PRODUCTION PROBLEMS 1131 



of casing-head pressure at a constant rate of production. If the rate of 

 production is held constant, the bottom-hole pressure ptn is constant. After 

 equilibrium has been established, the rate of production is normally unaf- 

 fected by the value of the casing-head pressure because the rate of pro- 

 duction is dependent only upon the difference in pressure between the 

 bottom-hole producing pressure and the formation pressure. That is, for 

 any given rate of production, a change in casing-head pressure causes a 

 corresponding change in fluid elevation and after equilibrium has been 

 established there is no resultant change in bottom-hole pressure. Hence, 

 for a constant rate of production. Equation 1 may be written in the form : 



p + hd = pb^ = constant ( 5 ) 



where p equals pch + pg. 



When Equation 5 is solved for h explicitly, 



h = -^ ^ = height above datum plane 



Diflferentiation of the last equation with respect to p yields 



dh^_d_ /hA _ d_ /j\ 

 dp dp\d) dp \d J 



pbji is constant. Hence,— (pu) = and the last equation becomes 



dh__ hnA. /^N _ J_ 4- J_J_ r^\ 

 dp ~ d^ dp ^"^^ d d^ dp ^"^^ 



Also, if the fluid density does not depend on the total gas pressure, 



i«=0 and |^ = -i- 

 or 



dh dh ^ '' 



Equation 6 is a well known relationship for static-fluid surfaces and other 

 equilibrium systems. The equation states that the fluid density or fluid- 

 pressure gradient is equal to minus the slope of the total gas pressure 

 versus fluid-level curve, provided the fluid density does not depend on 

 the total gas pressure.* 



In practice, an average value for the absolute magnitude of ^ may be 



* The density might be expected to vary with gas pressure, temperature, producing 

 gas-oil ratio, and diameter of casing — or annular spacing. Practically, it is usually 

 observed that the fluid density is relatively constant provided the casing diameter does 

 not vary. However, if there is a change in the size of the annulus, the fluid density 

 changes almost proportionately to the effective areas of the annulus. 



