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{\rm VRR_{cum}} = \frac{B_w \, Q_{WI}}{B_w \, Q_W + B_o \, Q_O + B_g Q_G - B_g R_s Q_O}

{\rm VRR_{cur}} = \frac{B_w \, q_{WI}}{B_w \, q_W + B_o \, q_O + B_g (q_G - R_s Q_O)}

{\rm RF} = \frac{Q_O}{V_{STOIIP}}

{\rm Y_w} = \frac{q_W}{q_{LIQ}}

{\rm Y_{wm}} = \frac{1}{1 + \frac{K_{ro}}{K_{rw}} \cdot \frac{ \mu_w}{\mu_o}  \cdot \frac{B_w}{B_o} }

s_w = s_{wi} + (1-s_{wi}-s_{or}) \cdot \rm RF

{\rm GOR} = \frac{q_g}{q_o}

{\rm GOR_m} = R_s +  \frac{k_{rg}}{k_{ro}} 
\cdot \frac{\mu_o}{\mu_g} 
\cdot \frac{B_o }{B_g}

q_{LIQ} = q_O + q_W

{\rm PIR} = \frac{Q_O}{Q_{WI}}

{\rm PIR_m} = { \frac{1}{VRR} } \cdot { \frac{1-Y_w}{ Y_w + (1-Y_w) \bigg[ \frac{B_o}{B_w} - \frac{B_g}{B_w}(GOR - R_s) \bigg] } }

{\rm J_{O}} = \frac{q_O}{P_e - P_{wf}} {\quad \Rightarrow \quad} P_{wf} = P_e - \frac{1}{J_O} q_O

{\rm J_t} = \frac{q_t}{P_e - P_{wf}}

{\rm J_{tm} } = \frac{2 \pi \sigma}{\ln \frac{r_e}{r_w} +0.5 + S} 

{\rm j_t} = \frac{q_t}{h \cdot (P_e - P_{wf})}

{\rm j_{tm} } = \frac{2 \pi <k/\mu>}{\ln \frac{r_e}{r_w} +0.5 + S} 

VRR = \frac{B_w \, q_{WI}}{B_w \, q_W + B_o \, q_O + B_g \, [ q_G - R_s \, q_O] } =  \frac{B_w \, q_{WI}}{B_w \, q_W + B_o \, 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 }

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

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

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

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