Vogel Inflow Performance Relationship

Overview

The Vogel IPR (1968) is one of the most widely used correlations in petroleum engineering for predicting well performance when:

• Reservoir pressure is below bubble point (two-phase flow in reservoir)
• Drive mechanism is solution-gas drive
• Well is producing oil with dissolved gas

The Problem with Linear PI

For single-phase flow, the productivity index is constant:

But when pressure drops below bubble point:

• Gas evolves from solution in the reservoir
• Two-phase flow reduces oil mobility
• IPR becomes curved (non-linear)
• Productivity index is no longer constant

Vogel developed a dimensionless correlation to predict this curved IPR behavior.

Vogel's Dimensionless IPR Equation

Based on computer simulation of 21 reservoir conditions, Vogel derived:

Where:

• $q_o$ = oil production rate at bottom-hole pressure $p_{wf}$, STB/d
• $q_{o,max}$ = maximum (theoretical) rate at $p_{wf} = 0$ (abandoned), STB/d
• $p_{wf}$ = flowing bottom-hole pressure, psia
• $p_R$ = average reservoir pressure, psia

Physical Interpretation

Shape of curve:

• At $p_{wf} = p_R$ (well shut-in): $q_o = 0$
• As $p_{wf}$ decreases: rate increases, but not linearly
• At $p_{wf} = 0$ (theoretical): $q_o = q_{o,max}$

Curvature:

• Linear term (0.2): represents single-phase contribution
• Quadratic term (0.8): represents two-phase flow effect
• Stronger curvature than straight-line PI

Development Background

Computer Simulation Approach

Vogel used Weller's (1966) solution-gas drive reservoir simulation to calculate IPR curves for:

Key Finding

When IPR curves were plotted dimensionlessly ($q_o/q_{o,max}$ vs. $p_{wf}/p_R$), they all collapsed to a single curve shape, regardless of:

• Fluid properties
• Relative permeability characteristics
• Well spacing
• Time in reservoir life

Implication: A universal relationship exists for solution-gas drive IPR.

Using the Vogel Correlation

Method 1: Given One Test Point

If you have one stabilized well test (q₁, pwf₁) at current reservoir pressure pR:

1. Calculate qmax:

   $$q_{o,max} = \frac{q_1}{1 - 0.2(p_{wf,1}/p_R) - 0.8(p_{wf,1}/p_R)^2}$$

2. Predict rate at any pwf:

   $$q_o = q_{o,max} \left[1 - 0.2\frac{p_{wf}}{p_R} - 0.8\left(\frac{p_{wf}}{p_R}\right)^2\right]$$

Excel:

qmax = FlowRateSSVogel(q1, pwf1, pR, 0)
q_new = FlowRateSSVogel(qmax, pR, pR, pwf_new)

Method 2: Given Productivity Index Above Bubble Point

If reservoir pressure started above bubble point and you have:

• $J$ = productivity index measured above $p_b$
• Current $p_R < p_b$

Then:

Physical basis: Linear IPR above Pb, Vogel curve below Pb, matched at bubble point.

Method 3: Using Current Test with Future Forecast

Given test at (pR₁, pwf₁, q₁), predict future performance at pR₂:

1. Calculate current qmax: $q_{o,max,1}$ (Method 1)
2. Assume qmax changes proportionally to pressure: $$q_{o,max,2} = q_{o,max,1} \times \frac{p_{R,2}}{p_{R,1}}$$
3. Calculate new rate at pR₂, pwf₂ using Vogel equation

Caution: Assumes productivity doesn't change (no skin, permeability constant).

Applicability and Limitations

Valid When:

✅ Reservoir pressure below bubble point (two-phase flow)
✅ Solution-gas drive mechanism (no strong water/gas drive)
✅ Stabilized flow (transient effects minimal)
✅ Homogeneous reservoir (uniform properties near wellbore)
✅ Vertical well (not horizontal/deviated)
✅ Oil production (not gas or water wells)

Not Valid When:

❌ Pressure above bubble point → Use linear PI
❌ Strong water drive → Use modified Vogel or Fetkovich
❌ Gas cap drive → Use modified correlation
❌ High skin factor → IPR approaches straight line
❌ Horizontal wells → Use Bendakhlia-Aziz or others
❌ Gas wells → Use Darcy/non-Darcy equations
❌ Highly fractured → IPR may deviate

Accuracy Expectations

Best practice: Always validate with actual well tests when possible.

Extensions and Modifications

Composite IPR (Above and Below Bubble Point)

When $p_R > p_b$ but $p_{wf} < p_b$:

Use case: Reservoir initially above bubble point, now depleted below Pb.

Wiggins Modification (Water Drive)

For reservoirs with partial water drive:

Where $a$ varies from 0.2 (solution-gas) to 0.8 (strong water drive).

Standing Modification (Two-Phase)

Accounts for water production in IPR calculation (beyond scope here).

Functions Covered

The following functions implement Vogel's IPR for different flow regimes. See each function page for detailed parameter definitions, Excel syntax, and usage examples.

Note: Most applications use steady-state or pseudo-steady state. Transient Vogel is for buildup/drawdown analysis.

Related Documentation

• WellFlow Overview — Productivity models overview
• Productivity Index — Linear PI for single-phase
• Horizontal Wells — IPR for horizontal wells
• Gas Wells — Deliverability equations
• Vertical Flow Correlations — Bottomhole to wellhead

References

1. Vogel, J.V. (1968). "Inflow Performance Relationships for Solution-Gas Drive Wells." Journal of Petroleum Technology, 20(1), pp. 83-92. SPE-1476-PA.

2. Weller, W.T. (1966). "Reservoir Performance During Two-Phase Flow." Journal of Petroleum Technology, 18(2), pp. 240-246.

3. Standing, M.B. (1971). "Concerning the Calculation of Inflow Performance of Wells Producing from Solution Gas Drive Reservoirs." Journal of Petroleum Technology, 23(9), pp. 1141-1142.

4. Wiggins, M.L. (1994). "Generalized Inflow Performance Relationships for Three-Phase Flow." SPE Reservoir Engineering, 9(3), pp. 181-182.

5. Economides, M.J., Hill, A.D., Ehlig-Economides, C., and Zhu, D. (2013). Petroleum Production Systems, 2nd Edition. Upper Saddle River, NJ: Prentice Hall. Chapter 2: Inflow Performance.

6. Ahmed, T. (2019). Reservoir Engineering Handbook, 5th Edition. Cambridge, MA: Gulf Professional Publishing. Chapter 18: Oil Well Performance.