PRODUCTION PROBLEMS 1123 



A knowledge of producing zone pressures is a prime requisite for the 

 solution of numerous petroleum reservoir analysis and producing well 

 problems. For the former purpose the shut-in or static pressure (zero rate 

 of production) is generally of most importance, while for the latter, the 

 major concern usually involves the establishment of a relation between 

 pressure and rate of production. It is occasionally found that the deter- 

 mination of a shut-in pressure may not be feasible, in which event the use 

 of an extrapolation of the pressure-production rate relationship to a zero 

 rate of production can be resorted to. 



The rate of production Q depends on the difference between the reser- 

 voir pressure pr, and a "friction pressure" pf, plus the bottom-hole pressure 

 pin- At moderate rates of flow, the rate of production Q may be expressed 

 in the form 



Q = C[Pr-{Pf + P,n)] 

 where 



Q = rate of production 



C = a constant of proportionality 



pr — pressure of fluid in the formation at zero production 



pf = "friction pressure." This so-called friction pressure is a back 

 pressure in the formation and is a measure of the resistance 

 which the fluid must overcome while flowing through the oil 

 sand into the hole. 



If it is assumed that the "friction pressure" is proportional to the rate 

 of production (/>/= kQ, where k is a constant), the equation given above 

 for Q may be written in the form 



Q = Cpr-CkQ-Cp^n 

 On collecting the terms involving Q, one obtains 



Q(l + Ck)=C(Pr-p,n) 

 or 



Q = c(pr-Pi>n) (2) 



c 



where c = ., , „, = a constant. 



1 "T Ck 



The assumption that the pressure />/ varies linearly with the rate of 

 production leads to a theoretical production rate that is always greater 

 than the physical pumping rate. This is especially true at the higher rates 

 of production where turbulent flow occurs in the vicinity of the well. Thus, 

 at production rates approaching zero and also at high rates, where a trans- 

 ition takes place from stream-line to turbulent flow in the stratum adjacent 

 the hole, Equation 2 may break down. However, in problems con- 

 cerned with the theoretical maximum rates of production, it is customary 

 to assume that the assumptions made in deriving Equation 2 are fulfilled 

 approximately, and this convention will be followed here. 



