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Consider a well-reservoir system consisting of:

• producing well W₁ with total sandface flowrate 

  LaTeX Math Inline
  body q_1(t)>0

   and bhp  and BHP 

  LaTeX Math Inline
  body p_1(t)>0

  , draining the reservoir volume

  LaTeX Math Inline
  body V_{\phi, 1}

   
  
  
• water injecting well W₂ with total sandface flowrate 

  LaTeX Math Inline
  body q_2(t) <0

  , supporting pressure in reservoir volume

  LaTeX Math Inline
  body V_{\phi, 2}

   which includes the drainage volume 

  LaTeX Math Inline
  body V_{\phi, 1}

   of producer W₁ and potentially other producers. 

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