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title	PRIME Mathematics

Property Abbrevy

Property Name

Formula

VRR_cum

Cumulative Voidage Replacement Ratio

LaTeX Math Block

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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}

VRRcur

Current Voidage Replacement Ratio

(month over month)

LaTeX Math Block

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

RF

Recovery Factor

LaTeX Math Block

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

Y_w

Watercut (production)

LaTeX Math Block

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

Y_wm

Watercut (proxy-model)

LaTeX Math Block

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

LaTeX Math Block

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s_w = \frac{Q_o \, B_o}{V_\phi}

GOR

Gas-Oil Ratio (production)

LaTeX Math Block

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

GOR_m

Gas-Oil Ratio (proxy-model)

LaTeX Math Block

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

q_LIQ

Liquid rate

LaTeX Math Block

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q_{LIQ} = q_O + q_W

PIR

Production Injection Ratio (production)

LaTeX Math Block

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

PIR_m

Production Injection Ratio (model)

LaTeX Math Block

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{\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] } }

J_O

Oil Productivity Index

LaTeX Math Block

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

J_PI

Total Productivity Index (production)

LaTeX Math Block

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

J_PIm

Total Productivity Index (model)

LaTeX Math Block

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

_jPI

Total Specific Productivity Index

LaTeX Math Block

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

_jPIm

Total Specific Productivity Index (model)

LaTeX Math Block

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

Expand

title	Derivation of PIR equation

LaTeX Math Block

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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 }

LaTeX Math Block

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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 }

LaTeX Math Block

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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} \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] }

LaTeX Math Block

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PIR=\frac{q_W}{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] }

› PRIME Diagnostics

Sample Case 1 – Waterflood Sector Analysis

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Fig. 1.1. Production History Map	Fig. 1.2. Recovery Map


Fig. 2.1. Cross-section & PLT, permeability, GOC, OWC	Fig. 2.2. Cross-section & PLT, permeability, GOC, OWC


Fig. 3.1. Production History Graphs	Fig. 3.2. Production Forecasts


Fig. 4.1. Recovery History	Fig. 4.2. Recovery Forecasts


Fig. 5.1. WOR Diagnostic	Fig. 5.2.. GOR Diagnostic


Fig. 6. 71. Productivity Index Diagnostic	Fig. 6. 72. Injectivity Index Diagnostic


Fig. 67. Injection Efficiency Diagnostics

Sample Case 2 – Oil Producer Analysis

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Fig. 1. Production History Map	Fig. 2. Recovery Map	Fig. 3. Cross-section & PLT


Fig. 4. Production History Graphs	Fig. 5. Decline Curve Analysis	Fig. 6. Recovery Diagnostic


Fig. 7. Watercut Diagnostic	Fig. 8. GOR Diagnostic	Fig. 9. Injection Efficiency Diagnostics


Fig. 10. Well Performance Analysis (VFP + IPR)	Fig. 11. Productivity Index Diagnostic	Fig. 12. Well Completion & PLT

Sample Case 3 – Water Injector Analysis

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Fig. 1. Production History Map	Fig. 2. Recovery Map	Fig. 3. Cross-section & PLT


Fig. 4. Injection History Graphs


Fig. 10. Well Performance Analysis (VFP + IPR)	Fig. 11. Injectivity Index Diagnostic	Fig. 12. Well Completion & PLT