Page 282 - Air and Gas Drilling Manual
P. 282

The flow down the annulus is three phase flow (gas, fluid, and solids).  The exit
                               pressure for stable foam drilling operations is the back pressure, P bp, upstream of the
                               valve in the flow line from the annulus.  Equation 6-26 becomes
                                      ∫  P bh  dP  =  ∫  H  dh  Chapter 6: Direct Circulation Models    6-27
                                                                                       (6-88)
                                            P
                                       P bp  B ()  0
                                          a
                                                                    2
                               where P bp is the back pressure on the annulus (lb/ft , abs), and
                                                                 
                                                                 
                                                       w ˙       
                                      B () =            t        
                                         P
                                       a
                                                   P    T av    
                                                  g
                                                        Q +  Q f 
                                                            g
                                                   P    T    
                                                       g
                                                                            2  
                                                           P g    T av    
                                                             P        Q g  + Q f     
                                                                T
                                        
                                        1 +       f          π   g           
                                        
                                            2 (  h  − )       ( D  2  − )   
                                                                        2
                                                     D
                                              gD
                                                                      D
                                                       p
                                                            4   h     p     
                                                                             
                                        
                               where the total weight rate of flow,  ˙ w ,  in  the annulus from the bottom  of the well
                                                           t
                               to the surface is (see Equation 6-13)
                                       ˙ w =  ˙ w  +  ˙ w  +  ˙ w
                                       t     g    f    s
                                   The Fanning friction factor f given in  the above equation is  determined by the
                               standard  fluid  mechanics  empirical  expressions  relating  the  friction  factor  to  the
                               Reynolds number, diameter (or hydraulic diameter), and absolute pipe roughness.  In
                               general, the values for Reynolds number, diameter, and absolute pipe roughness are
                               known.  The classic expression for the Reynolds number is
                                                  D
                                            ( D h  − ) V
                                                   p
                                      N   =                                            (6-89)
                                                 ν
                                        R
                                   There are three flow conditions that can exist in the annulus.  These are laminar,
                               transitional, and turbulent flow conditions.
                                   The  empirical  expression  for  the  Fanning  friction  factor  for  laminar  flow
                               conditions is
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