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CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is a non-provisional of U.S. Application Ser. No. 60/786,328 filed on Mar. 27, 2006, which is incorporated by reference herein in its entirety. 
     
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
       [0002]    Not applicable. 
       BACKGROUND OF THE INVENTION 
       [0003]    1. Field of the Invention 
         [0004]    This invention relates to the field of expandable tubulars and more specifically to the field of expanding tubulars with multiple expansion swages. 
         [0005]    2. Background of the Invention 
         [0006]    Expandable tubulars have become a viable technology for well drilling, repair, and completion. In a conventional technique for expansion, an expansion swage is positioned inside a pre-expanded portion of a tubular that is sealed at the bottom with a plug. Hydraulic pressure is applied through the drill pipe into the pre-expanded portion of the tubular generating sufficient force to propagate the expansion swage and radially expand the unexpanded portion of the tubular. Drawbacks to such conventional technique include that the expansion pressure may be limited by the yield pressure of the expanded portion of the tubular, which may limit the degree of expansion. Further drawbacks include the ratio of the expandable tubular diameter to its wall thickness, which may be due to the maximum pressure available on drilling rigs. Consequently, conventional techniques may typically be limited to expansion ratios of 10-16% and to a collapse resistance of 3,000-4,000 psi. 
         [0007]    Other conventional techniques for expansion include using a hydraulic actuator to generate force for propagating an expansion swage and radially expanding a tubular. The force is applied against a front anchor or a back anchor, which results in compressive or tensile stresses in the tubular. The connectors in the expandable tubulars, due to geometrical constraints, are typically of flush or a near flush type, which typically results in a tensile efficiency of 50%. Drawbacks include that the expansion force may not be higher than 50% of the tubular body yield strength, which may limit the degree of tubular expansion to 25-28%. 
         [0008]    Another technique includes lowering the friction coefficient (i.e., by lubricants) between the tubular and the expansion swage, which may reduce the value of the friction factor. Drawbacks include the cost and efficiency of such a technique. 
         [0009]    Consequently, there is a need for a technique that provides expandable tubulars with significantly higher performance characteristics, including collapse resistance, and higher expansion ratios. 
       BRIEF SUMMARY OF SOME OF THE PREFERRED EMBODIMENTS 
       [0010]    These and other needs in the art are addressed in one embodiment by an apparatus for radially expanding a tubular. The apparatus includes at least two expansion swages. At least one expansion swage is axially movable relative to other expansion swages. In addition, the apparatus includes sealing means capable of providing fluid tight pressure chambers between the expansion swages and an expanded portion of the tubular. 
         [0011]    In another embodiment, these and other needs in the art are addressed by an apparatus for radially expanding a tubular. The apparatus includes at least two expansion swages. In addition, at least one expansion swage is axially movable relative to the other expansion swages. Moreover, the apparatus includes at least one actuator that is capable of providing a force for providing longitudinal movement of at least one of the expansion swages inside the tubular to plastically radially expand the tubular. 
         [0012]    An additional embodiment that addresses these and other needs in the art includes an apparatus for radially expanding a tubular. The apparatus includes at least two expansion swages. At least one expansion swage is axially movable relative to the other expansion swages. In addition, the apparatus includes a driving means capable of providing a force for providing sequential longitudinal movement of the expansion swages inside the tubular to plastically radially expand the tubular. 
         [0013]    The foregoing has outlined rather broadly the features and technical advantages of the present invention in order that the detailed description of the invention that follows may be better understood. Additional features and advantages of the invention will be described hereinafter that form the subject of the claims of the invention. It should be appreciated by those skilled in the art that the conception and the specific embodiments disclosed may be readily utilized as a basis for modifying or designing other embodiments for carrying out the same purposes of the present invention. It should also be realized by those skilled in the art that such equivalent embodiments do not depart from the spirit and scope of the invention as set forth in the appended claims. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0014]    For a detailed description of the preferred embodiments of the invention, reference will now be made to the accompanying drawings in which: 
           [0015]      FIG. 1  illustrates a fragmentary sectional view of a tubular expansion apparatus; 
           [0016]      FIGS. 2A-2C  illustrate a cross-sectional view of a tubular expansion apparatus shown in various stages of operation thereof, and 
           [0017]      FIG. 3  illustrates a fragmentary sectional view of a tubular expansion apparatus employing an actuator. 
       
    
    
     NOTATION AND NOMENCLATURE 
       [0018]    “Actuator” refers to a device comprising one or more annular pistons and a cylinder slidingly arranged over the pistons, having at least one pressure chamber per piston, and capable of providing a force to axially move an expansion swage inside the expandable tubular to plastically radially expand the tubular. 
         [0019]    “Anchor” refers to a device capable of being selectively engaged with the inner surface of the tubular and preventing movement of selected parts of the tubular expansion apparatus relative to the tubular under applied forces during the expansion process. 
         [0020]    “Driving mean” refers to a device such as a pressure chamber, an actuator, an electric motor, a mud motor, a mechanical pull, and the like, capable of providing a sufficient force to axially move the expansion swage inside the expandable tubular to plastically radially expand the tubular. 
         [0021]    “Expandable tubular” and “tubular” refer to a tubular member such as a liner, casing, borehole clad to seal a selected zone, and the like that is capable of being plastically radially expanded by the application of a radial expansion force. 
         [0022]    “Expansion swage” refers to a device that may generate sufficient radial forces to plastically increase tubular diameter when it is displaced in a longitudinal direction in the tubular. Without limitation, an example of a suitable expansion swage includes a tapered cone of a fixed or a variable diameter. 
         [0023]    “Sealing means” refers to a device such as a rubber O-ring, a polymer cup-seal, a differential fill-up collar, a metal-to-metal seal, a plug in the tubular, and the like for providing a pressure chamber. 
       DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0024]    In an embodiment, a tubular expansion apparatus comprises at least two expansion swages. It has been found through theoretical modeling and experimentation that expansion force, F exp., maybe evaluated by equation (1). 
         [0000]        F  exp.=π· k·Yp·to· ( Dc−Do )   (1) 
         [0025]    k is an experimentally defined factor depending on the coefficient of friction between the tubular and swage and shape of the swage, Yp is yield stress of tubular material, t O  is wall thickness of tubular in front of the swage, Dc is swage diameters and D O  is tubular inner diameter in front of the swage. 
         [0026]    The pressure for the swage propagation and expansion of the tubular may be calculated by dividing expansion force, equation (1), by the swage cross-sectional area as shown by equation (2). 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                      
                     
                         
                     
                      
                     
                       exp 
                       . 
                     
                   
                   = 
                   
                     4 
                     · 
                     k 
                     · 
                     Yp 
                     · 
                     to 
                     · 
                     
                       
                         ( 
                         
                           Dc 
                           - 
                           Do 
                         
                         ) 
                       
                       
                         
                           ( 
                           Dc 
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0027]    One of the drawbacks of conventional techniques of tubular expansion may be due to the limitation of rig pressure, which may result in limited performance of expanded tubular such as collapse resistance. Under normal operating conditions, due to safety reasons and equipment limitations, the maximum operational pressure on the rig may be limited to a certain value, P max. Thus, the maximum expansion pressure is limited to the expression of equation (3). 
         [0000]      P exp.≦P max 
         [0028]    The main parameter that controls tubular collapse resistance after expansion is the ratio of tubular outside diameter, ODexp., to its wall thickness, texp. To calculate this ratio, the tubular expansion ratio, ε, of equation (4) may be used. 
         [0000]    
       
         
           
             
               
                 
                   ɛ 
                   = 
                   
                     
                       ( 
                       
                         Dc 
                         - 
                         Do 
                       
                       ) 
                     
                     Do 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0029]    It is to be understood that when a tubular is expanded in the radial direction, it may shrink in the longitudinal direction, and its wall thickness becomes thinner depending on the boundary conditions. For the most constrained conditions, such as when the tubular is differentially stuck and constrained from longitudinal shrinkage, the deformation of wall thinning is equal to the radial deformation as shown by equation (5). 
         [0000]        t  exp.=(1−ε)· to    (5) 
         [0030]    texp. is tubular wall thickness after expansion. Using equations (2), (4) and (5) the condition of expression (9) may be written as equation (6). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       OD 
                        
                       
                           
                       
                        
                       
                         exp 
                         . 
                       
                     
                     
                       t 
                        
                       
                           
                       
                        
                       
                         exp 
                         . 
                       
                     
                   
                   ≥ 
                   
                     
                       
                         Yp 
                         
                           P 
                            
                           
                               
                           
                            
                           
                             max 
                             . 
                           
                         
                       
                       · 
                       4 
                       · 
                       k 
                       · 
                       
                         ɛ 
                         
                           ( 
                           
                             1 
                             - 
                             
                               ɛ 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     2 
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0031]    Where OD exp. is outside diameter of expandable tubular, ODexp. may be expressed as equation (7). 
         [0000]        OD  exp.= D  exp.+2 ·t  exp.   (7) 
         [0032]    Dexp. is inner tubular diameter after expansion, substantially equal to the swage diameter, D C . Equation (6) allows calculation of the minimum ratio of the expanded pipe diameter to its wall thickness, which is a parameter for calculation of the collapse resistance of the pipe. For example, for typical values of P max.=5,000·psi, k=1.85, Yp=80,000·psi, and 20% radial expansion, equation (6) yields equation (8). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       OD 
                        
                       
                           
                       
                        
                       
                         exp 
                         . 
                       
                     
                     
                       t 
                        
                       
                           
                       
                        
                       
                         exp 
                         . 
                       
                     
                   
                   ≥ 
                   26.7 
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0033]    Using an API 5C3 formula for collapse resistance, Pc, of the expanded tubular, we have the expression of (9). 
         [0000]      Pc≦2,500·psi   (9) 
         [0034]    Therefore, the maximum collapse resistance of tubulars expanded 20% by conventional techniques, due to 5,000 psi rig pressure restriction, may be limited to 2,500 psi. 
         [0035]    Another drawback on the degree of tubular radial expansion by conventional techniques is the limited efficiency of expandable tubular connectors. Due to geometrical constraints, the connectors of expandable tubulars are flush or near-flush, which may limit their tensile efficiency to 50% of the tubular body yield strength, Fy. Therefore, the expansion force may be limited to the constraint of (10). 
         [0000]        F  exp.≦0.5· Fy    (10) 
         [0036]    The tubular body yield strength may be estimated as equation (11). 
         [0000]    
       
         
           
             
               
                 
                   Fy 
                   = 
                   
                     
                       π 
                       4 
                     
                     · 
                     
                       [ 
                       
                         
                           
                             ( 
                             OD 
                             ) 
                           
                           2 
                         
                         - 
                         
                           
                             ( 
                             ID 
                             ) 
                           
                           2 
                         
                       
                       ] 
                     
                     · 
                     Yp 
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0037]    OD is outside diameter, and ID is inside diameter of unexpanded tubular. Using equations (1), (4), and (11), the constraint (10) yields expression (12). 
         [0000]    
       
         
           
             
               
                 
                   ɛ 
                   ≤ 
                   
                     
                       0.5 
                       k 
                     
                      
                     
                       ( 
                       
                         1 
                         + 
                         
                           to 
                           Do 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0038]    For expandable tubulars of practical interest with 10≦D O /t O ≦25 and k=1.85, equation (12) shows that the maximum expansion ratio due to connector efficiency may be limited to the expression of (13). 
         [0000]      ε≦30%   (13) 
         [0039]    The above analysis shows that the limitation on the maximum degree of radial expansion and performance characteristics of the expanded tubulars may be a result of high expansion forces or expansion pressures for tubular expansion by conventional techniques. The analysis also shows that reducing the expansion force by selecting low yield (Yp) tubulars may not eliminate the problem because both tubular body yield strength, equation (11), and expansion force, equation (1), linearly depend on Yp, and therefore the limitations may not be affected. Thus, the most effective way for overcoming the drawbacks discussed above is to employ multiple, sequential expansions of the tubular, each at a relative expansion ratio lower than the final degree of expansion. 
         [0040]      FIG. 1  illustrates an embodiment of a tubular expansion apparatus  5  that provides multiple expansions. Tubular expansion apparatus  5  includes expansion swages  34  and  35  working sequentially. First expansion swage  35  has diameter D 1 , which is less than the diameter D 2  of second expansion swage  34 . Expanded portion  32  of tubular  205  comprises a pressure plug  39 , and both expansion swages  34  and  35  are pressure sealed against the inside surface of tubular  205  providing two pressure chambers  37  and  38 . The pressure is applied sequentially either in both pressure chambers  37  and  38  or only in one chamber  38 . The alternating of pressure is accomplished by a valve (not shown). It is to be understood that in some embodiments the valve may be adapted to selectively control the flow of operating fluid to at least one of the pressure chambers  37 ,  38  and fluid outflow from chamber  37  depending on the relative positions of expansion swages  34 ,  35 . First expansion swage  35  may slide over shaft  31 , while second expansion swage  34  is permanently attached to shaft  31 . In an embodiment, shaft  31  has at least two longitudinal bores for flow of operating liquid to and from pressure chambers  37 ,  38 . If the pressure is applied to both chambers  37  and  38 , second expansion swage  34  has equal pressure in back  34   b  and in front  34   a  and, therefore, second expansion swage  34  does not move with regard to tubular  205 . Pressure in chamber  37  may be higher than or equal to the pressure in tubular annulus  33 . At a certain level of pressure differential, first expansion swage  35  is propelled in tubular  205  sliding over shaft  31  and expanding tubular  205  from its original inside diameter Do to the diameter D 1 . At the end of the stroke, the valve releases pressure from chamber  37  and allows free passage of the liquid from chamber  37 , while the pressure in chamber  38  is maintained. At a certain level of pressure, second expansion swage  34  is propelled expanding tubular  205  from diameter D 1  to diameter D 2  and moves shaft  31  through first expansion swage  35 , which is stationary relative to tubular  205 . 
         [0041]    To minimize the pressure for expanding tubular  205  from its original diameter Do to the final diameter D 2 , the diameters of first and second swages  35  and  34  may be selected such that the pressure for the propagation of first expansion swage  35  is equal to the pressure for the propagation of second expansion swage  34 . The force, F 1 , for the propagation of first expansion swage  35  may be calculated using equation (1) with Dc=D 1 , as shown by equation (14). 
         [0000]        F 1 =π·k·Yp·to· ( D 1 −Do )   (14) 
         [0042]    Then, the expansion pressure, P 1 , for the propagation of first expansion swage  35  is calculated by dividing propagation force F 1  by the cross-sectional area of first expansion swage  35  minus cross-sectional area of shaft  31  as shown by equation (15). 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                      
                     
                         
                     
                      
                     1 
                   
                   = 
                   
                     4 
                     · 
                     k 
                     · 
                     Yp 
                     · 
                     to 
                     · 
                     
                       
                         ( 
                         
                           
                             D 
                              
                             
                                 
                             
                              
                             1 
                           
                           - 
                           Do 
                         
                         ) 
                       
                       
                         ( 
                         
                           
                             
                               ( 
                               
                                 D 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             
                               ( 
                               Ds 
                               ) 
                             
                             2 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0043]    Ds is a diameter of shaft  31  over which first expansion swage  35  is sliding. The force, F 2 , to propagate second expansion swage  34  is also calculated using equation (1) with, Dc=D 2 , D O =D 1 , and t O =t 1 , where t 1  is wall thickness of tubular  205  after expansion by first expansion swage  35  as shown by equation (16). 
         [0000]        F 2 =π·k·Yp·t   1 ( D 2 −D 1)   (16) 
         [0044]    The corresponding expansion pressure, P 2 , for second expansion swage  34  is calculated by dividing expansion force F 2  by the fill cross-sectional area of second expansion swage  34  as shown by equation (17). 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                      
                     
                         
                     
                      
                     2 
                   
                   = 
                   
                     4 
                     · 
                     k 
                     · 
                     Yp 
                     · 
                     
                       t 
                       1 
                     
                     · 
                     
                       
                         ( 
                         
                           
                             D 
                              
                             
                                 
                             
                              
                             2 
                           
                           - 
                           
                             D 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                         ) 
                       
                       
                         
                           ( 
                           
                             D 
                              
                             
                                 
                             
                              
                             2 
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
         [0045]    Equating pressure P 1  from equation (15) and pressure P 2  from equation (17) (ignoring changes in wall thickness) yields the expression of equation (18). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           D 
                            
                           
                               
                           
                            
                           1 
                         
                         - 
                         Do 
                       
                       ) 
                     
                     
                       ( 
                       
                         
                           
                             ( 
                             
                               D 
                                
                               
                                   
                               
                                
                               1 
                             
                             ) 
                           
                           2 
                         
                         - 
                         
                           
                             ( 
                             Ds 
                             ) 
                           
                           2 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           D 
                            
                           
                               
                           
                            
                           2 
                         
                         - 
                         
                           D 
                            
                           
                               
                           
                            
                           1 
                         
                       
                       ) 
                     
                     
                       
                         ( 
                         
                           D 
                            
                           
                               
                           
                            
                           2 
                         
                         ) 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0046]    For a selected tubular with inside original diameter Do and selected final diameter after expansion D 2 , this equation (18) defines the diameter D 1  of the first swage. The expansion pressure may be defined by equations (15) or (17). Equations (2) and (17) show that the expansion pressure provided by tubular expansion apparatus  5  is significantly less than the expansion pressure of conventional methods. This allows expansion of pipes with significantly lower diameter to wall thickness ratios, which results in expanded tubulars with collapse resistance significantly higher than that of tubulars expanded by conventional methods. For instance, consider the instance in which expansion pressure is limited by the maximum available rig pressure, see equation (3). When the tubular is expanded by 20%, the expression of equation (19) is provided, 
         [0000]        D 2=1.2· Do    (19) 
         [0000]    and for the selected shaft diameter Ds=0.5·Do, equation (18) defines the diameter of first expansion swage D 1 =1.077·Do. Then, the condition of maximum available pressure, equation (3), using equation (17), may be written as equation (20). 
         [0000]    
       
         
           
             
               
                 
                   
                     Do 
                     to 
                   
                   ≥ 
                   
                     0.315 
                     · 
                     k 
                     · 
                     
                       Yp 
                       
                         P 
                          
                         
                             
                         
                          
                         
                           max 
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0047]    Assigning values of friction factor k=1.85, yield stress Yp=80·ksi, and maximum available pressure P max=5,000·psi, the same as in the example of conventional expansion methods, the expression of equation (21) has been found. 
         [0000]    
       
         
           
             
               
                 
                   
                     Do 
                     to 
                   
                   ≥ 
                   9.3 
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0048]    Therefore, the minimum ratio of outside diameter to the wall thickness of the pipe after 20% expansion is shown by equation (22). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ODexp 
                         . 
                       
                       
                         texp 
                         . 
                       
                     
                     ≥ 
                     
                       
                         
                           
                             ( 
                             
                               1 
                               + 
                               0.2 
                             
                             ) 
                           
                           · 
                           Do 
                         
                         
                           
                             ( 
                             
                               1 
                               - 
                               0.2 
                             
                             ) 
                           
                           · 
                           to 
                         
                       
                       + 
                       2 
                     
                   
                   = 
                   16 
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
         [0049]    Using an API 5C3 formula for collapse resistance, Pc, of the expanded tubular yields the expression of equation (23). 
         [0000]        Pc= 8,018·psi   (23) 
         [0050]    Thus, utilizing the same pressure as in the conventional methods, tubular expansion apparatus  5  allows expansion of tubulars with significantly thicker walls, which results in greater than 3 times higher collapse resistance of the expanded tubular than that achievable by conventional methods. 
         [0051]      FIGS. 2A-2C  illustrate cross-sectional views of tubular expansion apparatus  5  in various stages of operation. Tubular expansion apparatus  5  includes first expansion swage  45  and second expansion swage  47 . First expansion swage  45  has an elongated arm  43  and may slide along shaft  49 . Second expansion swage  47  is connected to shaft  49 . Expanded end  48  of tubular  40  is sealed with pressure plug  55 . Both first expansion swage  45  and second expansion swage  47  are sealed against tubular  40  and against shaft  49 , thus comprising two pressure chambers  53  and  54 . Tubular expansion apparatus  5  also includes a valve  42  capable of connecting and disconnecting pressure lines  51  and  52 , depending on the relative position of first expansion swage  45  and second expansion swage  47 . 
         [0052]    As shown in  FIGS. 2A-2C , the pressurized fluid is supplied through a conduit such as drill pipe or coiled tubing to pressure line  52 . When valve  42  is in its end position connecting pressure line  52  with line  51 , as shown in  FIG. 2A , the pressure is applied in both pressure chambers  53  and  54 . In this position, pressure is applied to both front side  47   a  and back side  47   b  of second expansion swage  47 , and it remains stationary with regard to tubular  40 . First expansion swage  45  is under high pressure on back side  45   b  by pressure chamber  53  and under low pressure on front side  45   a  equal to the pressure in annulus  41 . At a certain level of pressure differential applied to first expansion swage  45 , first expansion swage  45  starts sliding over shaft  49  expanding tubular  40  to provide expanded portion  46 . At the end of the stroke, first expansion swage  45  displaces valve  42  to the end position in which pressure lines  51  and  52  are disconnected, as shown in  FIG. 2B . Under theses conditions, liquid from front side  45   a  and back side  45   b  is communicating with annulus  41  through vents  44  and  50 , and therefore, first expansion swage  45  remains stationary with regard to tubular  40 . Second expansion swage  47  is exposed to high pressure on back side  47   b  from pressure chamber  54  and low pressure on front side  47   a , equal to the pressure in annulus  41 . At a certain pressure differential, second expansion swage  47  moves forward with shaft  49  sliding through first expansion swage  45  and expanding tubular  40  to provide expanded portion  48 . As shown in  FIG. 2C , at the end of the stroke, valve  42  is displaced to the end position in which pressure lines  51  and  52  are connected, and which is the same position as in the beginning of the cycle as shown in  FIG. 2A . Thus, tubular expansion apparatus  5  provides automatic sequential movement of expansion swages  45 ,  47  under continuous supply of pressurized fluid through pressure line  52 . By selecting diameters D 1 , D 2  of expansion swages  45 ,  47  by equation (24) the operational expansion pressure may be minimal and practically constant. 
         [0053]    As shown in  FIG. 2A , valve  42  is a hydraulic valve and includes a cylinder longitudinally slidably engaged with shaft  49  and forming an internal annular pressure chamber surrounding shaft  49 . Valve  42  is a two-position valve with a first position corresponding to a pressure supply to both pressure chambers  53  and  54 , and a second position corresponding to pressure supply to only pressure chamber  54  and allowing liquid flow from pressure chamber  53  to annulus  41 . In an embodiment, valve  42  includes a position control device (not illustrated) to selectively and releasably lock the cylinder in first or second positions This may be achieved, for example, by utilizing a C-ring locking mechanism. As shown in  FIG. 2A , C-ring  60  may be engaged or disengaged in grooves  61  or  62  under the action of an axial force applied to valve  42  through the action of springs  56  and  57 . It will be understood that C-ring  60  may bear against any suitable surfaces or any components having fixed relationship with shaft  49  and/or with the valve cylinder. C-ring  60  may be configured to operate primarily in tension or primarily in compression. It will also be understood that other position control devices, such as a collets and the like, capable of selectively and releasably securing a position of the valve cylinder on shaft  49  may be used. 
         [0054]    The shifting between the end positions of valve  42  is provided by the relative displacement of expansion swages  45  and  47 . The length of elongated arm  43  may generally be equal to the length of the total stroke displacement between expansion swages  45 ,  47 . Each spring  56 ,  57  is capable of displacing valve  42  from the first valve position to the second valve position and vice versa. It will be understood that springs  56  and  57  may bear against any suitable surfaces or any components having a fixed relationship with valve  42  and/or with elongated arm  43 . Springs  56  and  57  may be configured to operate primarily in tension or primarily in compression. It will also be understood that any other type of valve may be used that is suitable for alternating the pressure and liquid outflow from the chamber between expansion swages  45 ,  47  depending on relative position of expansion swages  45 ,  47 . 
         [0055]      FIG. 3  illustrates another embodiment of tubular expansion apparatus  5 , which shows a fragmentary sectional view of tubular expansion apparatus  5  with expansion swages  62  and  64 . Tubular expansion apparatus  5  also comprises anchors  63  and  65  capable of being selectively anchored to the inner surface of tubular  61 . Tubular expansion apparatus  5  also comprises an actuator  71  including a cylinder  72  attached to expansion swage  62  and a piston  68  attached to shaft  66  and a two position hydraulic valve  77 , for instance as disclosed in Application PCT/US2006/060624 which is incorporated by reference herein in its entirety, capable of alternating pressure and fluid outflow from pressure chambers  67  and  69 . When pressure is applied in pressure chamber  67 , fluid is vented from pressure chamber  69 , and anchor  65  is anchored against tubular  61  while anchor  63  is disengaged. At a certain level of pressure, first expansion swage  62  moves inside tubular  61  and expands it to a diameter substantially equal to the diameter, D 1 , of first expansion swage  62  while second expansion swage  64  remains stationary with regard to tubular  61 . At the end of the stroke, the pressure is applied to pressure chamber  69  while the fluid from pressure chamber  67  is vented, and anchor  63  is anchored to tubular  61  while anchor  65  is disengaged. At a certain level of pressure, second expansion swage  64  moves inside tubular  61  and expands it to a diameter substantially equal to the diameter, D 2 , of second expansion swage  64 , while first expansion swage  62  remains stationary with regard to tubular  61 . Thus, expansion swages  62 ,  64  move inside tubular  61  in sequential manner expanding tubular  61  from its original inside diameter Do to the diameter D 1  and then from D 1  to D 2 . To minimize expansion forces, for expansion of a selected tubular of unexpanded diameter Do to a final expanded diameter D 2 , the diameter, D 1 , of first expansion swage  62  may be defined from the condition that expansion forces for expansion by each swage should be equal. Equating forces F 1  from equation (14) and F 2  from equation (16) and ignoring changes in wall thickness, equation (24) is obtained. 
         [0000]    
       
         
           
             
               
                 
                   
                     D 
                      
                     
                         
                     
                      
                     1 
                   
                   = 
                   
                     
                       Do 
                       + 
                       
                         D 
                          
                         
                             
                         
                          
                         2 
                       
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
         [0056]    Equation (24) defines the relationship between diameters of first and second expansion swages  62  and  64 . Equation (24) also provides the minimum expansion force for tubular radial expansion by two swages. If diameters of the swages are selected according to equation (24), the expansion force calculated using equation (14) becomes equation (25). 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                      
                     
                         
                     
                      
                     1 
                   
                   = 
                   
                     π 
                     · 
                     k 
                     · 
                     to 
                     · 
                     Yp 
                     · 
                     
                       
                         ( 
                         
                           
                             D 
                              
                             
                                 
                             
                              
                             2 
                           
                           - 
                           Do 
                         
                         ) 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
         [0057]    The expansion force to expand the same tubular to the same diameter, D 2 , using a conventional swage technique, calculated by equation (1) with Dc=D 2  and D f =Do is shown by equation (26). 
         [0000]        F exp. =π·k·to·Yp· ( D 2 −Do )   (26) 
         [0058]    Comparison of equations (25) and (26) shows that the force for tubular expansion by tubular expansion apparatus  5  may be half of the force for expansion of the same tubular to the same degree of expansion by a conventional expansion technique. 
         [0059]    Selecting the diameters of swages according to equation (24) and using the expansion ratio defined as equation (27), 
         [0000]    
       
         
           
             
               
                 
                   ɛ 
                   = 
                   
                     
                       ( 
                       
                         
                           D 
                            
                           
                               
                           
                            
                           2 
                         
                         - 
                         Do 
                       
                       ) 
                     
                     Do 
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
         [0000]    the limitation on maximum degree of expansion due to the constraint of connector efficiency, shown by constraint (10), may be obtained by substituting expansion force from equation (25) in constraint (10) and shown by equation (28). 
         [0000]    
       
         
           
             
               
                 
                   ɛ 
                   ≤ 
                   
                     
                       1 
                       k 
                     
                      
                     
                       ( 
                       
                         1 
                         + 
                         
                           to 
                           Do 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
         [0060]    For the same values of k=1.85 and Do/to=10 as in the case of conventional expansion methods, shown by equation (13), the maximum degree of tubular expansion, equation (28), may be estimated as expression (29). 
         [0000]      ε≦60%   (29) 
         [0061]    Thus, the maximum degree of radial expansion of a tubular by tubular expansion apparatus  5  may be double the maximum degree of expansion by the conventional expansion techniques, see equation (19). 
         [0062]    It will be further appreciated by those skilled in the art that the tubular expansion apparatus  5  comprising multiple expansion swages working in a sequential manner described herein may employ any conventional swages such as, but not limited to, swages of fixed or variable diameters. Additionally, the driving means may employ hydraulic pressure, hydraulic actuators, electric motors, mud motors, mechanical pull force, or combinations thereof. 
         [0063]    It is to be understood that in some embodiments tubular expansion apparatus  5  has two or more actuators for providing suitable force for longitudinal movement of at least one of the expansion swages. It is to be further understood that expansion of the tubular may include plastic radial expansion of the tubular. 
         [0064]    Without being limited by theory, tubular expansion apparatus  5  provides an expansion pressure 35-40% less than the expansion pressure for the same degree of tubular expansion accorded to conventional expansion methods. Further, without being limited by theory, tubular expansion apparatus  5  allows expansion of the tubular with lower ratios of tubular diameter to tubular wall thickness, which may result in expanded tubulars with collapse resistance 2-3 times higher than the collapse resistance of tubulars expanded by conventional methods. 
         [0065]    Although the present invention and its advantages have been described in detail, it should be understood that various changes, substitutions and alterations may be made herein without departing from the spirit and scope of the invention as defined by the appended claims.

Summary:
A method and apparatus for tubular expansion are disclosed. In an embodiment, an apparatus for radially expanding a tubular comprises at least two expansion swages. At least one expansion swage is axially movable relative to other expansion swages. In addition, the apparatus includes sealing means capable of providing fluid tight pressure chambers between the expansion swages and an expanded portion of the tubular.