Theory

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

Key Features

Comprehensive Coverage

Previous (Honarpour 1982)Ibrahim-Koederitz (2000)
2 wettabilities4 wettabilities
No oil-wet equations✅ Oil-wet included
Water/oil + oil/gas only✅ + Gas/water + gas/condensate
651 data sets416 data sets (higher quality)

Modified Craig's Wettability Rules

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

WettabilitySwcSw @ crossoverkrw° @ (1-Sorw)
Strongly water-wet≥ 15%≥ 45%≤ 0.07
Water-wet≥ 10%≥ 45%0.07 < krw° ≤ 0.3
Intermediate-wet≥ 10% OR ≤ 15%45-55%0.3 < krw° < 0.5
Oil-wet≤ 15%≤ 55%≥ 0.5

Where:

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

Normalized Saturations

All equations use normalized saturations:

Oil-Water System

Sw∗=Sw−Swc1−Swc−SorwS_w^* = \frac{S_w - S_{wc}}{1 - S_{wc} - S_{orw}}

Where:

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

Gas-Oil System

Sg∗=Sg−Sgc1−Sgc−SlcS_g^* = \frac{S_g - S_{gc}}{1 - S_{gc} - S_{lc}} Sl∗=1−(Sg1−Slc)S_l^* = 1 - \left(\frac{S_g}{1 - S_{lc}}\right)

Where:

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

Oil-Water System: Sandstone

Strongly Water-Wet (Equations A1, A2)

Oil relative permeability:

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

Water relative permeability:

krw∗=0.0910(Sw∗)1.5−0.1841ϕ0.5(Sw∗)1.4−0.0002Sw∗kaSwc2.6k_{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} −1.1811Sorw3(Sw∗)1.5+0.6493Sorw5(Sw∗)2.5ϕ+2.1227ϕ6Sw∗+0.0138(ln⁡ka)Sw∗- 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:

krow∗=1−0.7233Sw∗−1.7721(Sw∗)2+1.5041(Sw∗)3k_{row}^* = 1 - 0.7233 S_w^* - 1.7721 (S_w^*)^2 + 1.5041 (S_w^*)^3

Water relative permeability:

krw∗=0.2848SorwSw∗−0.0325Sw∗+0.0711(Sw∗)1.5−2.2462SwcSorw1.7(Sw∗)2k_{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:

krow∗=1−3.091Sw∗+2.8670(Sw∗)1.6−0.7690(Sw∗)2k_{row}^* = 1 - 3.091 S_w^* + 2.8670 (S_w^*)^{1.6} - 0.7690 (S_w^*)^2

Water relative permeability (complex, 9 terms):

krw∗=0.2212(Sw∗)1.6+0.2493Sorw2(Sw∗)3ϕ+21.37(Sw∗)2Sorw5+…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:

krow∗=1−2.6525Sw∗+2.4721(Sw∗)2−0.8144(Sw∗)3k_{row}^* = 1 - 2.6525 S_w^* + 2.4721 (S_w^*)^2 - 0.8144 (S_w^*)^3

Water relative permeability:

krw∗=0.1164(Sw∗)4+2.6696(Sw∗)0.8(SwcSorw)2+0.4754Sw∗(Swcln⁡ka)2+…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)

Oil-Water System: Carbonate

Strongly Water-Wet (Equations A9, A10)

Oil relative permeability:

krow∗=1−2.9858Sw∗+3.1548(Sw∗)2−1.1715(Sw∗)3k_{row}^* = 1 - 2.9858 S_w^* + 3.1548 (S_w^*)^2 - 1.1715 (S_w^*)^3

Water relative permeability:

krw∗=0.2442Sw∗−0.3551(Sw∗)2+0.5118(Sw∗)3k_{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:

krow∗=1−4.9854Sw∗+21.322(Sw∗)2−29.046(Sw∗)2.5+11.724(Sw∗)3k_{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):

krw∗=0.4669(ln⁡ka)SwcSw∗−0.0590Sw∗−0.1939Sw∗ϕ1.5(Sorwln⁡ka)2+…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:

krow∗=1−3.2547Sw∗+3.8177(Sw∗)2−1.5632(Sw∗)3k_{row}^* = 1 - 3.2547 S_w^* + 3.8177 (S_w^*)^2 - 1.5632 (S_w^*)^3

Water relative permeability:

krw∗=0.3643Sw∗−0.7458(Sw∗)2+1.0609(Sw∗)3k_{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:

krow∗=1−8.6103Sw∗+87.942(Sw∗)2−207.04(Sw∗)2.5+187.10(Sw∗)3−60.389(Sw∗)3.5k_{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:

krw∗=0.2179(Sw∗)0.4+0.0054(ln⁡ka)2(Sw∗)0.5−9.7494(Sw∗)0.7(SorwSwc)2+…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)

Gas-Oil System

Sandstone (Equations A17, A18)

Oil relative permeability:

krog∗=0.1599Sl∗−1.0455(Sl∗)2+4.0844(Sl∗)3−5.4142(Sl∗)4+3.2103(Sl∗)5k_{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:

krg∗=0.9397(Sg∗)2−0.7742Sorg(Sg∗)2−1.2163Swc(Sg∗)2+1.1628ϕ(Sg∗)2−1.2482Sgc(Sg∗)2k_{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:

krog∗=4.4659(Sl∗)2−0.2528Sl∗−22.936(Sl∗)3+53.001(Sl∗)4−55.199(Sl∗)5+21.918(Sl∗)6k_{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:

krg∗=0.3297(Sg∗)2−0.0017ka(Sg∗)2+2.0568SwcSg∗+1.2314ϕ(Sg∗)2k_{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)

Gas-Water System (Equation A21, A22)

Gas relative permeability:

krgw∗=1.3047Sg∗−8.1596(Sg∗)2+25.510(Sg∗)3−31.538(Sg∗)4+13.884(Sg∗)5k_{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:

krw∗=0.9456Sl∗−1.2967(Sl∗)1.7+1.6959(Sl∗)3−0.0425Sgc(ln⁡ka)3(Sl∗)5+…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: krgw∗k_{rgw}^* normalized to krg = 1.0 at Swc (not at Slc = 0).

Gas-Condensate System (Equations A23, A24)

Condensate relative permeability:

krcg∗=0.1194Sl∗−0.0892(Sl∗)2+0.9607(Sl∗)3k_{rcg}^* = 0.1194 S_l^* - 0.0892 (S_l^*)^2 + 0.9607 (S_l^*)^3

Gas relative permeability:

krg∗=3.3393(Sg∗)1.2+6.7567(Sg∗)0.9Slc1.2−20.927Sg∗Slc2Sgc+101.65Sg∗Sgc4−7.3836Slc0.5Sg∗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).

Applicability and Limitations

Validated Range (All Systems)

ParameterMinimumMaximum
Porosity5%35%
Permeability0.01 md10,000 md
Swc0%45%
Sorw5%65%
Org4%60%

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

Functions Covered

Oil-Water System: Sandstone (8 functions)

FunctionWettabilityPhase
KrowIKSandStrongWaterWetStrongly WWOil (A1)
KrwIKSandStrongWaterWetStrongly WWWater (A2)
KrowIKSandWaterWetWater-wetOil (A3)
KrwIKSandWaterWetWater-wetWater (A4)
KrowIKSandInterWetIntermediateOil (A5)
KrwIKSandInterWetIntermediateWater (A6)
KrowIKSandOilWetOil-wetOil (A7)
KrwIKSandOilWetOil-wetWater (A8)

Oil-Water System: Carbonate (8 functions)

FunctionWettabilityPhase
KrowIKCarbStrongWaterWetStrongly WWOil (A9)
KrwIKCarbStrongWaterWetStrongly WWWater (A10)
KrowIKCarbWaterWetWater-wetOil (A11)
KrwIKCarbWaterWetWater-wetWater (A12)
KrowIKCarbInterWetIntermediateOil (A13)
KrwIKCarbInterWetIntermediateWater (A14)
KrowIKCarbOilWetOil-wetOil (A15)
KrwIKCarbOilWetOil-wetWater (A16)

Gas-Oil System (4 functions)

FunctionRock TypePhase
KrogIKGasOilSandSandstoneOil (A17)
KrgIKGasOilSandSandstoneGas (A18)
KrogIKGasOilCarbCarbonateOil (A19)
KrgIKGasOilCarbCarbonateGas (A20)

Gas-Water and Gas-Condensate (4 functions)

FunctionSystemPhase
KrgwIKGasWaterGas-waterGas (A21)
KrwIKGasWaterGas-waterWater (A22)
KrgIKGasCondGas-condensateGas (A24)
KrcgIKGasCondGas-condensateCondensate (A23)

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.

Special Core Analysis
special core analysisrelative permeabilityIbrahim-Koederitzlinear regressionsandstonecarbonatewater-wetstrongly water-wetintermediate-wetoil-wetoil-watergas-oilgas-watergas-condensate

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