Tarek Ahmed. Reservoir engineering handbook

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porosity (f), and permeability (k). It is believed that these three prop-

erties can be used to determine the recovery fraction (RF).

Core 1 Core 2 Core 3

f .185 .157 .484

0.476 .527 .637

k .614 .138 .799

Recovery factor .283 .212 .141

The recovery factor can be expressed by the following equation:

RF = a

f+a

+ a

where a

, a

, and a

are constants

Calculate RF if:

= .75, f=.20, and k = .85

REFERENCES

1. Calhoun, J. R., Fundamentals of Reservoir Engineering. University of Okla-

homa Press, 1976.

2. Cole, Frank, Reservoir Engineering Manual. Houston: Gulf Publishing

Company, 1969.

3. Dykstra, H., and Parsons, R. L., “The Prediction of Oil Recovery by Water

Flood,” In Secondary Recovery of Oil in the United States, 2nd ed., pp.

160–174. API, 1950.

4. Geertsma, J., “The Effect of Fluid Pressure Decline on Volumetric Changes

of Porous Rocks,” Trans. AIME, 1957, pp. 210, 331–340.

5. Hall, H. N., “Compressibility of Reservoir Rocks,” Trans. AIME, 1953, p. 309.

6. Hustad, O., and Holt, H., “Gravity Stable Displacement of Oil by Gas after

WaterFlooding,” SPE Paper 24116, SPE/DOE Symposium on EOR, Tulsa,

OK, April 22–24, 1972.

7. Johnson C. R., Careenkorn, R. A. and Woods, E. G., “Pulse Testing: A New

Method for Describing Reservoir Flow Properties Between Wells,” JPT,

Dec. 1966, pp. 1599–1604

8. Jones, S. C., “A Rapid Accurate Unsteady State Klinkenberg Parameter,”

SPEJ, 1972, Vol. 12, No. 5, pp. 383–397.

278 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 04b 2001-10-24 09:46 Page 278

9. Klinkenberg, L. J., “The Permeability of Porous Media to Liquids and

Gases,” API Drilling and Production Practice, 1941, p. 200.

10. Leverett, M. C., “Capillary Behavior in Porous Solids,” Trans. AIME, 1941.

11. McCardell, W. M., “A Review of the Physical Basis for the Use of the J-

Function,” Eighth Oil Recovery Conference, Texas Petroleum Research

Committee, 1955.

12. Miller, M. G., and Lents, M. R., “Performance of Bodcaw Reservoir, Cotton

Valley Field Cycling Project: New Methods of Predicting Gas-Condensate

Reservoir,” SPEJ, Sept. 1966, pp. 239.

13. Morris, R. L., and Biggs, W. P., “Using Log-Derived Values of Water Satu-

ration and Porosity.” SPWLA, Paper X, 1967.

14. Newman, G. H., “Pore-Volume Compressibility,” JPT, Feb. 1973, pp.

129–134

15. Schmalz, J. P., and Rahme, H. D., “The Variation of Waterflood Perfor-

mance With Variation in Permeability Profile,” Prod. Monthly, 1950, Vol.

15, No. 9, pp. 9–12.

16. Timur, A., “An Investigation of Permeability, Porosity, and Residual Water

Saturation Relationships,” AIME, June 1968.

17. Warren, J. E. and Price, H. S., “Flow in Heterogeneous Porous Media,”

SPEJ, Sept. 1961, pp. 153–169.

Fundamentals of Rock Properties 279

Reservoir Eng Hndbk Ch 04b 2001-10-24 09:46 Page 279

280

CHAPTER 5

RELATIVE PERMEABILITY

CONCEPTS

Numerous laboratory studies have concluded that the effective perme-

ability of any reservoir fluid is a function of the reservoir fluid saturation

and the wetting characteristics of the formation. It becomes necessary,

therefore, to specify the fluid saturation when stating the effective perme-

ability of any particular fluid in a given porous medium. Just as k is the

accepted universal symbol for the absolute permeability, k

, k

, and k

are the accepted symbols for the effective permeability to oil, gas, and

water, respectively. The saturations, i.e., S

, S

, and S

, must be specified

to completely define the conditions at which a given effective permeabil-

ity exists.

Effective permeabilities are normally measured directly in the labora-

tory on small core plugs. Owing to many possible combinations of satu-

ration for a single medium, however, laboratory data are usually summa-

rized and reported as relative permeability.

The absolute permeability is a property of the porous medium and is a

measure of the capacity of the medium to transmit fluids. 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, or:

Reservoir Eng Hndbk Ch 05 2001-10-24 09:51 Page 280

where k

= relative permeability to oil

= relative permeability to gas

= relative permeability to water

k = absolute permeability

= effective permeability to oil for a given oil saturation

= effective permeability to gas for a given gas saturation

= effective permeability to water at some given water

saturation

For example, if the absolute permeability k of a rock is 200 md and the

effective permeability k

of the rock at an oil saturation of 80 percent is

60 md, the relative permeability k

is 0.30 at S

= 0.80.

Since the effective permeabilities may range from zero to k, the rela-

tive permeabilities may have any value between zero and one, or:

0  k

, k

 1.0

It should be pointed out that when three phases are present the sum of

the relative permeabilities (k

+ k

) is both variable and always

less than or equal to unity. An appreciation of this observation and of its

physical causes is a prerequisite to a more detailed discussion of two-

and three-phase relative permeability relationships.

It has become a common practice to refer to the relative permeability

curve for the nonwetting phase as k

and the relative permeability for

the wetting phase as k

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 charac-

teristic wetting and nonwetting phase relative permeabilities. Since the

wetting phase occupies the smaller pore openings at small saturations, and

these pore openings do not contribute materially to flow, it follows that

the presence of a small wetting phase saturation will affect the nonwetting

Relative Permeability Concepts 281

Reservoir Eng Hndbk Ch 05 2001-10-24 09:51 Page 281

phase permeability only to a limited extent. Since the nonwetting phase

occupies the central or larger pore openings which contribute materially to

fluid flow through the reservoir, however, a small nonwetting phase satu-

ration will drastically reduce the wetting phase permeability.

Figure 5-1 presents a typical set of relative permeability curves for a

water-oil system with the water being considered the wetting phase. Fig-

ure 5-1 shows the following four distinct and significant points:

282 Reservoir Engineering Handbook

Figure 5-1. Typical two-phase flow behavior.

Reservoir Eng Hndbk Ch 05 2001-10-24 09:51 Page 282

• Point 1

Point 1 on the wetting phase relative permeability shows that a small

saturation of the nonwetting phase will drastically reduce the relative

permeability of the wetting phase. The reason for this is that the non-

wetting phase occupies the larger pore spaces, and it is in these large

pore spaces that flow occurs with the least difficulty.

• Point 2

Point 2 on the nonwetting phase relative permeability curve shows

that the nonwetting phase begins to flow at the relatively low saturation

of the nonwetting phase. The saturation of the oil at this point is called

critical oil saturation S

• Point 3

Point 3 on the wetting phase relative permeability curve shows that

the wetting phase will cease to flow at a relatively large saturation. This

is because the wetting phase preferentially occupies the smaller pore

spaces, where capillary forces are the greatest. The saturation of the

water at this point is referred to as the irreducible water saturation S

wir

or connate water saturation S

—both terms are used interchangeably.

• Point 4

Point 4 on the nonwetting phase relative permeability curve shows

that, at the lower saturations of the wetting phase, changes in the wet-

ting phase saturation have only a small effect on the magnitude of the

nonwetting phase relative permeability curve. The reason for the phe-

nomenon at Point 4 is that at the low saturations the wetting phase fluid

occupies the small pore spaces which do not contribute materially to

flow, and therefore changing the saturation in these small pore spaces

has a relatively small effect on the flow of the nonwetting phase.

This process could have been visualized in reverse just as well. It

should be noted that this example portrays oil as nonwetting and water

as wetting. The curve shapes shown are typical for wetting and nonwet-

ting phases and may be mentally reversed to visualize the behavior of

an oil-wet system. Note also that the total permeability to both phases,

+ k

, is less than 1, in regions B and C.

The above discussion may be also applied to gas-oil relative perme-

ability data, as can be seen for a typical set of data in Figure 5-2. Note

that this might be termed gas-liquid relative permeability since it is plot-

ted versus the liquid saturation. This is typical of gas-oil relative perme-

ability data in the presence of connate water. Since the connate (irre-

ducible) water normally occupies the smallest pores in the presence of oil

Relative Permeability Concepts 283

Reservoir Eng Hndbk Ch 05 2001-10-24 09:51 Page 283

and gas, it appears to make little difference whether water or oil that

would also be immobile in these small pores occupies these pores. Con-

sequently, in applying the gas-oil relative permeability data to a reservoir,

the total liquid saturation is normally used as a basis for evaluating the

relative permeability to the gas and oil.

Note that the relative permeability curve representing oil changes

completely from the shape of the relative permeability curve for oil in the

water-oil system. In the water-oil system, as noted previously, oil is nor-

mally the nonwetting phase, whereas in the presence of gas the oil is the

wetting phase. Consequently, in the presence of water only, the oil rela-

tive permeability curve takes on an S shape whereas in the presence of

gas the oil relative-permeability curve takes on the shape of the wetting

phase, or is concave upward. Note further that the critical gas saturation

is generally very small.

284 Reservoir Engineering Handbook

Figure 5-2. Gas-oil relative permeability curves.

Reservoir Eng Hndbk Ch 05 2001-10-24 09:51 Page 284

Another important phenomenon associated with fluid flow through

porous media is the concept of residual saturations. As when one immis-

cible fluid is displacing another, it is impossible to reduce the saturation

of the displaced fluid to zero. At some small saturation, which is pre-

sumed to be the saturation at which the displaced phase ceases to be con-

tinuous, flow of the displaced phase will cease. This saturation is often

referred to as the residual saturation. This is an important concept as it

determines the maximum recovery from the reservoir. Conversely, a fluid

must develop a certain minimum saturation before the phase will begin

to flow. This is evident from an examination of the relative permeability

curves shown in Figure 5-1. The saturation at which a fluid will just

begin to flow is called the critical saturation.

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 satura-

tion. Thus, the saturation histories of the two measurements are different.

As was discussed for capillary-pressure data, there is also a saturation

history effect for relative permeability. The effect of saturation history on

relative permeability is illustrated in Figure 5-3. If the rock sample is ini-

tially saturated with the wetting phase (e.g., water) and relative-perme-

ability data are obtained by decreasing the wetting-phase saturation while

flowing nonwetting fluid (e.g., oil) in the core, the process is classified as

drainage or desaturation.

If the data are obtained by increasing the saturation of the wetting

phase, the process is termed imbibition or resaturation. The nomencla-

ture is consistent with that used in connection with capillary pressure.

This difference in permeability when changing the saturation history is

called hysteresis. Since relative permeability measurements are subject to

hysteresis, it is important to duplicate, in the laboratory, the saturation

history of the reservoir.

Drainage Process

It is generally agreed that the pore spaces of reservoir rocks were orig-

inally filled with water, after which oil moved into the reservoir, displac-

ing some of the water, and reducing the water to some residual satura-

tion. When discovered, the reservoir pore spaces are filled with a connate

water saturation and an oil saturation. If gas is the displacing agent, then

gas moves into the reservoir, displacing the oil.

Relative Permeability Concepts 285

Reservoir Eng Hndbk Ch 05 2001-10-24 09:51 Page 285

This same history must be duplicated in the laboratory to eliminate the

effects of hysteresis. The laboratory procedure is to first saturate the core

with water, then displace the water to a residual, or connate, water satura-

tion with oil after which the oil in the core is displaced by gas. This flow

process is called the gas drive, or drainage, depletion process. In the gas

drive depletion process, the nonwetting phase fluid is continuously

increased, and the wetting phase fluid is continuously decreased.

Imbibition Process

The imbibition process is performed in the laboratory by first saturating

the core with the water (wetting phase), then displacing the water to its

irreducible (connate) saturation by injection oil. This “drainage” procedure

is designed to establish the original fluid saturations that are found when

the reservoir is discovered. The wetting phase (water) is reintroduced into

the core and the water (wetting phase) is continuously increased. This is

the imbibition process and is intended to produce the relative permeability

data needed for water drive or water flooding calculations.

Figure 5-3 schematically illustrates the difference in the drainage and

imbibition processes of measuring relative permeability. It is noted that

the imbibition technique causes the nonwetting phase (oil) to lose its

mobility at higher values of water saturation than does the drainage

process. The two processes have similar effects on the wetting phase

(water) curve. The drainage method causes the wetting phase to lose its

mobility at higher values of wetting-phase saturation than does the imbi-

bition method.

Two-phase Relative Permeability Correlations

In many cases, relative permeability data on actual samples from the

reservoir under study may not be available, in which case it is necessary

to obtain the desired relative permeability data in some other manner.

Field relative permeability data can usually be calculated, and the proce-

dure will be discussed more fully in Chapter 6. The field data are

unavailable for future production, however, and some substitute must be

devised. Several methods have been developed for calculating relative

permeability relationships. Various parameters have been used to calcu-

late the relative permeability relationships, including:

• Residual and initial saturations

• Capillary pressure data

286 Reservoir Engineering Handbook

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In addition, 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

1 - S

(5 - 2)

-1

(5 -1)

Relative Permeability Concepts 287

Figure 5-3. Hysteresis effects in relative permeability.

Reservoir Eng Hndbk Ch 05 2001-10-24 09:51 Page 287