3 Phase IPR

Three-phase Inflow Performance Relationship

3 Phase IPR Curve

3 Phase IPR calculates IPR curve for oil wells producing water at various watercuts.

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 .

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

Math and Physics

Total flow rate equations:

For P_b < P_wf < P_r

For pressures between reservoir pressure and bubble point pressure:

$q_t =J (P_r - P_{wf})$

For P_wfG < P_wf < P_b

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

$q_t =\frac{-C+\sqrt{C^2-4B^2D}}{2B^2}\ for B \ne 0$

$q_t =D/C\ for B = 0$

where:

$A=\frac{P_{wf}+0.125F_oP_b-F_wP_r}{0.125F_oP_b}$

$B=\frac{F_w}{0.125F_oP_bJ}$

$C=2AB+\frac{80}{q_{o_{max}}-q_b}$

$D=A^2-80\frac{q_b}{q_{o_{max}}-q_b}-81$

For 0 < P_wf < P_wfG

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

where:

$tan(\beta) = CD/CG$

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

$CG = 0.001 q_{o_{max}}$

And

$P_{wfG}=F_w \left ( P_r - \frac{q_{o_{max}}}{J}\right )$

$q_{o_{max}}=q_b+\frac{JP_b}{1.8}$

3 Phase IPR calculation example

Following the example problem #21, page 33 :

Given:

$P_r$ = 2550 psi

$P_b$ = 2100 psi

Test data:

$P_{wf}$ = 2300 psi

$q_t$ = 500 b/d

Calculate:

Determine the 3 Phase IPR curves for F_w=0, 0.25, 0.5, 0.75, and 1.

Solution:

The problem was run through PQplot software for different values of watercut.

Result 3 Phase IPR curves are shown on Fig.1. Points indicate results obtained by Brown .

The PQplot model from this example is available online by the following link: 3 Phase IPR calculation example

Nomenclature

$A, B, C, D, tan(\beta), CD, CG$ = calculation variables

$F_o$ = oil fraction, fraction

$F_w$ = water fraction, fraction

$J$ = productivity index, stb/d/psia

$P$ = pressure, psia

$q$ = flowing rate, stb/d

Subscripts

b = at bubble point

max = maximum

o = oil

r = reservoir

t = total

wf = well flowing bottomhole pressure

wfG = well flowing bottomhole pressure at point G

See also

IPR

Vogel's IPR

Darcy's law