3 Phase IPR

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Three-phase Inflow Performance Relationship

3 Phase IPR Curve [1]

3 Phase IPR calculates IPR curve for oil wells producing water.

3 Phase IPR equation was derived by Petrobras based on combination of Vogel's IPR equation for oil flow and constant productivity for water flow [1].

3 Phase IPR curve is determined geometrically from those equations considering the fractional flow of oil and water [1].

Math and Physics

Total flow rate equations:

For Pb < Pwf < Pr

For pressures between reservoir pressure and bubble point pressure:

 q_t =J (P_r - P_{wf}) [1]

For PwfG < Pwf < Pb

For pressures between the bubble point pressure and the flowing bottom-hole pressures:

 q_t =\frac{-C+\sqrt{C^2-4B^2D}}{2B^2}\ for B \ne 0[1]
 q_t =D/C\ for B = 0[1]

where:

 A=\frac{P_{wf}+0.125F_oP_b-F_wP_r}{0.125F_oP_b}[1]
 B=\frac{F_w}{0.125F_oP_bJ}[1]
 C=2AB+\frac{80}{q_{o_{max}}-q_b}[1]
 D=A^2-80\frac{q_b}{q_{o_{max}}-q_b}-81[1]

For 0 < Pwf < PwfG

 q_t =\frac{P_{wfG}+q_{o_{max}}tan(\beta)-P_{wf}}{tan(\beta)}[1]

where:

 tan(\beta) = CD/CG [1]
 CD = F_w\frac{0.001q_{o_{max}}}{J}+F_o0.125P_b \left ( -1+\sqrt{81-80 \frac{0.999q_{o_{max}}-q_b}{q_{o_{max}}-q_b}} \right)[1]
 CG = 0.001 q_{o_{max}}[1]

And

 P_{wfG}=F_w \left ( P_r - \frac{q_{o_{max}}}{J}\right )[1]
 q_{o_{max}}=q_b+\frac{JP_b}{1.8}[1]

IPR calculator software

Nomenclature

 B = formation volume factor, bbl/stb
 J_D = dimensionless productivity index, dimensionless
 kh = permeability times thickness, md*ft
 \bar{P} = average reservoir pressure, psia
 P_{\bar{P}} = average reservoir pseudopressure, psia2/cP
 P_{wf} = well flowing pressure, psia
 P_{P_{wf}} = average well flowing pseudopressure, psia2/cP
 q = flowing rate, stb/d
 q_g = gas rate, MMscfd
 T = temperature, °R

Greek symbols

 \mu = viscosity, cp

References

  1. 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 Brown, Kermit (1984). The Technology of Artificial Lift Methods. Volume 4. Production Optimization of Oil and Gas Wells by Nodal System Analysis. Tulsa, Oklahoma: PennWellBookss. 

See also

141.2 derivation
Darcy's law
JD
Production Potential