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Consider a well-reservoir system consisting of:

  • producing well W1 draining  with total sandface flowrate 
    LaTeX Math Inline
    bodyq_1(t)>0
     and bhp 
    LaTeX Math Inline
    bodyp_1(t)>0
    , draining the reservoir volume
    LaTeX Math Inline
    bodyV_{\phi, 1}
     
  • water injecting well W2 supporting  with total sandface flowrate 
    LaTeX Math Inline
    bodyq_2(t) <0
    , supporting pressure in reservoir volume
    LaTeX Math Inline
    bodyV_{\phi, 2}
     which includes the drainage volume 
    LaTeX Math Inline
    bodyV_{\phi, 1}
     of producer W1 and potentially other producers. 

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Case #2 – Constant BHP: 
LaTeX Math Inline
bodyp_1 = \rm const


Assume that the flowrate flowrate in producer W1 is being automatically adjusted by

LaTeX Math Inline
body\delta q_1(t)
in producer W1 is being adjusted to  to compensate the bottom-hole pressure variation 
LaTeX Math Inline
body\delta p_1(t)
in response to the the  total sandface flowrate variation 
LaTeX Math Inline
body\delta q_2
 in injector W2 so that bottom-hole pressure in producer W1 stays constant at all times
LaTeX Math Inline
body\delta p_1(t) = \delta p_1 = \rm const
. In petroleum practice this happens when the formation is capable to deliver more fluid than the current lift settings in producer so that the bottom-hole pressure in producer is constantly kept at minimum value defined by the lift design..

In this case, flowrate response 

LaTeX Math Inline
body\delta q_1
in producer W1 to the flowrate variation 
LaTeX Math Inline
body\delta q_2
 in injector W2 is going to be:

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