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titlePIR equation deduction



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VRR = \frac{B_w \, q_{WI}}{B_w \, q_W + B_Oo \, q_O + B_g \, [ q_G - R_s \, q_O] } =  \frac{B_w \, q_{WI}}{B_w \, q_W + B_Oo \, q_O + B_g \, [ GOR - R_s] q_O } = \frac{B_w \, q_{WI}}{B_w \, q_W + [ B_o  + B_g \, ( GOR - R_s) ] \, q_O }
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VRR = \frac{q_{WI}}{q_W + \bigg[ \frac{B_o}{B_w}  + \frac{B_g}{B_w} \, ( GOR - R_s) \bigg] \, q_O }
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Y_w=\frac{q_W}{q_W + q_O} \rightarrow \frac{q_O}{q_W} = \frac{1-Y_w}{Y_w} \rightarrow q_W =  \frac{Y_w}{1-Y_w} \, q_O
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VRR =  \frac{q_{WI}}{q_O} \frac{1}{\frac{Y_w}{1-Y_w}  + \bigg[ \frac{B_o}{B_w}  + \frac{B_g}{B_w} \, ( GOR - R_s) \bigg] } =
\frac{q_{WI}}{q_O} \frac{1-Y_w}{Y_w  + (1-Y_w) \, \bigg[ \frac{B_o}{B_w}  + \frac{B_g}{B_w} \, ( GOR - R_s) \bigg] }



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PIR=\frac{q_W}{q_{WI}} = \frac{1}{VRR} \, \frac{1-Y_w}{Y_w  + (1-Y_w) \, \bigg[ \frac{B_o}{B_w}  + \frac{B_g}{B_w} \, ( GOR - R_s) \bigg] }

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