IPR Vogel Darcy Fetkovich Formula

The Inflow Performance Relationship (IPR) describes the relationship between flowing bottomhole pressure (Pwf) and production rate (q) for a well. It defines how much fluid the reservoir can deliver to the wellbore at various drawdown pressures. IPR curves are essential for nodal analysis, artificia...

Overview

Theory

The shape of the IPR curve depends on whether flow is single-phase (above bubble point) or two-phase (below bubble point):

Single-phase (Darcy): Linear relationship — q = J × (Pr - Pwf)
Two-phase (Vogel): Curved relationship due to gas liberation reducing effective oil permeability
Empirical (Fetkovich): Multi-rate test-based, captures non-Darcy effects

Formulas

Vogel's IPR (1968) — Saturated Oil Reservoirs

q/qmax = 1 - 0.2*(Pwf/Pr) - 0.8*(Pwf/Pr)^2

Rearranging for qmax from a single test point (qtest, Pwf_test):

qmax = qtest / (1 - 0.2*(Pwf_test/Pr) - 0.8*(Pwf_test/Pr)^2)

Absolute Open Flow (AOF): q at Pwf = 0 → AOF = qmax

Darcy / Productivity Index (PI)

q = J * (Pr - Pwf)

where J = productivity index (STB/d/psi). From a test:

J = qtest / (Pr - Pwf_test)

AOF = J × Pr

Composite (Standing's Extension)

Above bubble point: q = J × (Pr - Pwf)

Below bubble point:

qb = J * (Pr - Pb)
qVogel_max = J * Pb / 1.8
q = qb + qVogel_max * (1 - 0.2*(Pwf/Pb) - 0.8*(Pwf/Pb)^2)

Fetkovich (1973) — Empirical

q = C * (Pr^2 - Pwf^2)^n

where C = flow coefficient, n = deliverability exponent (0.5 ≤ n ≤ 1.0).

From a test point:

C = qtest / (Pr^2 - Pwf_test^2)^n

When n = 1.0, Fetkovich reduces to a simplified Vogel-like form. When n = 0.5, it captures high-turbulence non-Darcy effects.

Worked Example

Given: Pr = 3,000 psia, test rate = 400 STB/d at Pwf = 2,000 psia, Pb = 2,500 psia.

Vogel (assume Pr < Pb, fully saturated):

qmax = 400 / (1 - 0.2*(2000/3000) - 0.8*(2000/3000)^2)
     = 400 / (1 - 0.1333 - 0.3556)
     = 400 / 0.5111
     = 783 STB/d (AOF)

Rate at Pwf = 1,000 psia:

q = 783 * (1 - 0.2*(1000/3000) - 0.8*(1000/3000)^2)
  = 783 * (1 - 0.0667 - 0.0889)
  = 783 * 0.8444
  = 661 STB/d

Fetkovich (n = 0.85):

C = 400 / (3000^2 - 2000^2)^0.85
  = 400 / (5e6)^0.85
  = 400 / 1,412,537
  = 2.83e-4
AOF = C * (3000^2)^0.85 = 2.83e-4 * (9e6)^0.85 = 717 STB/d

Valid Ranges

Model	Application	Limitations
Vogel	Saturated oil (Pr ≤ Pb), solution-gas drive	Not for gas wells, horizontal wells, or wells with significant skin
Darcy/PI	Undersaturated oil (Pwf > Pb)	Only valid for single-phase liquid flow
Composite	Pr > Pb but Pwf may drop below Pb	Most realistic for many field scenarios
Fetkovich	Any reservoir with multi-rate test data	Requires n from testing; n outside 0.5–1.0 is suspect

References

Vogel, J.V. (1968). "Inflow Performance Relationships for Solution-Gas Drive Wells." JPT, 20(1), 83–92.
Fetkovich, M.J. (1973). "The Isochronal Testing of Oil Wells." SPE-4529.
Standing, M.B. (1970). "Inflow Performance Relationships for Damaged Wells Producing by Solution-Gas Drive." JPT, 22(11), 1399–1400.
PetroWiki — Inflow performance: https://petrowiki.spe.org/Inflow_performance

IPR Vogel Darcy Fetkovich