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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 drainage volume 
    LaTeX Math Inline
    bodyV_{\phi, 1}
     of producer W1 and potentially other producers {W1 .. WN}


The drainage volume difference volumes 

LaTeX Math Inline
bodyV_{\phi, 0}
 and 
LaTeX Math Inline
bodyV_{\phi, 1}
 may not be equal and the difference
LaTeX Math Inline
body\delta V_{\phi} = V_{\phi, 0} - V_{\phi, 1} >0
 may be  maybe related to the fact that water injection injection in W20 is shared between
LaTeX Math Inline
bodyV_{\phi, 1}
 and another reservoir  or with another producerother drainage areas around other producers

Image Modified

Fig. 1. Well locations.

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

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