Corey and LET Relative Permeability Models

Overview

Analytical relative permeability models provide mathematical formulations for kr curves without requiring empirical correlations. These models are widely used when:

• Laboratory data unavailable — screening studies and concept evaluations
• Sensitivity analysis — testing impact of wettability and rock parameters
• Quick estimates — preliminary reservoir simulation inputs
• Interpolation/extrapolation — extending measured data beyond tested ranges

This document covers two fundamental analytical models:

1. Corey (1954) — Power-law model with saturation exponents
2. LET (2005) — Three-parameter model with flexible endpoint and shape control

Theory

Normalized Saturation

Both models use normalized saturation ($S^*$) to account for irreducible and residual saturations:

$S_w^* = \frac{S_w - S_{wi}}{1 - S_{wi} - S_{or}}$

$S_o^* = \frac{S_o - S_{or}}{1 - S_{wi} - S_{or}} = 1 - S_w^*$

Where:

• $S_w$ = water saturation, fraction
• $S_{wi}$ = irreducible water saturation, fraction
• $S_{or}$ = residual oil saturation, fraction
• $S_w^*$ = normalized water saturation, fraction
• $S_o^*$ = normalized oil saturation, fraction

Brooks-Corey Model (1964)

The Brooks-Corey model is the foundational work that led to the simplified "Corey" correlations. Brooks and Corey developed equations relating relative permeability to capillary pressure data through a pore-size distribution parameter $\lambda$.

Relative Permeability Equations

Based on capillary pressure correlations, Brooks and Corey derived:

$k_{rw} = (S_w^*)^{\frac{2 + 3\lambda}{\lambda}}$

$k_{rnw} = (1 - S_w^*)^2 \left[1 - (S_w^*)^{\frac{2 + \lambda}{\lambda}}\right]$

Where:

• $k_{rw}$ = water relative permeability, fraction
• $k_{rnw}$ = non-wetting phase (oil or gas) relative permeability, fraction
• $S_w^* = \frac{S_w - S_{wr}}{1 - S_{wr}}$ = normalized water saturation
• $\lambda$ = pore-size distribution index (lithology factor) from capillary pressure data, dimensionless

Lithology Factor ($\lambda$)

The parameter $\lambda$ characterizes the pore volume structure and is obtained from capillary pressure measurements:

$S_w^* = \left(\frac{P_e}{P_c}\right)^{\lambda}$

Where:

• $P_c$ = capillary pressure, psi
• $P_e$ = pore entry pressure (from log-log plot intercept at $S_w^* = 1$), psi

Physical Interpretation:

• High $\lambda$ (>2): Uniform pore sizes, well-sorted rock
• Low $\lambda$ (<1): Widely varying pore sizes, poorly sorted rock

Typical $\lambda$ Values

Simplified Corey Model

The simplified Corey model uses power-law equations without the capillary pressure linkage. This is the form most commonly implemented in reservoir simulators:

General Form

$k_{rw} = k_{rw}^{\circ} \times (S_w^*)^{n_w}$

$k_{ro} = k_{ro}^{\circ} \times (S_o^*)^{n_o}$

Where:

• $k_{rw}^{\circ}$ = endpoint water relative permeability at $S_w = 1 - S_{or}$, fraction
• $k_{ro}^{\circ}$ = endpoint oil relative permeability at $S_o = 1 - S_{wi}$, fraction
• $n_w$ = water saturation exponent (Corey exponent), dimensionless
• $n_o$ = oil saturation exponent (Corey exponent), dimensionless

Relationship to Brooks-Corey:

• For water-wet systems, $n_w \approx \frac{2 + 3\lambda}{\lambda}$
• The simplified form allows independent tuning of $n_w$ and $n_o$

Typical Parameter Ranges

Notes:

• Higher $n$ values → more curved kr relationship
• Water-wet systems: typically $n_w < n_o$
• Oil-wet systems: typically $n_w > n_o$

LET Three-Parameter Model (2005)

Status: ⚠️ Limited Information — General model structure documented; specific parameter guidance requires primary reference.

General Form

The LET model provides more flexible curve shapes using three parameters (L, E, T):

$k_{rw} = k_{rw}^{\circ} \times \frac{(S_w^*)^{L_w}}{(S_w^*)^{L_w} + E_w (1-S_w^*)^{T_w}}$

$k_{ro} = k_{ro}^{\circ} \times \frac{(S_o^*)^{L_o}}{(S_o^*)^{L_o} + E_o (1-S_o^*)^{T_o}}$

Where:

• $L$ = Low-saturation exponent (controls shape near irreducible saturation)
• $E$ = Elevation parameter (controls vertical position of inflection point)
• $T$ = Top-saturation exponent (controls shape at high saturation)

LET Parameter Interpretation

Advantages over Corey:

• Better fit to laboratory data with only 3 parameters
• Independent control of curve shape at both endpoints
• Smoother transitions between linear and curved regions
• Can capture S-shaped curves

Primary Reference (to be obtained):

• Lomeland, F., Ebeltoft, E., and Thomas, W.H. (2005). "A New Versatile Relative Permeability Correlation." SCA2005-32, International Symposium of the Society of Core Analysts.
• $E$ = Elevation parameter (vertical position of inflection point)
• $T$ = Top-saturation exponent (controls shape at high saturation)

LET Parameter Interpretation

Advantages over Corey:

• Better fit to laboratory data with only 3 parameters
• Independent control of curve shape at both endpoints
• Smoother transitions between linear and curved regions

Blocking Reference

• Lomeland, F., Ebeltoft, E., and Thomas, W.H. (2005). "A New Versatile Relative Permeability Correlation." SCA2005-32, International Symposium of the Society of Core Analysts.

Comparison of Models

Functions Covered

Note: Excel function syntax and parameter details are available on individual function pages.

Applicability and Limitations

When to Use Corey Model

✅ Recommended:

• Screening studies with no SCAL data
• Sensitivity analysis (varying $n$ values)
• Analytical solutions requiring simple kr forms
• Historical models requiring Corey formulation

❌ Not Recommended:

• Precise history matching (insufficient flexibility)
• Complex wettability systems
• Fractured reservoirs with dual porosity

When to Use LET Model

✅ Recommended:

• History matching with limited SCAL data
• Better fit to measured kr curves
• Interpolation between measured points
• Uncertainty quantification with parametric variations

❌ Not Recommended:

• First-pass screening (overparameterization)
• No calibration data available

Related Topics

• SCAL Overview — Relative permeability correlation selection guide
• Honarpour Correlations — Empirical kr for sandstone/carbonate
• Ibrahim-Koederitz Correlations — Comprehensive empirical kr database

References

1. Brooks, R.H. and Corey, A.T. (1964). "Hydraulic Properties of Porous Media." Hydrology Papers, No. 3, Colorado State University, Fort Collins, Colorado. [Available: theory/references/articles/Empirical_Capillary_Relationship.md]

2. Brooks, R.H. and Corey, A.T. (1966). "Properties of porous media affecting fluid flow." J. Irrig. Drain. Div., 6, p61.

3. Corey, A.T. (1954). "The Interrelation Between Gas and Oil Relative Permeabilities." Producers Monthly, 19(1), 38-41. (To be obtained)

4. Lomeland, F., Ebeltoft, E., and Thomas, W.H. (2005). "A New Versatile Relative Permeability Correlation." SCA2005-32, International Symposium of the Society of Core Analysts. (To be obtained)

Related Reading

For background on analytical kr models:

• Honarpour, M., Koederitz, L., and Harvey, A.H. (1986). Relative Permeability of Petroleum Reservoirs. CRC Press.
• Lake, L.W. (1989). Enhanced Oil Recovery. Prentice Hall.
• Ahmed, T. (2019). Reservoir Engineering Handbook. Gulf Professional Publishing.

Document Status

Status: 👀 Ready for Review — Brooks-Corey equations documented from reference materials. LET model requires primary reference for complete parameter guidance.