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Effective and Relative Permeability Instructional Objectives:

- List 3 uses of relative permeability data. - Define absolute permeability, effective permeability, and relative permeability. - List 4 parameters that affect relative permeability. - Explain hysteresis in two-phase relative permeability data. - Explain how the use of relative permeability curves is tied with the reservoir mechanism

and/or the depletion process. - Apply Corey correlations to calculate relative permeability data. - List the common methods to measure two-phase relative permeability.

Uses of Effective and Relative Permeability Data:

- Reservoir simulation. - Flow calculations that involve multi-phase flow in reservoirs. - Estimation of residual oil (and/or gas) saturation.

Effective and relative permeability data are used in almost all reservoir engineering calculations that involve movements of several fluids together. Relative permeability data is an important input to reservoir simulation models. Reservoir simulation is used to study the reservoir behavior under a variety of conditions. Among the many uses of reservoir simulation models are:

- Prediction of reservoir performance. - Development planning. - Alternative production plans evaluation (water injection, gas injection, EOR… etc). - Alternative well configurations (fractured wells, horizontal wells … etc).

Relative permeability is also an input to simple models that calculate flow of more than one fluid (e.g. water flooding models). Relative permeability can also be used to estimate residual hydrocarbon saturation. Definitions: Permeability: is a property of the porous medium and is a measure of the capacity of the medium to transmit fluids. Absolute Permeability: When the medium is completely saturated with one fluid, then the permeability measurement is often referred to as specific or absolute permeability. Absolute permeability is often calculated from the steady-state flow equation:

L

pAkq

Effective Permeability: When the rock pore spaces contain more than one fluid, then the permeability to a particular fluid is called the effective permeability.

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Effective permeability is a measure of the fluid conductance capacity of a porous medium to a particular fluid when the medium is saturated with more than one fluid. Calculating effective permeability:

Oil: L

pAkq

o

oeoo

Water: L

pAkq

w

weww

Gas: L

pAkq

g

geg

g

Relative Permeability: It is defined as the ratio of the effective permeability to a fluid at a given saturation to a base permeability. The base permeability is commonly taken as the effective permeability to the fluid at 100% saturation (absolute permeability) or the effective non-wetting phase permeability at irreducible wetting phase saturation. Calculating relative permeability:

Oil: k

kk eoro

Water: k

kk ewrw

Gas: k

kk

eg

rg

Effective permeability is normalized to some base permeability to calculate relative permeability. The common base permeabilities include:

- Air permeability. - Absolute permeability. - Effective non-wetting phase permeability at irreducible wetting phase saturation.

Fundamental Concepts: Water phase: Water is usually located in smaller pore spaces and along sand grains. Therefore, relative permeability to water is a function of water saturation only (i.e., it does not matter what the relative amount of oil and gas is). Thus, we can plot relative permeability to water against water saturation on Cartesian coordinate paper.

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Oil phase: Oil is located between water and gas in the pore spaces, and to a certain extent, in the smaller pores. Thus, relative permeability to oil is a function of oil, water, and gas saturations. If the water saturation can be considered constant (i.e., the minimum interstitial water saturation), then kro can be plotted against So on Cartesian coordinate paper. Gas phase: Gas is located in the center of the larger pores. Therefore, the relative permeability to gas is a function of gas saturation only (i.e., it does not matter what the relative amounts of oil and water are). Thus, we can plot krg against Sg (or Sw + So) on Cartesian coordinate paper. Common Multi-phase Flow Systems:

- Water-oil systems. - Oil-gas systems. - Water-gas systems. - Three phase systems (water, oil, and gas).

Multi-phase flow is common in most petroleum reservoirs. In such multi-phase systems, we need to quantify the flow of each phase in the presence of other phases. This is done through effective and relative permeability data. We use sets of relative permeability data that correspond to the fluids moving in the reservoir. Example: We need to use a water-gas relative permeability set to perform reservoir engineering calculations when we study dry gas reservoirs under water influx from an aquifer. Exercise: What are the relative permeability data sets we need to use for the following situations?

- Water flooding an oil reservoir above the bubble point. - Production from an oil reservoir with a gas-cap and water aquifer.

Water flooding is the process of injecting water to displace the oil to the producing wells. If water injection starts when the reservoir pressure is above the bubble point pressure of the oil, gas does not come out of solution (i.e. free gas phase does not form in the reservoir). In a situation where oil is produced from a reservoir with a gas-cap and water aquifer, we anticipate that all three phases may flow. There will be places in the reservoir where only oil and gas flow together, places where only oil and water flow together, and places where all three phases may flow together. Solution: For water flooding an oil reservoir above the bubble point:

- Water-oil relative permeability For three phase flow:

- Water-oil relative permeability.

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- Gas-oil (or gas-liquid) relative permeability. - 3 phase relative permeability.

Oil-Water Relative Permeability:

The figure represents typical oil-water relative permeability data. Usually the experiment is done in the direction of increasing water saturation to simulate water injection in the reservoir. The base used to normalize the relative permeability data is the effective oil permeability at the irreducible water saturation. As water saturation increases, the relative permeability to oil decreases and the water relative permeability increases until it reaches a maximum at the residual oil saturation.

40

0

20

400 1006020 80

Water Saturation (%)

Rela

tive P

erm

eab

ilit

y (

%)

100

60

80

Waterkrw @ Sor

Oil

Two-Phase Flow

Region

Irreducible

Water

Saturation

kro @ Swi

Residual Oil

Saturation

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Oil-Gas Relative Permeability:

Oil-gas relative permeability data are usually plotted against total liquid saturation. They are sometimes plotted against gas saturation. As liquid saturation decreases, the relative permeability to oil decreases and the gas relative permeability increases until it reaches a maximum at the residual liquid saturation. In case of gas displacing oil, the residual liquid saturation will be the summation of irreducible water saturation and residual oil saturation. Exercise:

The figure above shows water-oil relative permeability data for a water-wet system.

40

0

20

400 1006020 80

Total Liquid Saturation - % of Pore Volume

Re

lati

ve P

erm

eab

ilit

y (

%)

100

60

80

Gaskro

Oil

krg

SL = So + Swi

0.4

0

0.2

400 1006020 80

Water Saturation (% PV)

Re

lati

ve

Pe

rmea

bil

ity

, F

racti

on

1.0

0.6

0.8 (1)

(2)

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What do curves (1) and (2) represent? Estimate the following:

- Irreducible water saturation. - Residual oil saturation. - Relative oil permeability at irreducible water saturation. - Relative water permeability at residual oil saturation.

Solution: Curve (1) is oil relative permeability Curve (2) is water relative permeability Values are:

- Swi = 15% - Sor = 20% - kro @ Swi = 1.0 - krw @ Sor = 0.35

Importance of Relative Permeability Data:

- Relative permeability data affect the flow characteristics of reservoir fluids. - Relative permeability data affect the recovery of oil and/or gas.

Relative permeability data influence the flow of fluids in the reservoir. Relative permeability curves determine how much oil, gas, and water are flowing relative to each other. Example: Effect of Relative Permeability Data:

0

20

40

60

80

100

0 20 40 60 80 100

Water Saturation (% )

Re

lati

ve

Pe

rme

ab

ilit

y (

%)

Rock Type 2

Rock Type 1

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The figure shows two sets of water-oil relative permeability data. One set for rock type 1 is drawn with triangles, and another set for rock type 2 is drawn with squares. The oil relative permeability, kro, is almost the same for the two sets, however, the water relative permeability, krw, is different. In the next figure, we see the impact of these two different sets of relative permeability data on oil recovery of a linear water flood.

The y-axis shows oil recovery as a percent of the movable oil in the reservoir. The figure clearly shows that water flooding in rock type 1 is more efficient than rock type 2. In rock type 1, water breakthrough (water reaches the producer) occurs later in time than rock type 2 because of the shape of the relative permeability data. Oil recovery for a process of water displacing oil can be 25 to 65% of the original oil-in-place. In the graph above, the recovery is a percentage of the recoverable oil, meaning that the irreducible oil saturation has been excluded from the calculation. Factors Affecting Effective and Relative Permeabilities:

- Fluid saturations. - Geometry of the rock pore spaces and grain size distribution. - Rock wettability. - Fluid saturation history (i.e., imbibition or drainage).

The effect of fluid saturations was shown on the figures above. In general, relative permeability to a particular fluid increases as the saturation of that fluid increases.

0

20

40

60

80

100

0 2 4 6 8 10

Pore Volum es In jected

Pe

rce

nt

of

Re

co

ve

rab

le O

il

Rock Type 1

Rock Type 2

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The geometry of the rock pore spaces and grain size distribution also affect both the shape of the relative permeability curves and their end points. Different rock characteristics are expected to produce different relative permeability curves. The effects of wettability and saturation history are shown in the following figures. Effect of Wettability: In a strongly oil-wet system, water is expected to flow easier than in a strongly water-wet system. In addition, we generally expect that the residual oil saturation will be higher for strongly oil-wet systems.

Effect of Saturation History: Types of relative permeability curves: Drainage curve Wetting phase is displaced by the non-wetting phase, i.e., the wetting phase saturation is decreasing. Imbibition Curve Non-wetting phase is displaced by wetting phase, i.e., the wetting phase saturation is increasing.

0.4

0

0.2

400 1006020 80

Water Saturation (% PV)

Rela

tive P

erm

eab

ilit

y,

Fra

cti

on

1.0

0.6

0.8

Water

Oil

Strongly Water-Wet Rock

0.4

0

0.2

400 1006020 80

Water Saturation (% PV)

Rela

tive P

erm

eab

ilit

y,

Fra

cti

on

1.0

0.6

0.8

WaterOil

Strongly Oil-Wet Rock

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Process begins with rock completely saturated with wetting phase, i.e., Swp = 100%. The wetting phase is displaced with the non-wetting phase (i.e., drainage process) until wetting phase ceases to flow. At this point, the wetting phase saturation equals the minimum interstitial wetting phase saturation. Then, the non-wetting phase is displaced with the wetting phase, (i.e., imbibition process) until the non-wetting phase ceases to flow. At this point, the non-wetting phase saturation equals the equilibrium or residual non-wetting phase saturation. The term “hysteresis” describes the process in which the relative permeabilities are different when measurements are made in different directions. Choosing the Correct Curve:

- When simulating the waterflood of a water-wet reservoir rock, imbibition relative permeability curves should be used.

- When modeling gas injection into an oil reservoir, drainage relative permeability curves should be used.

The procedure (i.e., drainage or imbibition) used to obtain relative permeability data in the laboratory must correspond to the dominant process in the reservoir

0

20

40

60

80

100

0 20 40 60 80 100

Drainage

Imbibition

Wetting Phase Saturation, % PV

Re

lati

ve

Pe

rme

ab

ilit

y,

%

Residual non-wetting

phase saturation

Interstitial wettingphase saturation

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Relative Permeability Correlations:

In many cases, relative permeability data on actual samples from the reservoir under study may not be available. In such cases, it is necessary to obtain the desired relative permeability data in some other manner.

Various parameters have been used to calculate the relative permeability relationships. Relative permeability data are usually correlated with end points and/or capillary pressure data. Rose and Bruce method uses capillary pressure data to generate relative permeability curves.

The most commonly applied correlation is probably that of Corey, which was published in 1954.

Gas-Oil Relative Permeability Corey Correlations:

Corey proposed a simple mathematical expression for generating the relative permeability data of the gas-oil system. The approximation is good for drainage processes, i.e., gas-displacing oil. He also proposed simple mathematical expressions for water-oil relative permeability.

( )

( ) (

)

( )

Water-Oil Relative Permeability Corey Correlations:

In his original work, Corey suggested that exponent for water-oil relative permeability should be 4. It should be pointed out that Corey’s equations apply only to well-sorted homogeneous rocks. To account for the degree of consolidation, the exponent of the relationships can be expressed in a more generalized way such as proposed below.

(

)

(

)

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Three-Phase Relative Permeability:

Three fluid phases (i.e., oil, water, and gas) are often present in petroleum reservoirs. A ternary (triangular) diagram is used to represent a three-phase system. Three phase relative permeability data are mostly used in reservoir simulation studies to describe the flow characteristics of the three phases when they are flowing together. Three-phase relative permeability data are very hard to measure in the lab. Reservoir simulators often use correlations to calculate three-phase relative permeability from two-phase relative permeability data. Relative Permeability to Water in a Three-Phase System:

100% Gas

100% Oil100% Water

100% Gas

100% Water 100% Oil

0%

10%

20%

40%

60%

krw = 80%

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Straight lines on the ternary diagram indicate that relative permeability to water is a function of water saturation only. Because of this straight-line behavior, krw can be plotted on Cartesian coordinates as a function of Sw. Relative Permeability to Oil in a Three-Phase System:

In a water-wet system:

- The oil phase has a greater tendency than gas to wet the rock. - The interfacial tension between water and oil is less than that between water and gas. - Oil occupies portions of the pore spaces adjacent to the water. - At lower water saturation, the oil occupies more of the smaller pore spaces.

Laboratory Methods for Measuring Relative Permeability Data:

- Steady-state flow method. - Displacement (unsteady-state) method.

100% Gas

100% Water 100% Oil

5%

10%

20%

30%

40%

kro = 50%

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Nomenclature: A = flow area L = flow length k = permeability ke = effective permeability kr = relative permeability q = volumetric flow rate S = saturation

p = pressure drop

= fluid viscosity Subscripts g = gas o = oil w = water References: 1- Amyx, J.W., Bass, D.M., and Whiting, R.L.: Petroleum Reservoir Engineering, McGrow-Hill

Book Company New York, 1960. 2- Tiab, D. and Donaldson, E.C.: Petrophysics, Gulf Publishing Company, Houston, TX. 1996. 3- Core Laboratories, Inc. “A course in the fundamentals of Core analysis, 1982. 4- Donaldson, E.C., Thomas, R.D., and Lorenz, P.B.: “Wettability Determination and Its Effect

on Recovery Efficiency,” SPEJ (March 1969) 13-20. 5- Ahmed, T.: Reservoir Engineering Handbook, 4th Ed., Elsevier, 2010. 6- Corey, A. T.: “The Interrelation Between Gas and Oil Relative Permeabilities,” Prod. Mon.,

1954, pp. 19-38, 1954.