Patent Publication Number: US-7712522-B2

Title: Expansion cone and system

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
   This application claims the benefit of the filing date of U.S. provisional patent application Ser. No. 60/746,813, filed on May 9, 2006, the disclosure of which is incorporated herein by reference. 
   This application is a continuation in part of application Ser. No. 10/571,086, filed on Mar. 6, 2006, which is a national stage PCT application number PCT/US2004/028889, filed on Sep. 7, 2004, which claims the benefit of application 60/500,435, filed on Sep. 5, 2003, the disclosures of which are incorporated herein by reference. 
   This application is related to the following co-pending applications: (1) U.S. Pat. No. 6,497,289, which was filed as U.S. patent application Ser. No. 09/454,139, filed on Dec. 3, 1999, which claims priority from provisional application 60/111,293, filed on Dec. 7, 1998, (2) U.S. patent application Ser. No. 09/510,913, filed on Feb. 23, 2000, which claims priority from provisional application 60/121,702, filed on Feb. 25, 2000, (3) U.S. patent application Ser. No. 09/502,350, filed on Feb. 10, 2000, which claims priority from provisional application 60/119,611, filed on Feb. 11, 1999, (4) U.S. Pat. No. 6,328,113, which was filed as U.S. patent application Ser. No. 09/440,338, filed on Nov. 15, 1999, which claims priority from provisional application 60/108,558, filed on Nov. 16, 1998, (5) U.S. patent application Ser. No. 10/169,434, filed on Jul. 1, 2002, which claims priority from provisional application 60/183,546, filed on Feb. 18, 2000, (6) U.S. Pat. No. 6,640,903 which was filed as U.S. patent application Ser. No. 09/523,468, filed on Mar. 10, 2000, which claims priority from provisional application 60/124,042, filed on Mar. 11, 1999, (7) U.S. Pat. No. 6,568,471, which was filed as patent application Ser. No. 09/512,895, filed on Feb. 24, 2000, which claims priority from provisional application 60/121,841, filed on Feb. 26, 1999, (8) U.S. Pat. No. 6,575,240, which was filed as patent application Ser. No. 09/511,941, filed on Feb. 24, 2000, which claims priority from provisional application 60/121,907, filed on Feb. 26, 1999, (9) U.S. Pat. No. 6,557,640, which was filed as patent application Ser. No. 09/588,946, filed on Jun. 7, 2000, which claims priority from provisional application 60/137,998, filed on Jun. 7, 1999, (10) U.S. patent application Ser. No. 09/981,916, filed on Oct. 18, 2001 as a continuation-in-part application of U.S. Pat. No. 6,328,113, which was filed as U.S. patent application Ser. No. 09/440,338, filed on Nov. 15, 1999, which claims priority from provisional application 60/108,558, filed on Nov. 16, 1998, (11) U.S. Pat. No. 6,604,763, which was filed as application Ser. No. 09/559,122, filed on Apr. 26, 2000, which claims priority from provisional application 60/131,106, filed on Apr. 26, 1999, (12) U.S. patent application Ser. No. 10/030,593, filed on Jan. 8, 2002, which claims priority from provisional application 60/146,203, filed on Jul. 29, 1999, (13) U.S. provisional patent application Ser. No. 60/143,039, filed on Jul. 9, 1999, (14) U.S. patent application Ser. No. 10/111,982, filed on Apr. 30, 2002, which claims priority from provisional patent application Ser. No. 60/162,671, filed on Nov. 1, 1999, (15) U.S. provisional patent application Ser. No. 60/154,047, filed on Sep. 16, 1999, (16) U.S. provisional patent application Ser. No. 60/438,828, filed on Jan. 9, 2003, (17) U.S. Pat. No. 6,564,875, which was filed as application Ser. No. 09/679,907, on Oct. 5, 2000, which claims priority from provisional patent application Ser. No. 60/159,082, filed on Oct. 12, 1999, (18) U.S. patent application Ser. No. 10/089,419, filed on Mar. 27, 2002, which claims priority from provisional patent application Ser. No. 60/159,039, filed on Oct. 12, 1999, (19) U.S. patent application Ser. No. 09/679,906, filed on Oct. 5, 2000, which claims priority from provisional patent application Ser. No. 60/159,033, filed on Oct. 12, 1999, (20) U.S. patent application Ser. No. 10/303,992, filed on Nov. 22, 2002, which claims priority from provisional patent application Ser. No. 60/212,359, filed on Jun. 19, 2000, (21) U.S. provisional patent application Ser. No. 60/165,228, filed on Nov. 12, 1999, (22) U.S. provisional patent application Ser. No. 60/455,051, filed on Mar. 14, 2003, (23) PCT application US02/2477, filed on Jun. 26, 2002, which claims priority from U.S. provisional patent application Ser. No. 60/303,711, filed on Jul. 6, 2001, (24) U.S. patent application Ser. No. 10/311,412, filed on Dec. 12, 2002, which claims priority from provisional patent application Ser. No. 60/221,443, filed on Jul. 28, 2000, (25) U.S. patent application Ser. No. 10/, filed on Dec. 18, 2002, which claims priority from provisional patent application Ser. No. 60/221,645, filed on Jul. 28, 2000, (26) U.S. patent application Ser. No. 10/322,947, filed on Jan. 22, 2003, which claims priority from provisional patent application Ser. No. 60/233,638, filed on Sep. 18, 2000, (27) U.S. patent application Ser. No. 10/406,648, filed on Mar. 31, 2003, which claims priority from provisional patent application Ser. No. 60/237,334, filed on Oct. 2, 2000, (28) PCT application US02/04353, filed on Feb. 14, 2002, which claims priority from U.S. provisional patent application Ser. No. 60/270,007, filed on Feb. 20, 2001, (29) U.S. patent application Ser. No. 10/465,835, filed on Jun. 13, 2003, which claims priority from provisional patent application Ser. No. 60/262,434, filed on Jan. 17, 2001, (30) U.S. patent application Ser. No. 10/465,831, filed on Jun. 13, 2003, which claims priority from U.S. provisional patent application Ser. No. 60/259,486, filed on Jan. 3, 2001, (31) U.S. provisional patent application Ser. No. 60/452,303, filed on Mar. 5, 2003, (32) U.S. Pat. No. 6,470,966, which was filed as patent application Ser. No. 09/850,093, filed on May 7, 2001, as a divisional application of U.S. Pat. No. 6,497,289, which was filed as U.S. patent application Ser. No. 09/454,139, filed on Dec. 3, 1999, which claims priority from provisional application 60/111,293, filed on Dec. 7, 1998, (33) U.S. Pat. No. 6,561,227, which was filed as patent application Ser. No. 09/852,026, filed on May 9, 2001, as a divisional application of U.S. Pat. No. 6,497,289, which was filed as U.S. patent application Ser. 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   BACKGROUND OF THE INVENTION 
   The present disclosure relates generally to wellbore casings and/or pipelines, and in particular to wellbore casings and/or pipelines that are formed using expandable tubing. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is an illustration of a conventional method for drilling a borehole in a subterranean formation. 
       FIG. 2  is an illustration of a device for coupling an expandable tubular member to an existing tubular member. 
       FIG. 3  is an illustration of a hardenable fluidic sealing material being pumped down the device of  FIG. 2 . 
       FIG. 4  is an illustration of the expansion of an expandable tubular member using the expansion device of  FIG. 2 . 
       FIG. 5  is an illustration of the completion of the radial expansion and plastic deformation of an expandable tubular member. 
       FIG. 6  is a side view of an exemplary embodiment of an expansion device of  FIG. 2 . 
       FIGS. 7 and 7   a  are cross sections of the exemplary embodiment of the expansion device of  FIG. 6 . 
       FIG. 8  is a side view of another exemplary embodiment of an expansion device of  FIG. 2 . 
       FIGS. 9 and 9   a  are cross sections of the exemplary embodiment of the expansion device of  FIG. 8 . 
       FIG. 10  is a longitudinal cross section of a seamless expandable tubular member. 
       FIG. 11  is a radial cross section of the seamless expandable tubular member of  FIG. 10 . 
       FIG. 12  is an illustration of the expansion of the seamless expandable tubular member of  FIG. 10  using the expansion device of  FIG. 6 . 
       FIGS. 13 and 13   a  are top views of the expansion of the seamless expandable tubular member as shown in  FIG. 12 . 
       FIGS. 14 and 14   a  are the top views of another embodiment of the expansion of the seamless expandable tubular member of  FIG. 10  using an expansion device. 
       FIG. 15   a  is a side view of another embodiment of an expansion device. 
       FIGS. 15   b  and  15   c  are cross sectional views of the expansion device of  FIG. 15   a.    
       FIG. 16   a  is a side view of another embodiment of an expansion device. 
       FIGS. 16   b  and  16   c  are cross sectional views of the expansion device of  FIG. 16   a.    
       FIGS. 17   a  and  17   b  are illustrations of a computer model of a tapered expansion device and an expandable tubular member. 
       FIG. 17   c  is an illustration of experimental data for the length of the tapered expansion device surface versus the taper angle of the expansion device for the computer model of  FIGS. 17   a  and  17   b.    
       FIG. 17   d  is an illustration of the true stress-strain curve for the expandable tubular member in the computer model of  FIGS. 17   a  and  17   b.    
       FIG. 18  is an illustration of the total axial expansion force versus the friction shear factor for the computer model of  FIGS. 17   a  and  17   b.    
       FIG. 19  is an illustration of the influence of the taper angle of an expansion device on the ideal work, frictional work, and redundant work, during the expansion of the expandable tubular member of the computer model of  FIGS. 17   a  and  17   b.    
       FIG. 20  is an illustration of the total axial expansion force versus the taper angle of an expansion device, during the expansion of the expandable tubular member of the computer model of  FIGS. 17   a  and  17   b.    
       FIG. 21  is an illustration of a free body diagram of various forces acting on the tapered expansion device of the computer model of  FIGS. 17   a  and  17   b.    
       FIG. 22  is an illustration of the influence of the taper angle on the radial force acting on the expansion device of the computer model of  FIGS. 17   a  and  17   b.    
       FIG. 23  is an illustration of the effective strain in the expandable tubular member versus the taper angle of an expansion device one of the computer model of  FIGS. 17   a  and  17   b.    
       FIGS. 24   a  and  24   b  are illustrations of a computer model of a polynomial curvature expansion device and expandable tubular member. 
       FIG. 25  is an illustration of experimental data for the location of an inflection point in the expansion surface of the polynomial curvature expansion device of the computer model of  FIGS. 24   a  and  24   b.    
       FIG. 26  is an illustration of polynomial curvature expansion device surface shapes with different ratios of L f /L of the computer model of  FIGS. 24   a  and  24   b.    
       FIG. 27  is an illustration of the axial expansion force required for the polynomial curvature expansion device with different L f /L ratios and a constant length of the polynomial curvature expansion surface (L) and for a shear friction factor of m=0.05 of the computer model of  FIGS. 24   a  and  24   b.    
       FIG. 28  is a comparison of the axial expansion force for the polynomial curvature expansion device for different L f /L ratios at various shear friction factors for a given length of the expansion surface of the computer model of  FIGS. 24   a  and  24   b.    
       FIG. 29  is a comparison of the axial expansion force for the polynomial curvature expansion device for different lengths of the expansion surface at various shear friction factors for the optimum L f /L ratio of 0.6 of the computer model of  FIGS. 24   a  and  24   b.    
       FIG. 30  is a comparison of the axial expansion force between the optimum tapered angle expansion device of the computer model of  FIGS. 17   a  and  17   b  and the optimum polynomial curvature expansion device of the computer model of  FIGS. 24   a  and  24   b  for a friction shear factor of m=0.10. 
       FIG. 31  is a comparison of the axial expansion force between the optimum tapered angle expansion device of the computer model of  FIGS. 17   a  and  17   b  and the optimum polynomial curvature expansion device of the computer model of  FIGS. 24   a  and  24   b  for a friction shear factor of m=0.05 
       FIG. 32  is a comparison of the steady state radial force between the optimum tapered angle expansion device of the computer model of  FIGS. 17   a  and  17   b  and the optimum polynomial curvature expansion device of the computer model of  FIGS. 24   a  and  24   b  for a friction shear factor of m=0.10. 
       FIG. 33  is a comparison of the steady state radial force between the optimum tapered angle expansion device of the computer model of  FIGS. 17   a  and  17   b  and the optimum polynomial curvature expansion device of the computer model of  FIGS. 24   a  and  24   b  for a friction shear factor of m=0.05. 
       FIG. 34  is an illustration of the total axial expansion force versus expansion device displacement for the optimum tapered expansion device of the computer model of  FIGS. 17   a  and  17   b  and a friction shear factor of m=0.10. 
       FIG. 35  is an illustration of the total axial expansion force versus expansion device displacement for the optimum polynomial expansion device of the computer model of  FIGS. 24   a  and  24   b  and a friction shear factor of m=0.10. 
       FIG. 36  is an illustration of the total axial expansion force versus expansion device displacement for the optimum tapered expansion device of the computer model of  FIGS. 17   a  and  17   b  and a friction shear factor of m=0.05. 
       FIG. 37  is an illustration of the total axial expansion force versus expansion device displacement for the optimum polynomial curvature expansion device of the computer model of  FIGS. 24   a  and  24   b  and a friction shear factor of m=0.05. 
       FIG. 38  is a comparison of the maximum effective strain between the optimum tapered angle expansion device of the computer model of  FIGS. 17   a  and  17   b  and the optimum polynomial curvature expansion device of the computer model of  FIGS. 24   a  and  24   b  for a friction shear factor of m=0.10. 
       FIG. 39  is a comparison of the maximum effective strain between the optimum tapered angle expansion device of the computer model of  FIGS. 17   a  and  17   b  and the optimum polynomial curvature expansion device of the computer model of  FIGS. 24   a  and  24   b  for a friction shear factor of m=0.05. 
   

   DETAILED DESCRIPTION 
   Referring initially to  FIG. 1 , a conventional device  100  for drilling a borehole  102  in a subterranean formation  104  is shown. The borehole  102  may be lined with a casing  106  at the top portion of its length. An annulus  108  formed between the casing  106  and the formation  104  may be filled with a sealing material  110 , such as, for example, cement. In an exemplary embodiment, the device  100  may be operated in a conventional manner to extend the length of the borehole  102  beyond the casing  106 . 
   Referring now to  FIG. 2 , a device  200  for coupling an expandable tubular member  202  to an existing tubular member, such as, for example, the existing casing  106 , is shown. The device  200  includes a shoe  206  that defines a centrally positioned valveable passage  206   a  adapted to receive, for example, a ball, plug or other similar device for closing the passage. An end of the shoe  206   b  is coupled to a lower tubular end  208   a  of a tubular launcher assembly  208  that includes the lower tubular end, an upper tubular end  208   b , and a tapered tubular transition member  208   c . The lower tubular end  208   a  of the tubular launcher assembly  208  has a greater inside diameter than the inside diameter of the upper tubular end  208   b . The tapered tubular transition member  208   c  connects the lower tubular end  208   a  and the upper tubular end  208   b . The upper tubular end  208   b  of the tubular launcher assembly  208  is coupled to an end of the expandable tubular member  202 . One or more seals  210  are coupled to the outside surface of the other end of the expandable tubular member  202 . 
   An expansion device  212  is centrally positioned within and mates with the tubular launcher assembly  208 . The expansion device  212  defines a centrally positioned fluid pathway  212   a , and includes a lower section  212   b , a middle section  212   c , and an upper section  212   d . The lower section  212   b  of the expansion device  212  includes an inclined expansion surface  212   ba  that supports the tubular launcher assembly  208  by mating with the tapered tubular transition member  208   c  of the tubular launcher assembly. The upper section  212   d  of the expansion device  212  is coupled to an end of a tubular member  218  that defines a fluid pathway  218   a . The fluid pathway  218   a  of the tubular member  218  is fluidicly coupled to the fluid pathway  212   a  defined by the expansion device  212 . One or more spaced apart cup seals  220  and  222  are coupled to the outside surface of the tubular member  218  for sealing against the interior surface of the expandable tubular member  202 . In an exemplary embodiment, cup seal  222  is positioned near a top end of the expandable tubular member  202 . A top fluid valve  224  is coupled to the tubular member  218  above the cup seal  222  and defines a fluid pathway  226  that is fluidicly coupled to the fluid pathway  218   a.    
   During operation of the device  200 , as illustrated in  FIG. 2 , the device  200  is initially lowered into the borehole  102 . In an exemplary embodiment, during the lowering of the device  200  into the borehole  102 , a fluid  228  within the borehole  102  passes upwardly through the device  200  through the valveable passage  206   a  into the fluid pathway  212   a  and  218   a  and out of the device  200  through the fluid pathway  226  defined by the top fluid valve  224 . 
   Referring now to  FIG. 3 , in an exemplary embodiment, a hardenable fluidic sealing material  300 , such as, for example, cement, is then pumped down the fluid pathway  218   a  and  212   a  and out through the valveable passage  206   a  into the borehole  102  with the top fluid valve  224  in a closed position. The hardenable fluidic sealing material  300  thereby fills an annular space  302  between the borehole  102  and the outside diameter of the expandable tubular member  202 . 
   Referring now to  FIG. 4 , a plug  402  is then injected with a fluidic material  404 . The plug thereby fits into and closes the valveable passage  206   a  to further fluidic flow. Continued injection of the fluidic material  404  then pressurizes a chamber  406  defined by the shoe  206 , the bottom of the expansion device  212 , and the walls of the launcher assembly  208  and the expandable tubular member  202 . Continued pressurization of the chamber  406  then displaces the expansion device  212  in an upward direction  408  relative to the expandable tubular member  202  thereby causing radial expansion and plastic deformation of the launcher assembly  208  and the expandable tubular member. 
   Referring now to  FIG. 5 , the radial expansion and plastic deformation of the expandable tubular member  202  is then completed and the expandable tubular member is coupled to the existing casing  106 . The hardenable fluidic sealing material  300 , such as, for example, cement fills the annulus  302  between the expandable tubular member  202  and the borehole  102 . The device  200  has been withdrawn from the borehole and a conventional device  100  for drilling the borehole  102  may then be utilized to drill out the shoe  206  and continue drilling the borehole  102 , if desired. 
   Referring now to  FIGS. 6 ,  7  and  7   a , an expansion cone  600  includes an upper cone  602 , a middle cone  604 , and a lower tubular end  606 . The upper cone  602  has a leading surface  608  and an outer inclined surface  610  that defines an angle α 1 . The middle cone  604  has an outer inclined surface  612  that defines an angle α 2 . In an exemplary embodiment, the angle α 1  is greater than the angle α 2 . The outer inclined surfaces  610  and  612  together form the expansion surfaces  614  that upon displacement of the expansion cone  600  relative to the expandable tubular member  202  radially expand and plastically deform the expandable tubular member. In an exemplary embodiment, the expansion cone  600  defines one or more outer inclined expansion faceted surfaces  616 . In an exemplary embodiment, one or more contact points  618  are formed at the intersection of the one or more outer inclined expansion faceted surfaces  616 . 
   Referring now to  FIGS. 8 ,  9  and  9   a , an exemplary embodiment of an expansion cone  800  with an outside expansion surface  802  defining a parabolic equation, is shown. The expansion cone  800  has an upper expansion section  804  and a lower tubular end  806 . The upper expansion section  804  has a leading surface  808  and the outside expansion surface  802  is defined by a parabolic equation. In an exemplary embodiment, the expansion cone  800  defines one or more outer inclined expansion faceted surfaces  810 . In an exemplary embodiment, one or more contact points  812  are formed at the intersection of the outer inclined expansion faceted surfaces  810 . 
   In an exemplary embodiment, the expansion device  212  consists of one or more of the expansion devices  600  and  800 . 
   Referring now to  FIGS. 10 and 11 , an exemplary embodiment of a seamless expandable tubular member  1000  is shown. The seamless expandable tubular member  1000  includes a wall thickness t 1  and t 2  where t 1  is not equal to t 2 . In an exemplary embodiment, the seamless expandable tubular member  1000  has a non-uniform wall thickness. 
   In an exemplary embodiment, the expandable tubular member  202  consists of one or more of the seamless expandable tubular members  1000 . 
   Referring now to  FIGS. 12 ,  13  and  13   a , in an exemplary embodiment the expansion cone  600  is displaced by a conventional expansion device, such as, for example, the expansion devices commercially available from Baker Hughes Inc., Enventure Global Technology, or Weatherford International, in an upward direction  1200  relative to the seamless expandable tubular member  1000  thereby causing radial expansion and plastic deformation of the seamless expandable tubular member. In an exemplary embodiment, stress concentrations  1300  are formed within the seamless expandable tubular member  1000  where the contact point  618  of the expansion cone  600  is displaced into the seamless expandable tubular member. 
   The use of seamless expandable tubular members, such as, for example the seamless expandable tubular member  100 , with a variable wall thickness may require higher expansion forces when the expansion device encounters areas of increased wall thickness. An expansion device may take the path of least resistance when the expansion device encounters an area of increased wall thickness t 1  and over-expand the corresponding area of thin wall thickness t 2  of the seamless expandable tubular member in comparison to the thicker wall section t 1 . The use of a faceted expansion cone, such as, for example, the expansion cone  600  creates areas of stress concentrations in the seamless expandable tubular member, which may assist in maintaining a proportional wall thickness during the radial expansion and plastic deformation process. In addition, the use of a faceted expansion cone, such as, for example, the expansion cone  600  creates areas of stress concentrations in the seamless expandable tubular member, which may result in reduced expansion and initiation forces. 
   Referring to  FIGS. 14 and 14   a , in an exemplary embodiment, an expansion cone  1400  includes a plurality of outer inclined expansion faceted surfaces  1402 , having corresponding widths (W), that intersect to form contact points  1404 . Several factors may be considered when determining the appropriate number of outer inclined expansion faceted surfaces  1402 , such as, for example, the coefficient of friction between the expansion cone and the expandable tubular member  1000 , pipe quality, and data from lubrication tests. In an exemplary embodiment, for an expandable tubular member with uniform thickness, the number of circumferential spaced apart contact points may be infinity. In an exemplary experimental embodiment, the dimensions of the final design of an expansion cone may ultimately be refined by performing an empirical study. 
   In an exemplary embodiment, the following equations may be used to make a preliminary calculation of the optimum number of outer inclined expansion faceted surfaces  1402  on an expansion cone  1400  for expanding an expandable tubular member  1000 :
 
 R =( D   1   +D   exp )/2;  (1)
 
Sin(α/2)=1−( H/R ); and  (2)
 
 N= 360°/α;  (3)
 
where,
 
D 1 =Original tubular member inside diameter;
 
D exp =Expanded tubular member inside diameter;
 
H=Gap between gap surface and tubular member inside diameter;
 
R=Radius of polygon at midpoint of expansion cone;
 
α=Angle between circumferential spaced apart contact points of polygon; and
 
N=Number of polygon flat surfaces.
 
In an exemplary embodiment, expandable tubular member  1000  has an original inside diameter of 4.77″ that is expanded to an inside diameter of 5.68″ utilizing an expansion cone  1400 . In an exemplary embodiment, there is a lubricant gap depth of 0.06″. The optimum number of outer inclined expansion faceted surfaces  1402  is determined as follows:
 
 R= ( D   1   +D   exp )/2=(4.77−5.68)/2=0.42;
 
Sin(α/2)=1−( H/R )=1−(0.06/42);
 
α/2=12.3°;
 
α=24.6°;
 
 N= 360°/α=360°/24.6°=15;
 
Accordingly, the theoretical number (N) of outer inclined expansion faceted surfaces  1402 , on an expansion cone  1400  having a tapered faceted polygonal outer expansion surface is 15, but the actual number that may result from an empirical analysis may depend on tubular member quality, coefficient of friction, and data from lubrication tests. In an exemplary embodiment, a range for the actual number (N) of outer inclined expansion faceted surfaces  1402  necessary to expand an expandable tubular member having an original inside diameter of 4.77″ to an inside diameter of 5.68″ may range from 12 to 15.
 
   Referring to  FIGS. 15   a ,  15   b  and  15   c , in an exemplary embodiment, expansion cone  1500  includes tapered faceted polygonal outer expansion surfaces  1510 , a front end  1500   a , a rear end  1500   b , recesses  1512 , internal passage  1530  for drilling fluid, internal passages  1514  for lubricating fluids, and radial passageways  1516 . The width  1520  of tapered faceted polygonal outer expansion surfaces  1510  of expansion cone  1500  may be constant for the length of the cone, resulting in trapezoidal shaped lubricant gap  1522  between each contact surface  1510 . The following equations may be used for calculating the width (W)  1520  of the contact surface:
 
 W=[ 2 R  sin(α/2)]/ K;   (4)
 
 R= ( D 1 +D 2)/4;  (5)
 
α=360 degrees/ N;   (6)
 
where:
 
W=Width of contact point;
 
D 1 =initial tubular member diameter;
 
D 2 =expanded diameter;
 
N=Number of polygon flat surfaces; and
 
K=System friction coefficient that must be determined.
 
In an exemplary embodiment, K is between 3 to 5 for an expandable tubular member having an original inside diameter of 4.77″ and an expanded inside diameter of 5.68″. N may range from 12 to 15. In an exemplary embodiment, K is 4.2.
 
   Referring now to  FIGS. 16   a ,  16   b  and  16   c , in an exemplary embodiment, expansion cone  1600  has a tapered faceted polygonal outer expansion surface  1610 , a front end  1600   a , a rear end  1600   b , recesses  1612 , internal passage  1630  for drilling fluid, internal passages  1614  for lubricating fluids, and radial passageways  1616 . The width  1620  of tapered faceted polygonal outer expansion surfaces  1610  of expansion cone  1600  may vary the length of the cone. In an exemplary embodiment, width  1620  of tapered faceted polygonal outer expansion surfaces  1610  may be larger at the front end W 1  and become smaller toward the rear end W 2 . 
   In several exemplary embodiments, the tapered faceted polygonal outer expansion surface of an expansion cone may be implemented in any expansion cone, including one or more of expansion cones  600 ,  800 ,  1404 ,  1500 , and  1600 . Furthermore, it may be implemented in any expansion device including one or more expansion surfaces. 
   The optimum taper angle θ of the tapered portion of each expansion cone, including the tapered portions in expansion cones  600 ,  800 ,  1400 ,  1500 , and  1600 , may be dependant on the amount of friction between the tapered portion of the expansion cone and the inside diameter of the tubular member. In an exemplary experimental embodiment, a cone angle of 8.5° to 12.5° was shown to be sufficient to expand an expandable tubular member having an original inside diameter of 4.77″ to an inside diameter of 5.68″. The optimum taper angle θ may be determined after testing the lubricant system to determine the exact coefficient of friction. A cone angle greater than 10° may be required to minimize the effect of thinning the tubular member wall during expansion and may potentially reduce failures related to collapsing. 
   Referring to  FIGS. 17   a  and  17   b , in an exemplary experimental embodiment  1700 , using finite element analysis (“FEA”), the radial expansion and plastic deformation of an expandable tubular member  1702  by a tapered expansion device  1704  displaced in direction  1706  relative to the expandable tubular member, was modeled using commercially available FEA software DEFORM-2D in order to predict the actual performance of a corresponding actual tapered expansion device during the radial expansion and plastic deformation of an actual expandable tubular member. The FEA optimized the taper angle θ of the tapered expansion device  1704  for minimum expansion forces. The tapered expansion device surface  1708  of the tapered expansion device  1704  has a length L. The tapered expansion device  1704  has an initial diameter D 0  and a final diameter D 1 . Since the initial diameter D 0  and the final diameter D 1  are fixed in the tapered expansion device  1704 , any increase in the taper angle θ would result in an increase in the length L of the expansion surface  1708 . 
   Referring to  FIG. 17   c , in the exemplary experimental embodiment  1700  using FEA, the length L of the expansion surface  1708  versus the taper angle θ is shown. The length L of the expansion surface  1708  increases as the taper angle θ decreases. 
   Referring to  FIG. 17   d , in the exemplary experimental embodiment  1700  using FEA, a true stress-strain curve  1710  for the expandable tubular member  1702  with a modulus of elasticity of E=30×10 6  psi and a Poisson&#39;s ratio of 0.3, is provided. In the FEA, the expansion device  1704  was modeled as rigid body while the expandable tubular member  1702  was modeled as an elastic-plastic object. 
   In an exemplar embodiment, friction conditions at the interface  1712  between the expansion device  1704  and the expandable tubular member  1702  influence metal flow and stresses acting on the expansion device. Interface friction conditions may be expressed quantitatively in terms of a factor or coefficients. The friction shear stress, f s , may be expressed using Coulomb or shear friction. If Coulomb friction is assumed, the friction shear stress takes the following form
 
f s =up  (7)
 
p being a compressive normal stress at the interface and u being the coefficient of friction. However, if shear friction is assumed, the friction shear stress takes the form of
 
                   f   s     =     mk   =       m     3       ⁢     σ   _                 (   8   )               
k being the instantaneous shear strength of the material and m being the friction shear factor, 0≦m≦1. The instantaneous shear strength can be expressed as a function of instantaneous yield strength, δ, assuming the material obeys a von Mises yield criterion.
 
   When contact pressures at the interface  1712  become large, the shear stress predicted by Coulomb friction can exceed the shear strength of the material. Therefore, shear friction should be used to model the interface friction conditions for operations that produce high contact stresses. Since there is potential for large contact stress in the radial expansion and plastic deformation of the expandable tubular member  1702  by the expansion device  1704 , the shear friction model was used in all experimental embodiments. 
   Referring to  FIG. 18 , in the exemplary experimental embodiment  1700  using FEA, a total axial expansion force curve  1800  shows axial expansion force as a function of the friction shear factor (m) for a given tapered expansion device surface  1708  angle of 10°. The total axial expansion force curve  1800  increases with increasing friction shear factor (m). In an exemplary embodiment, in cold forming of steels with lubrication, the friction shear factor (m) falls in the range 0.05≦m≦0.15. 
   In an exemplary embodiment, the actual work w a  required to cause radial expansion and plastic deformation of the expandable tubular member  1702  is comprised of three components, a) ideal work w i , b) frictional work w f  and c) redundant work w r . The actual work w a  required to cause deformation is the sum of the three components, w a =w i +w f +w r . Ideal work w i , is the work required for homogeneous deformation, which exists only when plane sections remain plane during the deformation. Frictional work w f , is consumed at the interface between the deforming metal and the tool faces that constrain the metal. Redundant work w r , is due to internal shearing and bending that causes distortion of plane sections as they pass through the deformation zone, which increases the strain in the deforming metal. 
   Referring to  FIG. 19 , in the exemplary experimental embodiment  1700  using FEA, the influence of the taper angle θ of the tapered expansion device surface  1708  on the actual work w a , ideal work w i , frictional work w f , and redundant work w r  is shown. The actual work w a  is the sum of the frictional work w f , the redundant work w r , and the ideal work w i . The ideal work w i  remains constant and does not depend on the taper angle θ of the tapered expansion device surface  1708 . However, the frictional work w f  and redundant work w r  largely depend on the taper angle θ of the tapered expansion device surface  1708 . The frictional work w f  increases with decreasing taper angle θ of the tapered expansion device surface  1708 , while the redundant work w r  increases with increasing taper angle θ of the tapered expansion device surface. The actual work w a  is minimized, thereby minimizing the required total axial expansion force, at the low point θ−1 on the actual work w a  curve. The low point θ−1 on the actual work w a  curve thereby determines the optimum taper angle θ of the tapered expansion device surface  1708 . 
   Referring to  FIG. 20 , in the exemplary experimental embodiment  1700  using FEA, total axial expansion force curves  2002 ,  2004 , and  2006  are shown as a function of taper angle θ for three different friction shear factors (m), is shown. Axial expansion force curve  2002  has a friction shear factor of m=0.10 and a minimum axial expansion force at a taper angle of 8°. Axial expansion force curve  2004  has a friction shear factor of m=0.05 and a minimum axial expansion force at a taper angle of 7°. Axial expansion force curve  2006  has a friction shear factor of m=0.0 and a minimum axial expansion force at a taper angle of 5°. 
   Referring to  FIG. 21 , in the exemplary experimental embodiment  1700  using FEA, a free-body diagram  2100  illustrates the forces acting on the tapered expansion device  1704  including the force required to deform the expandable tubular member  1702  F N , the axial force component F z , the radial force component F r , and the friction force F f . The following equations explain the forces acting on the tapered expansion device  1704 :
 
 F   r   =F   N  cos(θ)− F   f  sin(θ) and  (9)
 
 F   z   =F   N  sin(θ)+ F   f  cos(θ);  (10)
 
where
 
F N =Normal force during deformation
 
F f =Frictional Force
 
F r =Radial force acting on the tapered expansion device  1704 
 
F z =Axial force acting on the tapered expansion device  1704 
 
The axial force component F z  increases with increase in the taper angle θ of the tapered expansion device surface  1708 , while the contribution from friction force F f  to the axial force component decreases with increase in the taper angle θ of the tapered expansion device surface  1708 . This is because, with increase in taper angle θ, the cos(θ) term decreases while the sin(θ) term increase. In an exemplary embodiment, however, the initial increase in the axial force for small taper angles in the presence of friction is due to the contribution from the friction force because for smaller angles the cos(θ) is approximately one, while the sin(θ) term is negligible.
 
   Referring to  FIG. 22 , in the exemplary experimental embodiment  1700  using FEA, radial reaction force curve  2202  shows the radial reaction force F r  on the expansion device  1704  as a function of taper angle θ and friction shear factor (m). In an exemplary embodiment, the radial reaction force F r  decreases with increase in the taper angle θ, and the radial reaction force F r  was independent of the friction shear factor (m). The radial reaction force curve  2202  was approximately linear for taper angles of 15 degrees or greater, and non-linear for taper angles less than 15 degrees. 
   Referring to  FIG. 23 , in the exemplary experimental embodiment  1700  using FEA, effective strain curve  2302  in the expandable tubular member  1702  as a function of taper angle θ for three different friction shear factors (m), is shown. In an exemplary embodiment, the maximum effective strain in the expandable tubular member  1702  increased with increasing taper angle θ, and was independent of friction shear factor (m). In an exemplary embodiment, the increase in the maximum effective strain with increasing taper angle θ is due to increased redundant deformation w r  in the expandable tubular member  1702  for large taper angles. In an exemplary embodiment, taper angles of approximately 15 degrees or greater were more effective at straining the expandable tubular member  1702 . 
   Referring to  FIGS. 24   a  and  24   b , in an exemplary experimental embodiment  2400  using finite element analysis (“FEA”), the radial expansion and plastic deformation of an expandable tubular member  1702  by a polynomial curvature expansion device  2402  displaced in direction  1706  relative to the expandable tubular member, was modeled using commercially available FEA software DEFORM-2D in order to predict the actual performance of a corresponding actual polynomial curvature expansion device during the radial expansion and plastic deformation of an actual expandable tubular member. In an exemplary embodiment, the FEA optimized the shape and length L of the polynomial curvature expansion device  2402  for minimum expansion forces. Polynomial curvature expansion device surface  2404  has a length L. In an exemplary embodiment, the polynomial curvature expansion device  2402  has an initial diameter D 0  at one end and a final diameter D 1  at another end. 
   Referring to  FIG. 25 , in the exemplary experimental embodiment  2400  using FEA, the shape of a polynomial curvature expansion device surface  2502  is illustrated. The polynomial curvature expansion surface  2502  has a length L and an inflection point L f . In an exemplary embodiment, the ratio of L f /L determines the shape of the polynomial curvature expansion surface  2502 . 
   In the exemplary experimental embodiment  2400  using FEA, the polynomial curvature is expressed as:
 
 r ( z )= a   0   +a   1   z+a   2   z   2   +a   3   z   3   +a   4   z   4   (11)
 
a 0 =R 1   (12)
 
a 1 =0  (13)
 
a 2 =input  (14)
 
                   a   3     =       2   L     ⁡     [       a   2     +       2   ⁢     (       R   1     -     R   0       )         L   2         ]               (   15   )                 a   4     =       1     L   2       ⁡     [       a   2     +       2   ⁢     (       R   1     -     R   0       )         L   2         ]               (   16   )               
where
 
r(z)=radial distance from the centerline of the expansion cone; and
 
z=longitudinal distance along the polynomial curvature expansion surface
 
In an exemplary embodiment, the optimum polynomial curvature expansion surface for minimum axial expansion forces for a friction shear factor m=0.10 was r(z)=2.020−0.150z 2 −0.043z 3 +0.055z 4 . In an exemplary embodiment, the optimum polynomial curvature expansion surface for minimum axial expansion forces for a friction shear factor m=0.05 was r(z)=2.020−0.095z 2 −0.023z 3 +0.023z 4 .
 
   Referring to  FIG. 26 , in the exemplary experimental embodiment  2400  using FEA, five different polynomial curvature expansion device surfaces  2602 ,  2604 ,  2606 ,  2608 , and  2610 , are shown. Polynomial curvature expansion device surface  2602  has a L f /L=0.67. Polynomial curvature expansion device surface  2604  has a L f /L=0.60. Polynomial curvature expansion device surface  2606  has a L f /L=0.50. Polynomial curvature expansion device surface  2608  has a L f /L=0.40. Polynomial curvature expansion device surface  2610  has a L f /L=0.32. 
   Referring to  FIG. 27 , in the exemplary experimental embodiment  2400  using FEA, axial expansion force curves  2702 ,  2704 ,  2706 , and  2708  are shown for increasing ratios of L f /L for four different polynomial curvature expansion device surface lengths at a constant friction shear factor of m=0.05. In an exemplary embodiment, the axial expansion force curve  2702  has a polynomial curvature expansion device surface length of 0.75 inches and the minimum axial expansion force was found at a L f /L ratio of 0.6. In an exemplary embodiment, the axial expansion force curve  2704  has a polynomial curvature expansion device surface length of 1.1626 inches and the minimum axial expansion force was found at a L f /L ratio of 0.6. In an exemplary embodiment, the axial expansion force curve  2706  has a polynomial curvature expansion device surface length of 2.0 inches and the minimum axial expansion force was found at a L f /L ratio of 0.6. In an exemplary embodiment, the axial expansion force curve  2708  has a polynomial curvature expansion device surface length of 2.25 inches and the minimum axial expansion force was found at a L f /L ratio of 0.6. In an exemplary embodiment, the minimum axial expansion force for the four axial expansion force curves  2702 ,  2704 ,  2706 , and  2708 , was found to be at the L f /L ratio of about 0.6, thus, the ratio L f /L at which the minimum axial expansion force occurs was found to be independent of the length of the polynomial curvature expansion surface for a given shear friction factor (m). 
   Referring to  FIG. 28 , in the exemplary experimental embodiment  2400  using FEA, axial expansion force curves  2802 ,  2804 , and  2806  are shown for increasing L f /L ratios at three different friction shear factors (m) and a constant polynomial curvature expansion surface length of 1.1626 inches. Axial expansion force curve  2802  has a friction shear factor of m=0.1 and a minimum axial expansion force at a L f /L ratio of 0.6. Axial expansion force curve  2804  has a friction shear factor of m=0.05 and a minimum axial expansion force at a L f /L ratio of 0.6. Axial expansion force curve  2806  has a friction shear factor of m=0.0 and a minimum axial expansion force at a L f /L ratio of 0.6. For the three axial expansion force curves  2802 ,  2804 , and  2806 , the minimum axial expansion force was found to be at the L f /L ratio of 0.6, thus, the ratio L f /L at which the minimum axial expansion force occurs was found to be independent of the shear friction factor (m) for a given length of the polynomial curvature expansion surface. 
   Referring to  FIG. 29 , in the exemplary experimental embodiment  2400  using FEA, axial expansion force curves  2902 ,  2904 , and  2906  are shown for increasing lengths of the polynomial curvature expansion device surface  2404  with the optimum L f /L ratio of 0.6 for three different shear friction factors (m). Axial expansion force curve  2902  has a friction shear factor of m=0.1, the optimum length of the polynomial curvature expansion device surface  2404  was found to be 1.625 inches for a expansion cone that is to achieve a 0.25″ increase in diameter. Axial expansion force curve  2904  has a friction shear factor of m=0.05, the optimum length of the polynomial curvature expansion device surface  2404  was found to be 1.875 inches for a expansion cone that is to achieve a 0.25″ increase in diameter. Axial expansion force curve  2906  has a friction shear factor of m=0.0, the optimum length of the polynomial curvature expansion device surface  2404  was found to be 2.5 inches for a expansion cone that is to achieve a 0.25″ increase in diameter. 
   Referring to  FIG. 30 , in the exemplary experimental embodiments  1700  and  2400  using FEA, axial expansion force  3002  corresponding to an optimum taper angle of 8 degrees for the tapered expansion device surface  1708  is compared to the axial expansion force  3004  corresponding to an optimum polynomial curvature expansion device surface  2404  with an optimum L f /L ratio of 0.6 and a length of 1.625 inches, for a friction shear factor of m=0.10. The optimum tapered expansion device surface  1708  and the optimum polynomial curvature expansion device surface  2404  required approximately the same axial expansion force, for a friction shear factor of m=0.10. 
   Referring to  FIG. 31 , in the exemplary experimental embodiments  1700  and  2400  using FEA, axial expansion force  3102  corresponding to an optimum taper angle of 7 degrees for the tapered expansion device surface  1708  is compared to the axial expansion force  3104  corresponding to an optimum polynomial curvature expansion device surface  2404  with an optimum L f /L ratio of 0.6 and a length of 1.875 inches, for a friction shear factor of m=0.05. The optimum tapered expansion surface  1708  and the optimum polynomial curvature expansion surface  2404  required approximately the same axial expansion force, for a friction shear factor of m=0.05. 
   Referring to  FIG. 32 , in the exemplary experimental embodiments  1700  and  2400  using FEA, radial expansion force  3202  required for the optimum taper angle of 8 degrees for the tapered expansion surface  1708  is compared to the axial expansion force  3204  required for the optimum polynomial curvature expansion surface  2404  with the optimum L f /L ratio of 0.6 and a length of 1.625 inches, for a friction shear factor of m=0.10. The radial reaction force produced by the polynomial curvature expansion surface  2404  was 16.4% lower than that of the tapered expansion surface  1708 , for a friction shear factor of m=0.10. 
   Referring to  FIG. 33 , in the exemplary experimental embodiments  1700  and  2400  using FEA, radial expansion force  3302  required for the optimum taper angle of 7 degrees for the tapered expansion surface  1708  is compared to the axial expansion force  3304  required for the optimum polynomial curvature expansion surface  2404  with the optimum L f /L ratio of 0.6 and a length of 1.875 inches, for a friction shear factor of m=0.05. The radial reaction force produced by the polynomial curvature expansion surface  2404  was 5% lower than that of the tapered expansion surface  1708 , for a friction shear factor of m=0.05. 
   Referring to  FIG. 34 , in an exemplary experimental embodiment  1700  using FEA, total axial expansion force curve  3402  shows the total axial expansion force versus the displacement of the tapered expansion device  1704  with an optimum taper angle of 8 degrees for a friction shear factor of m=0.10. The total axial expansion force curve  3402  has transient force spike  3404  at the beginning of the displacement of the tapered expansion device  1704  and transient force spike  3406  at the end of the displacement of the tapered expansion device. 
   Referring to  FIG. 35 , in an exemplary experimental embodiment  2400  using FEA, total axial expansion force curve  3502  shows the total axial expansion force versus the displacement of the polynomial curvature expansion device  2402  with the optimum polynomial curvature expansion surface  2404  with the optimum L f /L ratio of 0.6 and a length of 1.625 inches for a friction shear factor of m=0.10. There are no transient force spikes at the beginning or at the end of the displacement of the polynomial curvature expansion device  2402  for a friction shear factor of m=0.10. The lack of transient force spikes may result in longer equipment life in comparison to the corresponding tapered expansion device  1704 . 
   Referring to  FIG. 36 , in an exemplary experimental embodiment  1700  using FEA, total axial expansion force curve  3602  shows the total axial expansion force versus the displacement of the tapered expansion device  1704  with an optimum taper angle of 7 degrees for a friction shear factor of m=0.05. The total axial expansion force curve  3602  has transient force spike  3604  at the beginning of the displacement of the tapered expansion device  1704  and transient force spike  3606  at the end of the displacement of the tapered expansion device. 
   Referring to  FIG. 37 , in an exemplary experimental embodiment  2400  using FEA, total axial expansion force curve  3702  shows the total axial expansion force versus the displacement of the polynomial curvature expansion device  2402  with the optimum polynomial curvature expansion surface  2404  with the optimum L f /L ratio of 0.6 and a length of 1.875 inches for a friction shear factor of m=0.05. There are no transient force spikes at the beginning or at the end of the displacement of the expansion device  2402  for a friction shear factor of m=0.05. The lack of transient force spikes may result in longer equipment life in comparison to the corresponding tapered expansion device  1704 . 
   Referring to  FIG. 38 , in an exemplary experimental embodiment using FEA, the maximum effective strain  3802  corresponding to an optimum taper angle of 7 degrees for the tapered expansion surface  1708  is compared to the maximum effective strain  3804  corresponding to an optimum polynomial curvature expansion surface  2404  with an optimum L f /L ratio of 0.6 and a length of 1.625 inches, for a friction shear factor of m=0.10. The maximum effective strain  3802  produced by the optimum tapered expansion surface  1708  was approximately the same as the maximum effective strain  3804  produced by the optimum polynomial curvature expansion surface  2404 , for a friction shear factor of m=0.10. 
   Referring to  FIG. 39 , in an exemplary experimental embodiment using FEA, the maximum effective strain  3902  corresponding to an optimum taper angle of 7 degrees for the tapered expansion surface  1708  is compared to the maximum effective strain  3904  corresponding to an optimum polynomial curvature expansion surface  2404  with an optimum L f /L ratio of 0.6 and a length of 1.875 inches, for a friction shear factor of m=0.05. The maximum effective strain  3902  produced by the optimum tapered expansion surface  1708  was approximately the same as the maximum effective strain  3904  produced by the optimum polynomial curvature expansion surface  2404 , for a friction shear factor of m=0.05. 
   An expansion device for radially expanding a tubular member has been described that includes a first tapered outer surface. 
   An expansion device for radially expanding a tubular member has been described that includes: a first tapered outer surface defined by a polynomial equation. 
   An expansion device for radially expanding a tubular member has been described that includes: a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67. 
   An expansion device for radially expanding a tubular member has been described that includes: a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; wherein the length of the first tapered outer surface ranges from 0.5 inches to 2.5 inches. 
   An expansion device for radially expanding a tubular member has been described that includes: a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; wherein the length of the first tapered outer surface ranges from 1.6 inches to 1.9 inches. 
   An expansion device for radially expanding a tubular member has been described that includes: a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; and wherein the first tapered outer surface comprises one or more facets in cross section. 
   An expansion device for radially expanding a tubular member has been described that includes: a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; wherein the first tapered outer surface comprises one or more facets in cross section; wherein the number of facets ranges from about 12 to 16. 
   An expansion device for radially expanding a tubular member has been described that includes: a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; wherein the first tapered outer surface comprises one or more facets in cross section; wherein the number of facets ranges from about 12 to 16; wherein the faceted surfaces are wider near the front of the expansion device and become narrower toward the rear end of the expansion device. 
   An expansion device for radially expanding a tubular member has been described that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees. 
   An expansion device for radially expanding a tubular member has been described that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; and wherein the first angle of attack is greater than the second angle of attack. 
   An expansion device for radially expanding a tubular member has been described that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; wherein the first angle of attack is greater than the second angle of attack; wherein the first angle of attack ranges from about 6 to 20 degrees; and wherein the second angle of attack ranges from about 4 to 15 degrees. 
   An expansion device for radially expanding a tubular member has been described that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; wherein the first angle of attack is greater than the second angle of attack; and one or more intermediate tapered outer surfaces coupled between the first and second tapered outer surfaces. 
   An expansion device for radially expanding a tubular member has been described that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; wherein the first angle of attack is greater than the second angle of attack; and one or more intermediate tapered outer surfaces coupled between the first and second tapered outer surfaces; wherein the angle of attack of the intermediate tapered outer surfaces continually decreases from the first tapered outer surface to the second tapered outer surface. 
   An expansion device for radially expanding a tubular member has been described that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; wherein the first angle of attack is greater than the second angle of attack; and one or more intermediate tapered outer surfaces coupled between the first and second tapered outer surfaces; wherein the angle of attack of the intermediate tapered outer surfaces decreases in steps from the first tapered outer surface to the second tapered outer surface. 
   An expansion device for radially expanding a tubular member has been described that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; wherein the first tapered outer surface comprises one or more facets in cross section. 
   An expansion device for radially expanding a tubular member has been described that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; wherein the first tapered outer surface comprises one or more facets in cross section; wherein the number of facets ranges from about 12 to 16. 
   An expansion device for radially expanding a tubular member has been described that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; wherein the first tapered outer surface comprises one or more facets in cross section; wherein the faceted surfaces are wider near the front of the expansion device and become narrower toward the rear end of the expansion device. 
   An expansion device for radially expanding a tubular member has been described that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; and wherein the first angle of attack is greater than the second angle of attack; wherein the first tapered outer surface and the second tapered outer surface comprise one or more facets in cross section. 
   An expansion device for radially expanding a tubular member has been described that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; and wherein the first angle of attack is greater than the second angle of attack; wherein the first tapered outer surface and the second tapered outer surface comprise one or more facets in cross section; wherein the number of facets ranges from about 12 to 16. 
   An expansion device for radially expanding a tubular member has been described that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; and wherein the first angle of attack is greater than the second angle of attack; wherein the first tapered outer surface and the second tapered outer surface comprise one or more facets in cross section; wherein the faceted surfaces are wider near the front of the expansion device and become narrower toward the rear end of the expansion device. 
   An expansion device for radially expanding a tubular member has been described that includes: a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; wherein the length of the tapered outer surface ranges from about 1.6 inches to 1.9 inches; wherein the tapered outer surface comprises one or more facets in cross section; wherein the number of facets ranges from about 12 to 16; wherein the faceted surfaces are wider near the front of the expansion device and become narrower toward the rear end of the expansion device. 
   An expansion system for radially expanding a tubular member has been described that includes a first tapered outer surface; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes: an expansion device that includes a first tapered outer surface defined by a polynomial equation; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes: an expansion device that includes a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; and means for displacing the expansion device relative to the expandable tubular member; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes: an expansion device that includes a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; wherein the length of the first tapered outer surface ranges from 0.5 inches to 2.5 inches; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes: an expansion device that includes a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; wherein the length of the first tapered outer surface ranges from 1.6 inches to 1.9 inches; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes: an expansion device that includes a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; and wherein the first tapered outer surface comprises one or more facets in cross section; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes: an expansion device that includes a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; wherein the first tapered outer surface comprises one or more facets in cross section; wherein the number of facets ranges from about 12 to 16; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes: an expansion device that includes a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; wherein the first tapered outer surface comprises one or more facets in cross section; wherein the number of facets ranges from about 12 to 16; wherein the faceted surfaces are wider near the front of the expansion device and become narrower toward the rear end of the expansion device; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes an expansion device that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes: an expansion device that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; and wherein the first angle of attack is greater than the second angle of attack; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes an expansion device that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; wherein the first angle of attack is greater than the second angle of attack; wherein the first angle of attack ranges from about 6 to 20 degrees; and wherein the second angle of attack ranges from about 4 to 15 degrees; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes an expansion device that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; wherein the first angle of attack is greater than the second angle of attack; and one or more intermediate tapered outer surfaces coupled between the first and second tapered outer surfaces; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes an expansion device that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; wherein the first angle of attack is greater than the second angle of attack; and one or more intermediate tapered outer surfaces coupled between the first and second tapered outer surfaces; wherein the angle of attack of the intermediate tapered outer surfaces continually decreases from the first tapered outer surface to the second tapered outer surface; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes an expansion device that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; wherein the first angle of attack is greater than the second angle of attack; and one or more intermediate tapered outer surfaces coupled between the first and second tapered outer surfaces; wherein the angle of attack of the intermediate tapered outer surfaces decreases in steps from the first tapered outer surface to the second tapered outer surface; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes an expansion device that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; wherein the first tapered outer surface comprises one or more facets in cross section; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes an expansion device that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; wherein the first tapered outer surface comprises one or more facets in cross section; wherein the number of facets ranges from about 12 to 16; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes an expansion device that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; wherein the first tapered outer surface comprises one or more facets in cross section; wherein the faceted surfaces are wider near the front of the expansion device and become narrower toward the rear end of the expansion device; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes an expansion device that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; and wherein the first angle of attack is greater than the second angle of attack; wherein the first tapered outer surface and the second tapered outer surface comprise one or more facets in cross section; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes an expansion device that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; and wherein the first angle of attack is greater than the second angle of attack; wherein the first tapered outer surface and the second tapered outer surface comprise one or more facets in cross section; wherein the number of facets ranges from about 12 to 16; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes an expansion device that includes a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; and wherein the first angle of attack is greater than the second angle of attack; wherein the first tapered outer surface and the second tapered outer surface comprise one or more facets in cross section; wherein the faceted surfaces are wider near the front of the expansion device and become narrower toward the rear end of the expansion device; and means for displacing the expansion device relative to the expandable tubular member. 
   An expansion system for radially expanding a tubular member has been described that includes: an expansion device that includes a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; wherein the length of the tapered outer surface ranges from about 1.6 inches to 1.9 inches; wherein the tapered outer surface comprises one or more facets in cross section; wherein the number of facets ranges from about 12 to 16; wherein the faceted surfaces are wider near the front of the expansion device and become narrower toward the rear end of the expansion device; and means for displacing the expansion device relative to the expandable tubular member. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface defined by a polynomial equation. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; wherein the length of the first tapered outer surface ranges from 0.5 inches to 2.5 inches. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; wherein the length of the first tapered outer surface ranges from 1.6 inches to 1.9 inches. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; and wherein the first tapered outer surface comprises one or more facets in cross section. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; wherein the first tapered outer surface comprises one or more facets in cross section; wherein the number of facets ranges from about 12 to 16. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; wherein the first tapered outer surface comprises one or more facets in cross section; wherein the number of facets ranges from about 12 to 16; wherein the faceted surfaces are wider near the front of the expansion device and become narrower toward the rear end of the expansion device. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; and wherein the first angle of attack is greater than the second angle of attack. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; wherein the first angle of attack is greater than the second angle of attack; wherein the first angle of attack ranges from about 6 to 20 degrees; and wherein the second angle of attack ranges from about 4 to 15 degrees. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; wherein the first angle of attack is greater than the second angle of attack; and one or more intermediate tapered outer surfaces coupled between the first and second tapered outer surfaces. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; wherein the first angle of attack is greater than the second angle of attack; and one or more intermediate tapered outer surfaces coupled between the first and second tapered outer surfaces; wherein the angle of attack of the intermediate tapered outer surfaces continually decreases from the first tapered outer surface to the second tapered outer surface. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; wherein the first angle of attack is greater than the second angle of attack; and one or more intermediate tapered outer surfaces coupled between the first and second tapered outer surfaces; wherein the angle of attack of the intermediate tapered outer surfaces decreases in steps from the first tapered outer surface to the second tapered outer surface. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; wherein the first tapered outer surface comprises one or more facets in cross section. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; wherein the first tapered outer surface comprises one or more facets in cross section; wherein the number of facets ranges from about 12 to 16. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; wherein the first tapered outer surface comprises one or more facets in cross section; wherein the faceted surfaces are wider near the front of the expansion device and become narrower toward the rear end of the expansion device. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; and wherein the first angle of attack is greater than the second angle of attack; wherein the first tapered outer surface and the second tapered outer surface comprise one or more facets in cross section. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; and wherein the first angle of attack is greater than the second angle of attack; wherein the first tapered outer surface and the second tapered outer surface comprise one or more facets in cross section; wherein the number of facets ranges from about 12 to 16. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface; wherein the first tapered outer surface comprises an angle of attack ranging from about 6 to 10 degrees; a second tapered outer surface comprising a second angle of attack coupled to the first tapered outer surface; and wherein the first angle of attack is greater than the second angle of attack; wherein the first tapered outer surface and the second tapered outer surface comprise one or more facets in cross section; wherein the faceted surfaces are wider near the front of the expansion device and become narrower toward the rear end of the expansion device. 
   A method of radially expanding a tubular member has been described that includes radially expanding at least a portion of the tubular member by extruding at least a portion of the tubular member off of an expansion device; wherein the expansion device comprises a first tapered outer surface defined by a polynomial equation; wherein the polynomial equation has a L f /L ratio ranging from about 0.32 to 0.67; wherein the length of the tapered outer surface ranges from about 1.6 inches to 1.9 inches; wherein the tapered outer surface comprises one or more facets in cross section; wherein the number of facets ranges from about 12 to 16; wherein the faceted surfaces are wider near the front of the expansion device and become narrower toward the rear end of the expansion device. 
   The teaching of the present disclosure may be applied to the construction and/or repair of wellbore casings, pipelines, and/or structural supports. 
   Although illustrative embodiments of the invention have been shown and described, a wide range of modification, changes and substitution is contemplated in the foregoing disclosure. In some instances, some features of the present invention may be employed without a corresponding use of the other features, and some steps of the present invention may be executed without a corresponding execution of other steps. Accordingly, all such modifications, changes and substitutions are intended to be included within the scope of this invention as defined in the following claims, and it is appropriate that the claims be construed broadly and in a manner consistent with the scope of the invention. In the claims, means-plus-function clauses are intended to cover the structures described herein as performing the recited function and not only structural equivalents, but also equivalent structures.