex curvilinear liner  300  from a side perspective view.  FIG. 3B  is a side plan view of the trumpet convex curvilinear liner  300  and  FIG. 3C  is a cross-sectional view of the trumpet convex curvilinear liner  300  along a line B-B in  FIG. 3B . As shown in those figures and previously described with respect to  FIGS. 2A-2C , the exemplary trumpet convex curvilinear liner  300  includes, among other things, a curvilinear exterior central edge  305   a , a curvilinear interior central edge  305   b , a curvilinear bottom edge  303  that is defined by the curvilinear interior edge  305   b , and a first wing  304  and a second wing  304 , wherein each of the first wing  304  and the second wing  304  includes a curvilinear exterior surface  354  that extends from the curvilinear exterior central edge  305   a  to an exterior peripheral edge  306  and a curvilinear interior surface  364  that extends from the curvilinear interior central edge  305   b  to an interior peripheral edge  308 . A face surface  307  spans between the exterior peripheral edge  306  and the interior peripheral edge  308 . Further, each of the first wing  304  and the second wing  304  has a varying thickness t 1 , t 2  and the first wing  304  and the second wing  304  converge toward an apex  305  of the curvilinear exterior central edge  305   a.    
     The curvilinear bottom edge  303  of the exemplary trumpet convex curvilinear liner  300  defines a bottom edge contour  310  of the trumpet convex curvilinear liner  300 . The bottom edge contour  310  is convex with respect to the line or plane i that includes a first end  303   a  of the curvilinear bottom edge  303  and a second end  303   b  of the curvilinear bottom edge  303 —i.e., an apex  303   c  of the bottom edge contour  310  is below the line i that includes the first end  303   a  and the second end  303   b  of the curvilinear bottom edge  303 . Thus, the arc represented by the bottom edge contour  310  extends in a downward direction from each boundary at the first end  303   a  and the second end  303   b  of the curvilinear bottom edge  303  to the apex  303   c.    
     With continuing reference to the trumpet convex curvilinear liner  300  shown in  FIGS. 3A-3C , the curvilinear exterior central edge  305   a  defines a central edge contour  320 . The central edge contour  320  is substantially arc-shaped and may be defined by, or span between, the convergence of the curvilinear exterior surface  354  of the first wing  304  and the curvilinear exterior surface  354  of the second wing  304  at the apex  305  of the curvilinear exterior central edge  305   a.    
     As previously discussed with respect to the exemplary embodiment shown in  FIGS. 2A-2C , the curvilinear exterior surface  354  and the curvilinear interior surface  364  of each of the first wing  304  and the second wing  304  in the exemplary trumpet convex curvilinear liner  300  shown in  FIGS. 3A-3C  are concave with respect to the corresponding line or plane c that includes the apex  305  of the curvilinear exterior central edge  305   a  and respectively the exterior peripheral edge  306  and interior peripheral edge  308 . 
     Continuing with reference to  FIGS. 3A-3C , the exterior peripheral edge  306  and the interior peripheral edge  308  define a face surface contour  340 . The face surface contour  340  may extend between a first end  307   a  of the face surface  307  and a second end  307   b  of the face surface  307 . In the exemplary trumpet convex curvilinear liner  300  shown in  FIGS. 3A-3C , the face surface contour  340  is convex with respect to the line or plane e that includes each of the first end  307   a  of the face surface  307  and the second end  307   b  of the face surface  307 ; that is, an apex  307   c  of, e.g., the exterior peripheral edge  306  is below the line e. 
     With reference now to  FIGS. 4A-4C , an exemplary embodiment of a “tulip concave” curvilinear liner  400  is shown. The “tulip” designation is indicative of the profile shown, for example, in  FIG. 4C , wherein an edge contour  450  of a first wing  404  and a second wing  404  arcs vertically or inward, resembling the profile of a tulip flower.  FIG. 4A  shows the tulip concave curvilinear liner  400  from a side perspective view.  FIG. 4B  is a side plan view of the tulip concave curvilinear liner  400  and  FIG. 4C  is a cross-sectional view of the tulip concave curvilinear liner  400  along a line C-C in  FIG. 4B . As shown in those figures and previously described with respect to  FIGS. 2A-2C , the exemplary tulip concave curvilinear liner  400  includes, among other things, a curvilinear exterior central edge  405   a , a curvilinear interior central edge  405   b , a curvilinear bottom edge  403  that is defined by the curvilinear interior edge  405   b , and the first wing  404  and the second wing  404 , wherein each of the first wing  404  and the second wing  404  includes a curvilinear exterior surface  454  that extends from the curvilinear exterior central edge  405   a  to an exterior peripheral edge  406  and a curvilinear interior surface  464  that extends from the curvilinear interior central edge  405   b  to an interior peripheral edge  408 . A face surface  407  spans between the exterior peripheral edge  406  and the interior peripheral edge  408 . Further, each of the first wing  404  and the second wing  404  has a varying thickness t 1 , t 2  and the first wing  404  and the second wing  404  converge toward an apex  405  of the curvilinear exterior central edge  405   a.    
     The curvilinear bottom edge  403  of the exemplary tulip concave curvilinear liner  400  defines a bottom edge contour  410  of the tulip concave curvilinear liner  400 . As previously discussed with respect to the exemplary embodiment shown in  FIGS. 2A-2C , the bottom edge  403  is concave with respect to the line i that includes a first end  403   a  of the curvilinear bottom edge  403  and a second end  403   b  of the curvilinear bottom edge  403 . 
     With continuing reference to the tulip concave curvilinear liner  400  shown in  FIGS. 4A-4C , the curvilinear exterior central edge  405   a  defines a central edge contour  420 . The central edge contour  420  is substantially arc-shaped and may be defined by, or span between, the convergence of the curvilinear exterior surface  454  of the first wing  404  and the curvilinear exterior surface  454  of the second wing  404  at the apex  405  of the curvilinear exterior central edge  405   a.    
     In addition, the curvilinear exterior surface  454  and/or the curvilinear interior surface  464  of each of the first wing  404  and the second wing  404  may define a wing contour  430  of the tulip concave curvilinear liner  400  shown in  FIGS. 4A-4C . For example, the curvilinear exterior surface  454  and the curvilinear interior surface  464  are convex with respect to the corresponding line or plane c that includes the apex  405  of the curvilinear exterior central edge  405   a  and respectively the exterior peripheral edge  406  and the interior peripheral edge  408 . In other words, an apex  404   c  of, e.g., the curvilinear interior surface  464  is below a line c that includes the apex  405  of the curvilinear exterior central edge  405   a  and the interior peripheral edge  408 . 
     Continuing with reference to  FIGS. 4A-4C , the exterior peripheral edge  406  and the interior peripheral edge  408  define a face surface contour  440 . The face surface contour  440  may extend between a first end  407   a  of the face surface  407  and a second end  407   b  of the face surface  407 . As previously discussed with respect to the exemplary embodiment shown in  FIGS. 2A-2C , the exterior peripheral edge  406  and the interior peripheral edge  408  are concave with respect to the line e that includes each of the first end  407   a  of the face surface  407  and the second end  407   b  of the face surface  407 . 
     With reference now to  FIGS. 5A-5C , an exemplary embodiment of a “tulip convex” curvilinear liner  500  is shown.  FIG. 5A  shows the tulip convex curvilinear liner  500  from a side perspective view.  FIG. 5B  is a side plan view of the tulip convex curvilinear liner  500  and  FIG. 5C  is a cross-sectional view of the tulip convex curvilinear liner  500  along a line D-D in  FIG. 5B . As shown in those figures and previously described with respect to  FIGS. 2A-2C , the exemplary tulip convex curvilinear liner  500  includes, among other things, a curvilinear exterior central edge  505   a , a curvilinear interior central edge  505   b , a curvilinear bottom edge  503  that is defined by the curvilinear interior edge  505   b , and a first wing  504  and a second wing  504 , wherein each of the first wing  504  and the second wing  504  includes a curvilinear exterior surface  554  that extends from the curvilinear exterior central edge  505   a  to an exterior peripheral edge  506  and a curvilinear interior surface  564  that extends from the curvilinear interior central edge  505   b  to an interior peripheral edge  508 . A face surface  507  spans between the exterior peripheral edge  506  and the interior peripheral edge  508 . Further, each of the first wing  504  and the second wing  504  has a varying thickness t 1 , t 2  and the first wing  504  and the second wing  504  converge toward an apex  505  of the curvilinear exterior central edge  505   a.    
     The curvilinear bottom edge  503  of the exemplary tulip convex curvilinear liner  500  defines a bottom edge contour  510  of the tulip convex curvilinear liner  500 . As previously discussed with respect to the exemplary embodiment shown in  FIGS. 3A-3C , the bottom edge  503  is convex with respect to the line i that includes a first end  503   a  of the curvilinear bottom edge  503  and a second end  503   b  of the curvilinear bottom edge  503 . 
     With continuing reference to the tulip convex curvilinear liner  500  shown in  FIGS. 5A-5C , the curvilinear exterior central edge  505   a  defines a central edge contour  520 . The central edge contour  520  is substantially arc-shaped and may be defined by, or span between, the convergence of the curvilinear exterior surface  554  of the first wing  504  and the curvilinear exterior surface  554  of the second wing  504  at the apex  505  of the curvilinear exterior central edge  505   a.    
     In addition, the curvilinear exterior surface  554  and/or the curvilinear interior surface  564  of each of the first wing  504  and the second wing  504  may define a wing contour  530  of the tulip convex curvilinear liner  500  shown in  FIGS. 5A-5C . For example, and as previously discussed with respect to the exemplary embodiment shown in  FIGS. 4A-4C , the curvilinear exterior surface  554  and the curvilinear interior surface  564  are convex with respect to the corresponding line or plane c that includes the apex  505  of the curvilinear exterior central edge  505   a  and respectively the exterior peripheral edge  506  and the interior peripheral edge  508 . 
     Continuing with reference to  FIGS. 5A-5C , the exterior peripheral edge  506  and the interior peripheral edge  508  define a face surface contour  540 . The face surface contour  540  may extend between a first end  507   a  of the face surface  507  and a second end  507   b  of the face surface  507 . As previously discussed with respect to the exemplary embodiment shown in  FIGS. 3A-3C , the exterior peripheral edge  506  and the interior peripheral edge  508  are convex with respect to the line e that includes each of the first end  507   a  of the face surface  507  and the second end  507   b  of the face surface  507 . 
     With reference now to  FIGS. 6A-6C , an exemplary embodiment of a “V-shape concave” curvilinear liner  600  is shown.  FIG. 6A  shows the V-shape concave curvilinear liner  600  from a side perspective view.  FIG. 6B  is a side plan view of the V-shape concave curvilinear liner  600  and  FIG. 6C  is a cross-sectional view of the V-shape concave curvilinear liner  600  along a line E-E in  FIG. 6B . As shown in those figures and previously described with respect to  FIGS. 2A-2C , the exemplary V-shape concave curvilinear liner  600  includes, among other things, a curvilinear exterior central edge  605   a , a curvilinear interior central edge  605   b , a curvilinear bottom edge  603  that is defined by the curvilinear interior edge  605   b , and a first wing  604  and a second wing  604 , wherein each of the first wing  604  and the second wing  604  includes an exterior surface  654  that extends from the curvilinear exterior central edge  605   a  to an exterior peripheral edge  606  and an interior surface  664  that extends from the curvilinear interior central edge  605   b  to an interior peripheral edge  608 . However, as shown in  FIG. 6C , the cross-sections of each of the exterior surface  654  and the interior surface  664  of each of the first wing  604  and the second wing  604  is substantially straight, extending in one direction in the exemplary V-shape concave curvilinear liner  600 . A face surface  607  spans between the exterior peripheral edge  606  and the interior peripheral edge  608 . Further, each of the first wing  604  and the second wing  604  has a varying thickness t 1 , t 2  and the first wing  604  and the second wing  604  converge toward an apex  605  of the curvilinear exterior central edge  605   a.    
     The curvilinear bottom edge  603  of the exemplary V-shape concave curvilinear liner  600  defines a bottom edge contour  610  of the V-shape concave curvilinear liner  600 . As previously discussed with respect to the exemplary embodiments shown in  FIGS. 2A-2C and 4A-4C , the bottom edge  603  is concave with respect to the line i that includes a first end  603   a  of the curvilinear bottom edge  603  and a second end  603   b  of the curvilinear bottom edge  603 . 
     With continuing reference to the V-shape concave curvilinear liner  600  shown in  FIGS. 6A-6C , the curvilinear exterior central edge  605   a  defines a central edge contour  620 . The central edge contour  620  is substantially arc-shaped and may be defined by, or span between, the convergence of the exterior surface  654  of the first wing  604  and the exterior surface  654  of the second wing  604  at the apex  605  of the curvilinear exterior central edge  605   a . The cross-sections  FIG. 6C  of the exterior surface  654  and the interior surface  664  are neither concave nor convex with respect to a line or plane c that includes the apex  605  of the exterior central edge  605   a  and respectively the exterior peripheral edge  606  and the interior peripheral edge  608 . 
     Further, the exterior peripheral edge  606  and the interior peripheral edge  608  define a face surface contour  640 . The face surface contour  640  may extend between a first end  607   a  of the face surface  607  and a second end  607   b  of the face surface  607 . As previously discussed with respect to the exemplary embodiments shown in  FIGS. 2A-2C and 4A-4C , the exterior peripheral edge  606  and the interior peripheral edge  608  are concave with respect to the line e that includes each of the first end  607   a  of the face surface  607  and the second end  607   b  of the face surface  607 . 
     With reference now to  FIGS. 7A-7C , an exemplary embodiment of a “V-shape convex” curvilinear liner  700  is shown.  FIG. 7A  shows the V-shape convex curvilinear liner  700  from a side perspective view.  FIG. 7B  is a side plan view of the V-shape convex curvilinear liner  700  and  FIG. 7C  is a cross-sectional view of the V-shape convex curvilinear liner  700  along a line F-F in  FIG. 7B . As shown in those figures and previously described with respect to  FIGS. 2A-2C , the exemplary V-shape convex curvilinear liner  700  includes, among other things, a curvilinear exterior central edge  705   a , a curvilinear interior central edge  705   b , a curvilinear bottom edge  703  that is defined by the curvilinear interior edge  705   b , and a first wing  704  and a second wing  704 , wherein each of the first wing  704  and the second wing  704  includes an exterior surface  754  that extends from the curvilinear exterior central edge  705   a  to an exterior peripheral edge  706  and an interior surface  764  that extends from the curvilinear interior central edge  705   b  to an interior peripheral edge  708 . However, as shown in  FIG. 7C , the cross-sections of the exterior surface  754  and the interior surface  764  of each of the first wing  704  and the second wing  704  is substantially straight, extending in one direction in the exemplary V-shape convex curvilinear liner  700 . A face surface  707  spans between the exterior peripheral edge  706  and the interior peripheral edge  708 . Further, each of the first wing  704  and the second wing  704  has a varying thickness t 1 , t 2  and the first wing  704  and the second wing  704  converge toward an apex  705  of the curvilinear exterior central edge  705   a.    
     The curvilinear bottom edge  703  of the exemplary V-shape convex curvilinear liner  700  defines a bottom edge contour  710  of the V-shape convex curvilinear liner  700 . As previously discussed with respect to the exemplary embodiments shown in  FIGS. 3A-3C and 5A-5C , the bottom edge  703  is convex with respect to the line i that includes a first end  703   a  of the curvilinear bottom edge  703  and a second end  703   b  of the curvilinear bottom edge  703 . 
     With continuing reference to the V-shape convex curvilinear liner  700  shown in  FIGS. 7A-7C , the curvilinear exterior central edge  705   a  defines a central edge contour  720 . The central edge contour  720  is substantially arc-shaped and may be defined by, or span between, the convergence of the exterior surface  754  of the first wing  704  and the exterior surface  754  of the second wing  704  at the apex  705  of the curvilinear exterior central edge  705   a . The cross-sections  FIG. 7C  of the exterior surface  754  and the interior surface  764  are neither concave nor convex with respect to a line or plane c that includes the apex  705  of the exterior central edge  705   a  and respectively the exterior peripheral edge  706  and the interior peripheral edge  708 . 
     Further, the exterior peripheral edge  706  and the interior peripheral edge  708  define a face surface contour  740 . The face surface contour  740  may extend between a first end  707   a  of the face surface  707  and a second end  707   b  of the face surface  707 . As previously discussed with respect to the exemplary embodiments shown in  FIGS. 3A-3C and 5A-5C , the exterior peripheral edge  706  and the interior peripheral edge  708  are convex with respect to the line e that includes each of the first end  707   a  of the face surface  707  and the second end  707   b  of the face surface  707 . 
     The present disclosure, in various embodiments, configurations and aspects, includes components, methods, processes, systems and/or apparatus substantially developed as depicted and described herein, including various embodiments, sub-combinations, and subsets thereof. The present disclosure, in various embodiments, configurations and aspects, includes providing devices and processes in the absence of items not depicted and/or described herein or in various embodiments, configurations, or aspects hereof, including in the absence of such items as may have been used in previous devices or processes, e.g., for improving performance, achieving ease and/or reducing cost of implementation. 
     The phrases “at least one”, “one or more”, and “and/or” are open-ended expressions that are both conjunctive and disjunctive in operation. For example, each of the expressions “at least one of A, B and C”, “at least one of A, B, or C”, “one or more of A, B, and C”, “one or more of A, B, or C” and “A, B, and/or C” means A alone, B alone, C alone, A and B together, A and C together, B and C together, or A, B and C together. 
     In this specification and the claims that follow, reference will be made to a number of terms that have the following meanings. The terms “a” (or “an”) and “the” refer to one or more of that entity, thereby including plural referents unless the context clearly dictates otherwise. As such, the terms “a” (or “an”), “one or more” and “at least one” can be used interchangeably herein. Furthermore, references to “one embodiment”, “some embodiments”, “an embodiment” and the like are not intended to be interpreted as excluding the existence of additional embodiments that also incorporate the recited features. Approximating language, as used herein throughout the specification and claims, may be applied to modify any quantitative representation that could permissibly vary without resulting in a change in the basic function to which it is related. Accordingly, a value modified by a term such as “about” is not to be limited to the precise value specified. In some instances, the approximating language may correspond to the precision of an instrument for measuring the value. Terms such as “first,” “second,” “upper,” “lower” etc. are used to identify one element from another, and unless otherwise specified are not meant to refer to a particular order or number of elements. 
     As used herein, the terms “may” and “may be” indicate a possibility of an occurrence within a set of circumstances; a possession of a specified property, characteristic or function; and/or qualify another verb by expressing one or more of an ability, capability, or possibility associated with the qualified verb. Accordingly, usage of “may” and “may be” indicates that a modified term is apparently appropriate, capable, or suitable for an indicated capacity, function, or usage, while taking into account that in some circumstances the modified term may sometimes not be appropriate, capable, or suitable. For example, in some circumstances an event or capacity can be expected, while in other circumstances the event or capacity cannot occur—this distinction is captured by the terms “may” and “may be.” 
     As used in the claims, the word “comprises” and its grammatical variants logically also subtend and include phrases of varying and differing extent such as for example, but not limited thereto, “consisting essentially of” and “consisting of.” Where necessary, ranges have been supplied, and those ranges are inclusive of all sub-ranges therebetween. It is to be expected that variations in these ranges will suggest themselves to a practitioner having ordinary skill in the art and, where not already dedicated to the public, the appended claims should cover those variations. 
     The terms “determine”, “calculate” and “compute,” and variations thereof, as used herein, are used interchangeably and include any type of methodology, process, mathematical operation or technique. 
     The foregoing discussion of the present disclosure has been presented for purposes of illustration and description. The foregoing is not intended to limit the present disclosure to the form or forms disclosed herein. In the foregoing Detailed Description for example, various features of the present disclosure are grouped together in one or more embodiments, configurations, or aspects for the purpose of streamlining the disclosure. The features of the embodiments, configurations, or aspects of the present disclosure may be combined in alternate embodiments, configurations, or aspects other than those discussed above. This method of disclosure is not to be interpreted as reflecting an intention that the present disclosure requires more features than are expressly recited in each claim. Rather, as the following claims reflect, the claimed features lie in less than all features of a single foregoing disclosed embodiment, configuration, or aspect. Thus, the following claims are hereby incorporated into this Detailed Description, with each claim standing on its own as a separate embodiment of the present disclosure. 
     Advances in science and technology may make alternatives and substitutions possible that are not now contemplated by reason of the imprecision of language; these variations should be covered by the appended claims.