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 $n_w > n_o$

Corey Relative Permeability Curves

Water-wet sandstone example: $S_{wi}$ = 0.20, $S_{or}$ = 0.25, $k_{rw}^{\circ}$ = 0.3, $k_{ro}^{\circ}$ = 1.0, $n_w$ = 3, $n_o$ = 3:

---

LET Three-Parameter Model (2005)

Development and Background

Lomeland, Ebeltoft, and Thomas (2005) developed the LET correlation to overcome limitations of traditional models (Corey, Sigmund-McCaffery, Chierici) when modeling relative permeability across the entire saturation range. The model addresses:

• S-behavior at low water saturations observed in mixed-wet to weakly water-wet systems
• Flexibility to match steady-state experimental data without creating breaks in the curve
• Smooth representation suitable for both SCAL interpretation and reservoir simulation

Mathematical Formulation

Water-Oil System

For water injection with oil production:

$$S_{wn} = \frac{S_w - S_{wi}}{1 - S_{wi} - S_{orw}}$$

$$k_{row} = k_{ro}^x \times \frac{(1 - S_{wn})^{L_o^w}}{(1 - S_{wn})^{L_o^w} + E_o^w \cdot S_{wn}^{T_o^w}}$$

$$k_{rw} = k_{rw}^o \times \frac{S_{wn}^{L_w^o}}{S_{wn}^{L_w^o} + E_w^o \cdot (1 - S_{wn})^{T_w^o}}$$

Where:

• $S_{wn}$ = normalized water saturation, fraction
• $S_{wi}$ = irreducible water saturation, fraction
• $S_{orw}$ = residual oil saturation to water, fraction
• $k_{ro}^x$ = oil endpoint relative permeability at $S_w = S_{wi}$, fraction
• $k_{rw}^o$ = water endpoint relative permeability at $S_w = 1 - S_{orw}$, fraction
• $L_o^w$, $E_o^w$, $T_o^w$ = LET parameters for oil (subscript = phase, superscript = displacing phase)
• $L_w^o$, $E_w^o$, $T_w^o$ = LET parameters for water

LET Parameter Interpretation

Parameter Behavior:

1. L-parameter (Lower):

   • Describes the lower part of the curve
   • By experience, L-values are comparable to Corey exponents
   • Higher L → steeper rise from irreducible saturation

2. E-parameter (Elevation):

   • A value of E = 1 is neutral (slope position governed by L and T)
   • E > 1 pushes the slope toward the high end of the curve
   • E < 1 pushes the slope toward the lower end of the curve

3. T-parameter (Top):

   • Describes the upper part of the curve in a manner similar to L
   • Controls approach to maximum kr value
   • Higher T → more gradual approach to endpoint

S-Behavior Capability

The LET model successfully captures S-shaped oil relative permeability at low water saturations, commonly observed in:

• Mixed-wet systems — water in small pores/corners, oil in larger pores
• Weakly water-wet systems — spontaneous imbibition into medium pores before larger pore flooding

Physical explanation:

1. Initial water invasion enters water-wet small/medium pores (low impact on oil kr)
2. Small negative slope of $k_{ro}$ at low $S_w$
3. As water enters larger pores, slope steepens
4. Wettability, pore shape, and pore-size distribution create S-behavior

Advantages over Corey

✅ Flexibility: 3 parameters provide independent control over curve shape at low, middle, and high saturations

✅ Accuracy: Successfully reconciles steady-state experimental data (differential pressure and production) across entire saturation range

✅ Smoothness: Maintains smooth, physically meaningful curves without breaks

✅ S-behavior: Captures complex wettability effects that power-law models cannot represent

✅ Field-scale impact: Significant differences in water breakthrough timing and production forecasting vs. Corey model

---

Extensions to Other Fluid Systems

Gas-Oil System

Normalized gas saturation:

$$S_{gn} = \frac{S_g}{1 - S_{wi} - S_{org}}$$

Relative permeabilities:

$$k_{rog} = k_{ro}^x \times \frac{(1 - S_{gn})^{L_o^g}}{(1 - S_{gn})^{L_o^g} + E_o^g \cdot S_{gn}^{T_o^g}}$$

$$k_{rg} = k_{rg}^o \times \frac{S_{gn}^{L_g^o}}{S_{gn}^{L_g^o} + E_g^o \cdot (1 - S_{gn})^{T_g^o}}$$

Where:

• $S_{org}$ = residual oil saturation to gas, fraction
• $k_{rg}^o$ = gas endpoint relative permeability, fraction

Water-Gas System

Normalized water saturation:

$$S_{wn} = \frac{S_w - S_{wi}}{1 - S_{wi} - S_{grw}}$$

Relative permeabilities:

$$k_{rgw} = k_{rg}^x \times \frac{(1 - S_{wn})^{L_g^w}}{(1 - S_{wn})^{L_g^w} + E_g^w \cdot S_{wn}^{T_g^w}}$$

$$k_{rw} = k_{rw}^g \times \frac{S_{wn}^{L_w^g}}{S_{wn}^{L_w^g} + E_w^g \cdot (1 - S_{wn})^{T_w^g}}$$

Where:

• $S_{grw}$ = residual gas saturation to water, fraction
• $k_{rw}^g$ = water endpoint relative permeability after gas production, fraction

---

Comparison of Models

---

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:

• SCAL data interpretation (steady-state experiments)
• History matching with limited measured data
• Systems exhibiting S-behavior (mixed-wet, weakly water-wet)
• Field-scale simulations requiring accurate breakthrough prediction
• Sensitivity analysis with better parameter physical meaning
• Automated optimization workflows (smooth derivatives)

❌ Not Recommended:

• First-pass screening with zero data (use Corey)
• Simple systems well-represented by power-law (unnecessary complexity)

Field Application Results

Lomeland et al. (2005) demonstrated significant impact on full-field simulation for Norwegian Sea gas-cap/aquifer field:

• Water breakthrough timing: LET model delayed breakthrough vs. Corey
• Production rates: Oil rate differences up to 2× in first 4 years
• Water production: Corey predicted almost 2× higher water rates than LET
• Gas production: Minimal difference (similar rapid breakthrough)

The LET model better honored well test results showing no early water production, while Corey correlation showed immediate water breakthrough.

---

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.

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.

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, Toronto, Canada.

Software Implementation

• Sendra (Petec Software & Services) — LET correlation included for SCAL interpretation with automated optimization
• Eclipse (Schlumberger) — LET keywords supported
• CMG (Computer Modelling Group) — LET formulation available

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.