Relative Permeability in Petroleum Engineering: Concepts and Factors, Lecture notes of Engineering

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Relative Permeability

Concepts

C.D. Adenutsi, Ph.D. Department of Petroleum Engineering, KNUST Office: Petroleum Building, PB 318 January, 2024

Introduction

• When two or more fluids flow at the same time , the relative permeability of each phase at a specific saturation is the ratio of the effective permeability of the phase to the absolute permeability 𝑘 𝑟𝑜

𝑜 𝑘 𝑘𝑟𝑔 =

𝑔 𝑘 𝑘 𝑟𝑤

𝑤 𝑘

Introduction

• Since the effective permeabilities may range from zero to 𝒌 , the relative permeabilities may have any value between zero and one
• It should be pointed out that when three phases are present the sum of the relative permeabilities ( 𝒌 𝒓𝒐

𝒓𝒈

𝒓𝒘 ) is both variable and always less than or equal to unity

• It has become a common practice to refer to the relative permeability curve for the nonwetting phase as 𝒌𝒏𝒘 and the relative permeability for the wetting phase as 𝒌 𝒘

Two-Phase Relative Permeability

• When a wetting and a nonwetting phase flow together in a reservoir rock, each phase follows separate and distinct paths.
• The distribution of the two phases according to their wetting characteristics results in characteristic wetting and nonwetting phase relative permeabilities.

Two-Phase Relative Permeability

• Fig. 1 presents a typical set of relative permeability curves for a water-oil system with the water being considered the wetting phase.
• The range of water saturation is from the initial water saturation ( 𝑺 𝒘𝒊 ) , to water saturation at residual oil saturation 𝟏 − 𝑺 𝒐𝒓𝒘

• Oil relative permeability 𝒌 𝒓𝒐 is highest at 𝑺𝒘𝒊 and declines to zero at 𝟏 − 𝑺𝒐𝒓𝒘. (^) Fig. 1 Oil-water relative permeability curves.

Two-Phase Relative Permeability

• Water relative permeability 𝒌 𝒓𝒘 increases from zero at 𝑺 𝒘𝒊 to its highest value at 𝟏 − 𝑺𝒐𝒓𝒘.
• Figure 1 shows the location of critical water saturation ( 𝑺 𝒘𝒄

• Critical water saturation is the level of water saturation at which water starts to flow in the reservoir. Fig. 1 Oil-water relative permeability curves..

Two-Phase Relative Permeability

• Oil relative permeability ( 𝒌 𝒓𝒐 ) is highest at 𝑺 𝒈𝒊 and declines to zero at 𝟏 − 𝑺 𝒐𝒓𝒈

• Gas relative permeability (𝒌 𝒓𝒈

increases from zero at 𝑺𝒈𝒊 to its highest value at 𝟏 − 𝑺𝒐𝒓𝒈.

• The critical gas saturation 𝑺 𝒈𝒄 is the level of gas saturation at which gas starts to flow in the reservoir. Fig. 2 Gas-oil relative permeability curves..

Two-Phase Relative Permeability

• Theoretically, the critical saturation and the residual saturation should be exactly equal for any fluid ; however, they are not identical.
• Critical saturation is measured in the direction of increasing saturation, while irreducible saturation is measured in the direction of reducing saturation.
• Thus, the saturation histories of the two measurements are different.

Two-Phase Relative Permeability Correlations

• Most of the proposed correlations use the effective phase saturation as a correlating parameter.
• The effective phase saturation is defined by the following set of relationships: 𝑺 𝒐 ∗ =

𝒐 𝟏 − 𝑺𝒘𝒊𝒓𝒓

𝒘 ∗ =

𝒘

𝒘𝒊𝒓𝒓 𝟏 − 𝑺 𝒘𝒊𝒓𝒓

∗

𝒈 𝟏 − 𝑺 𝒘𝒊𝒓𝒓

Two-Phase Relative Permeability Correlations

𝒐 ∗ , 𝑺 𝒘 ∗ , 𝑺 𝒈 ∗ = effective oil, water, and gas saturation, respectively

• 𝑺𝒐, 𝑺𝒘, 𝑺𝒈 = oil, water and gas saturation, respectively
• 𝑺 𝒘𝒊𝒓𝒓 = connate (irreducible) water saturation

Two-Phase Relative Permeability Correlations

• Wyllie and Gardner have also suggested the following two expressions that can be used when one relative permeability is available:
• Oil-Water System 𝒌 𝒓𝒘

𝒘 ∗ 𝟐 − 𝒌 𝒓𝒐

𝒘 ∗ 𝟏 − 𝑺 𝒘 ∗

• Gas-Oil System 𝒌 𝒓𝒐

𝒐 ∗ − 𝒌 𝒓𝒈

𝒐 ∗ 𝟏 − 𝑺𝒐 ∗

Two-Phase Relative Permeability Correlations

• Torcaso and Wyllie Correlation
• Torcaso and Wyllie developed a simple expression to determine permeability of the oil phase in a gas-oil system.
• The expression permits the calculation of 𝒌 𝒓𝒐 from the measurements of 𝒌 𝒓𝒈 . The equation has the following form: 𝒌 𝒓𝒐

𝒓𝒈

𝒐 ∗ 𝟒 𝟏 − 𝑺 𝒐 ∗ (^) 𝟐 𝟏 − 𝑺 𝒐 ∗ (^) 𝟐

Two-Phase Relative Permeability Correlations

• For the nonwetting phase
• Imbibition 𝑘 𝑟 𝑛𝑜𝑛𝑤𝑒𝑡𝑡𝑖𝑛𝑔

2 ( 14 )

• Drainage 𝑘 𝑟 𝑛𝑜𝑛𝑤𝑒𝑡𝑡𝑖𝑛𝑔=

𝑤 ∗ 1 − 𝑆 𝑤 ∗ 0. 25 𝑆 𝑤

1. 5 ( 15 ) where 𝑺𝒏𝒘 = saturation of the nonwetting phase; 𝑺𝒘 = water saturation 𝑺 𝒘 ∗ = effective water saturation as defined by Equation

Two-Phase Relative Permeability Correlations

• Corey’s Method
• Corey proposed a simple mathematical expression for generating the relative permeability data of gas-oil system. The approximation is good for drainage processes, i.e., gas-displacing oil 𝒌 𝒓𝒐

𝒈 ∗ 𝟒 (𝟏𝟔) 𝒌𝒓𝒈 = 𝑺𝒈 ∗ 𝟐 − 𝑺𝒈 ∗ (𝟏𝟕)

• where the effective gas saturation 𝑺 𝒈 ∗ is defined in Equation 3.