PRODUCTION PROBLEMS 1137 



tion Q may be expressed by the equation 



(5) The production is the same as that in a "dead well" ; that is, 



Q = c{Pr-p) (2) 



It follows from assumptions (1) and (2) and Equation 1 that the 



bottom-hole pressure is equal to the weight of the fluid, provided the 

 friction drop in the well bore is negligible ; that is, 



Phh = hd 



Or if pbh is set equal to p, 



p=pi>k = hd (10) 



On inserting this value of p into Equation 9 one obtains 



Equating the right-hand members of Equations 2 and 11, 



a dp 



The last equation may be written in the form 



where K is the equilibrium value of h and Hq is the value of h at the instant 

 chosen to represent f = 0. 



Combining Equations 12 and 13, 



Pr-p ^ K-h ^-^t 



Pr — Po K — ho 



or 



a , pr — po a . hr — ho /..x 



