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  • producing well W1 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 W0 with total sandface flowrate 

    LaTeX Math Inline
    bodyq_0(t) <0
    , supporting pressure in reservoir volume
    LaTeX Math Inline
    bodyV_{\phi, 0}
     which includes  


The injection drainage volume  

LaTeX Math Inline
bodyV_{\phi, 0}
 includes the drainage volume 
LaTeX Math Inline
bodyV_{\phi, 1}

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of producer W1

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 and may be equal to it 

The drainage volumes 
LaTeX Math Inline
bodyV_{\phi, 0}
 and 
LaTeX Math Inline
body= V_{\phi, 1}
 may not be equal and the difference or may be bigger  
LaTeX Math Inline
body\delta V_{\phi} = V_{\phi, 0} - > V_{\phi, 1} >0
 maybe related to the fact that water injection in W0 is shared between
 in case injector W0 supports other producers {W1 .. WN}: 
LaTeX Math Inline
bodyV_{\phi, 0} = \sum_{k=1}^N V_{\phi, k}
 and other drainage areas around other producers. 



Image Removed

Fig. 1. Well locations Location map of injector-producer pairing with 4 producers {W1, W2, W3W4} and one injector W0.

Case #1 –  Constant flowrate production: 
LaTeX Math Inline
bodyq_1 = \rm const >0

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