It can be shown that the superposition of solutions given in Equation 4 can be 

 expressed as: 



Po~P w = —I ln^- - + 



* " i (to + O)' 11 , ^ 

 I In [- ^ 



In J (n-j) H + 6 L ij (5) 



If the number of sub-periods, n, is made indefinitely large, then Equation 5 

 becomes : 



to 



4ttM L ^ J dt J 



o 



provided dq/dt exists as a continuous function of £. Equation 6 is the solution 

 for non-constant production rate, q. The following case is considered as an 

 illustration : 



Let: dq/dt = a = constant, 



The integral in Equation 6 may be evaluated 



P -P W = JL- In (-±±JiS) a + (q n - qi ) 



4*kh L V d J 



where </ a = ^ + — is the average production rate. 



The case where dq/dt is a constant is a fair approximation to many DST 

 curves which have a varying production rate, q. The error may be expressed as 



v (qn - qi) * . t + 20 f t + 6 \ "1 



E =—4M I -IT ln \ )-\ 



(8) 



6l \ o j 



which is of the form: 



*= i± l^|> + 8 >'< 1 + T)-'] ;S = r (9) 



756 



