Ibrahim-Koederitz Relative Permeability Correlations

Overview

The Ibrahim and Koederitz (2000) correlations represent the most comprehensive empirical relative permeability prediction tool available, covering:

• 4 wettability states: Strongly water-wet, water-wet, intermediate-wet, oil-wet
• 2 rock types: Sandstone and carbonate
• 4 fluid systems: Oil-water, gas-oil, gas-water, gas-condensate
• 416 data sets from SPE literature and industry sources (1950-1998)

This represents a significant expansion over earlier correlations (Honarpour 1982: 651 sets covering only 2 wettabilities).

Applications

• Reservoir simulation — When core kr data unavailable
• Sensitivity analysis — Understanding wettability effects
• Regional correlations — Comparing with local core measurements
• Screening studies — Preliminary waterflood or gas injection design

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Key Features

Comprehensive Coverage

Modified Craig's Wettability Rules

Ibrahim-Koederitz refined Craig's rules to categorize kr curves:

Where:

• Swc = connate water saturation, %
• Crossover = saturation where krw = kro
• krw° = endpoint water relative permeability

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Normalized Saturations

All equations use normalized saturations:

Oil-Water System

$$S_w^* = \frac{S_w - S_{wc}}{1 - S_{wc} - S_{orw}}$$

Where:

• $S_{wc}$ = critical (connate) water saturation
• $S_{orw}$ = residual oil saturation (oil-water system)

Gas-Oil System

$$S_g^* = \frac{S_g - S_{gc}}{1 - S_{gc} - S_{lc}}$$$$S_l^* = 1 - \left(\frac{S_g}{1 - S_{lc}}\right)$$

Where:

• $S_{gc}$ = critical gas saturation
• $S_{lc}$ = critical liquid saturation (= $S_{wc} + S_{org}$)
• $S_{org}$ = residual oil saturation (gas-oil system)

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Oil-Water System: Sandstone

Strongly Water-Wet (Equations A1, A2)

Oil relative permeability:

$$k_{row}^* = 1 - 2.1529 S_w^* + 0.6389 (S_w^*)^2 + 1.4345 (S_w^*)^3 - 0.9197 (S_w^*)^4$$

Water relative permeability:

$$k_{rw}^* = 0.0910 (S_w^*)^{1.5} - 0.1841 \phi^{0.5} (S_w^*)^{1.4} - 0.0002 S_w^* k_a S_{wc}^{2.6}$$$$- 1.1811 S_{orw}^3 (S_w^*)^{1.5} + 0.6493 \frac{S_{orw}^5 (S_w^*)^{2.5}}{\phi} + 2.1227 \phi^6 S_w^* + 0.0138 (\ln k_a) S_w^*$$

Data basis: R² = 0.918 (krow), R² = 0.908 (krw)

Water-Wet (Equations A3, A4)

Oil relative permeability:

$$k_{row}^* = 1 - 0.7233 S_w^* - 1.7721 (S_w^*)^2 + 1.5041 (S_w^*)^3$$

Water relative permeability:

$$k_{rw}^* = 0.2848 S_{orw} S_w^* - 0.0325 S_w^* + 0.0711 (S_w^*)^{1.5} - 2.2462 S_{wc} S_{orw}^{1.7} (S_w^*)^2$$

Data basis: R² = 0.935 (krow), R² = 0.815 (krw)

Intermediate-Wet (Equations A5, A6)

Oil relative permeability:

$$k_{row}^* = 1 - 3.091 S_w^* + 2.8670 (S_w^*)^{1.6} - 0.7690 (S_w^*)^2$$

Water relative permeability (complex, 9 terms):

$$k_{rw}^* = 0.2212 (S_w^*)^{1.6} + 0.2493 \frac{S_{orw}^2 (S_w^*)^3}{\phi} + 21.37 (S_w^*)^2 S_{orw}^5 + \ldots$$

Data basis: R² = 0.929 (krow), R² = 0.873 (krw)

Oil-Wet (Equations A7, A8)

Oil relative permeability:

$$k_{row}^* = 1 - 2.6525 S_w^* + 2.4721 (S_w^*)^2 - 0.8144 (S_w^*)^3$$

Water relative permeability:

$$k_{rw}^* = 0.1164 (S_w^*)^4 + 2.6696 (S_w^*)^{0.8} (S_{wc} S_{orw})^2 + 0.4754 S_w^* (S_{wc} \ln k_a)^2 + \ldots$$

Data basis: R² = 0.945 (krow), R² = 0.862 (krw)

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Oil-Water System: Carbonate

Strongly Water-Wet (Equations A9, A10)

Oil relative permeability:

$$k_{row}^* = 1 - 2.9858 S_w^* + 3.1548 (S_w^*)^2 - 1.1715 (S_w^*)^3$$

Water relative permeability:

$$k_{rw}^* = 0.2442 S_w^* - 0.3551 (S_w^*)^2 + 0.5118 (S_w^*)^3$$

Data basis: R² = 0.942 (krow), R² = 0.865 (krw)

Water-Wet (Equations A11, A12)

Oil relative permeability:

$$k_{row}^* = 1 - 4.9854 S_w^* + 21.322 (S_w^*)^2 - 29.046 (S_w^*)^{2.5} + 11.724 (S_w^*)^3$$

Water relative permeability (complex, 7 terms):

$$k_{rw}^* = 0.4669 (\ln k_a) S_{wc} S_w^* - 0.0590 S_w^* - 0.1939 S_w^* \phi^{1.5} (S_{orw} \ln k_a)^2 + \ldots$$

Data basis: R² = 0.956 (krow), R² = 0.891 (krw)

Intermediate-Wet (Equations A13, A14)

Oil relative permeability:

$$k_{row}^* = 1 - 3.2547 S_w^* + 3.8177 (S_w^*)^2 - 1.5632 (S_w^*)^3$$

Water relative permeability:

$$k_{rw}^* = 0.3643 S_w^* - 0.7458 (S_w^*)^2 + 1.0609 (S_w^*)^3$$

Data basis: R² = 0.921 (krow), R² = 0.899 (krw)

Oil-Wet (Equations A15, A16)

Oil relative permeability:

$$k_{row}^* = 1 - 8.6103 S_w^* + 87.942 (S_w^*)^2 - 207.04 (S_w^*)^{2.5} + 187.10 (S_w^*)^3 - 60.389 (S_w^*)^{3.5}$$

Water relative permeability:

$$k_{rw}^* = 0.2179 (S_w^*)^{0.4} + 0.0054 (\ln k_a)^2 (S_w^*)^{0.5} - 9.7494 (S_w^*)^{0.7} (S_{orw} S_{wc})^2 + \ldots$$

Data basis: R² = 0.975 (krow), R² = 0.905 (krw)

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Gas-Oil System

Sandstone (Equations A17, A18)

Oil relative permeability:

$$k_{rog}^* = 0.1599 S_l^* - 1.0455 (S_l^*)^2 + 4.0844 (S_l^*)^3 - 5.4142 (S_l^*)^4 + 3.2103 (S_l^*)^5$$

Gas relative permeability:

$$k_{rg}^* = 0.9397 (S_g^*)^2 - 0.7742 S_{org} (S_g^*)^2 - 1.2163 S_{wc} (S_g^*)^2 + 1.1628 \phi (S_g^*)^2 - 1.2482 S_{gc} (S_g^*)^2$$

Data basis: R² = 0.941 (krog), R² = 0.903 (krg)

Carbonate (Equations A19, A20)

Oil relative permeability:

$$k_{rog}^* = 4.4659 (S_l^*)^2 - 0.2528 S_l^* - 22.936 (S_l^*)^3 + 53.001 (S_l^*)^4 - 55.199 (S_l^*)^5 + 21.918 (S_l^*)^6$$

Gas relative permeability:

$$k_{rg}^* = 0.3297 (S_g^*)^2 - 0.0017 k_a (S_g^*)^2 + 2.0568 S_{wc} S_g^* + 1.2314 \phi (S_g^*)^2$$

Data basis: R² = 0.967 (krog), R² = 0.851 (krg)

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Gas-Water System (Equation A21, A22)

Gas relative permeability:

$$k_{rgw}^* = 1.3047 S_g^* - 8.1596 (S_g^*)^2 + 25.510 (S_g^*)^3 - 31.538 (S_g^*)^4 + 13.884 (S_g^*)^5$$

Water relative permeability:

$$k_{rw}^* = 0.9456 S_l^* - 1.2967 (S_l^*)^{1.7} + 1.6959 (S_l^*)^3 - 0.0425 S_{gc} (\ln k_a)^3 (S_l^*)^5 + \ldots$$

Data basis: R² = 0.965 (krgw), R² = 0.921 (krw)

Note: $k_{rgw}^*$ normalized to krg = 1.0 at Swc (not at Slc = 0).

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Gas-Condensate System (Equations A23, A24)

Condensate relative permeability:

$$k_{rcg}^* = 0.1194 S_l^* - 0.0892 (S_l^*)^2 + 0.9607 (S_l^*)^3$$

Gas relative permeability:

$$k_{rg}^* = 3.3393 (S_g^*)^{1.2} + 6.7567 (S_g^*)^{0.9} S_{lc}^{1.2} - 20.927 S_g^* S_{lc}^2 S_{gc} + 101.65 S_g^* S_{gc}^4 - 7.3836 S_{lc}^{0.5} S_g^*$$

Data basis: R² = 0.932 (krcg), R² = 0.878 (krg)

Normalization: Same as gas-oil (krl = 1.0 at Sg = 0, krg = 1.0 at Slc).

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Applicability and Limitations

Validated Range (All Systems)

Strengths

1. Most comprehensive coverage (4 wettabilities × 2 rock types × 4 systems)
2. Includes oil-wet equations (missing from Honarpour)
3. High R² values (0.815 to 0.975, most > 0.90)
4. Validated against 416 data sets from worldwide sources
5. Standardized normalization procedure

Limitations

1. Complex equations — Many terms, difficult for hand calculation
2. Input requirements — Need φ, ka, Swc, Sor for some equations
3. Wettability determination — Must classify using Craig's rules or measurement
4. Room temperature data — High-temperature effects not included
5. Imbibition focus — Oil-water equations for water displacing oil

When to Use Ibrahim-Koederitz

Best for:

• Unknown wettability → Use all 4 wettability curves for sensitivity
• Oil-wet reservoirs → Only comprehensive correlation available
• Gas-water or gas-condensate systems → Unique equations
• Comparison with Honarpour → Validate against 2 correlations

Alternatives:

• Honarpour (1982) — Simpler equations, fewer terms
• Corey — Analytical power-law model
• LET — Three-parameter flexible model
• Laboratory — Always preferred when available

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Related Documentation

• SCAL Overview — Correlation selection guide
• Honarpour Correlations — Earlier, simpler equations
• Corey and LET Models — Analytical kr models
• PVT Properties — Fluid properties for multiphase flow

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References

1. Ibrahim, M.N.M. and Koederitz, L.F. (2000). "Two-Phase Relative Permeability Prediction Using a Linear Regression Model." SPE-65631-MS, presented at SPE Eastern Regional Meeting, Morgantown, WV, 17-19 October.

2. Craig, F.F. Jr. (1971). The Reservoir Engineering Aspects of Waterflooding. Monograph Series, SPE, Richardson, TX. Vol. 3.

3. Honarpour, M., Koederitz, L.F., and Harvey, A.H. (1982). "Empirical Equations for Estimating Two-Phase Relative Permeability in Consolidated Rock." JPT, 34(12), pp. 2905-2908.

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