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:

$$q_o = J (p_R - p_{wf})$$

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.

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Vogel's Dimensionless IPR Equation

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

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

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

Vogel Dimensionless IPR Curve

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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.

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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:

$$q_{o,max} = J \left(p_R - p_b\right) + \frac{J p_b}{1.8}$$

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).

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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.

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Extensions and Modifications

Composite IPR (Above and Below Bubble Point)

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

$$q_o = \begin{cases} J(p_R - p_b) + \frac{J p_b}{1.8}\left[1 - 0.2\frac{p_{wf}}{p_b} - 0.8\left(\frac{p_{wf}}{p_b}\right)^2\right] & p_{wf} \leq p_b \\ J(p_R - p_{wf}) & p_{wf} > p_b \end{cases}$$

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

Wiggins Modification (Water Drive)

For reservoirs with partial water drive:

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

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).

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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

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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.