Tarek Ahmed. Reservoir engineering handbook

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Determine:

a. Water saturation profile after 10, 100, and 300 hr

b. Oil recovery at breakthrough

c. Time to breakthrough

d. Cumulative water injected at breakthrough

4. The following relative permeability data

are available on a rock sam-

ple taken from the R-Field:

0.25 0.00 0.495

0.35 0.015 0.327

0.40 0.030 0.260

0.45 0.068 0.200

0.50 0.110 0.148

0.55 0.149 0.102

0.60 0.213 0.064

0.65 0.277 0.032

0.70 0.350 0.000

Additional data:

= 25% S

= 75% S

= 0%

= 1.2 bbl/STB B

= 1.0 bbl/STB 

= 0.90 cp



= 10.0 cp A = 28,934 ft

= 100 bbl/day

= 100% E

= 100% Area = 10 acres

h = 31 ft k = 50 md

Distance between injector and producer = 467 ft.

a. Calculate the water saturation profile after 100, 200, 500, 1000,

2000, and 5000 days. Plot your results.

b. Calculate the mobility ratio.

c. Calculate the time to breakthrough

d. Calculate and plot N

, Q

, and W

inj

as a function of time.

1018 Reservoir Engineering Handbook

Data from Reservoir Engineering Manual, Cole, Houston, TX: Gulf Publishing Com-

pany, 1969.

Reservoir Eng Hndbk Ch 14 2001-10-25 17:38 Page 1018

5. An oil reservoir is under consideration for waterflooding. The relative

permeability data are given by the following expression:

where

Other reservoir data are given below:

Flood pattern = Five-spot

Flood area = 40 acres

Oil viscosity = 2 cp

Water viscosity = 0.5 cp

= 1.3 bbl/STB

= 1.05 bbl/STB

= 0.75

= 0.25

= 0.35

= 0.25

φ = 15%

k = 50 md

= 0.3 ft

h = 20 ft

= 1000 psi

= 100%

Assuming a constant water injection rate of 800 bbl/day, predict the

recovery performance and express results in a graphical form.

6. An oil reservoir is under consideration for further development by

water injection. The relative permeability data are given below:

0.10 0.20 0.30 0.40 0.70 0.85

0.00 0.00 0.02 0.05 0.35 0.60

1.00 0.93 0.60 0.35 0.05 0.00

wwi

or wi

−

−−1

rw WD

ro WD

=−

(

)

(

)

041

Principles of Waterflooding 1019

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Additional data are given below:

Flood pattern = Five-spot

Absolute permeability = 70 md

Thickness = 20 ft

Porosity = 15%

= 15%

= 20%



= 3.1 cp



= 1.0 cp

= 1.25 bbl/STB

= 1.01 bbl/STB

Pattern area = 40 acres

= 1.0 ft

p = (P

inj

– P

) = 1000 psi (constant)

a. Calculate and plot the reservoir performance during the following

stages:

• Start—interference

• Interference—fill-up

• Fill-up—breakthrough

• After breakthrough

b. Show on your graph time to: interference, fill-up, and breakthrough.

c. Plot water injectivity and areal sweep efficiency as a function of

time.

7. The following core analysis is available on a reservoir that is being

considered for a waterflooding project:

Sample h, ft k, md

1214

2239

3 1 108

4277

5228

6 1 212

7 1 151

8310

9220

10355

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Reservoir Eng Hndbk Ch 14 2001-10-25 17:38 Page 1020

Other data:

= 1000 bbl/day



= 9.0 cp



= 0.95 cp

M = 4.73

= 6 MMSTB

= 1.02 bbl/STB

= 1.00 bbl/STB

= 0.2

= 0.8

Using the simplified Dykstra–Parsons method, determine the follow-

ing recovery parameters as a function of time:

• Q

• WOR

• N

• W

Show your results graphically.

8. The following core and relative permeability data are given:

Depth, ft k, md

4100–4101 295

2 762

388

487

5 148

671

762

8 187

9 349

10 77

11 127

12 161

13 50

14 58

15 109

16 228

Principles of Waterflooding 1021

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Depth, ft k, md

17 282

18 776

19 87

20 47

21 16

22 35

23 47

24 54

25 273

26 454

27 308

28 159

29 178

0.10 0.00 0.96

0.20 0.00 0.89

0.24 0.00 0.80

0.30 0.01 0.60

0.40 0.03 0.30

0.50 0.06 0.12

0.60 0.12 0.06

0.70 0.19 0.015

0.76 0.25 0.00

0.80 0.32 0.00

0.90 0.65 0.00

= 0.59 µ

= 40.34 cp

= 0.24 µ

= 0.82 cp

= 0.17 B

= 1.073 bbl/STB

h = 29 ft B

= 1.00 bbl/STB

A = 40 acres  = 19%

Using the modified Dykstra–Parsons method, generate the perfor-

mance curves for this reservoir under a constant water-injection rate

of 700 bbl/day.

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Reservoir Eng Hndbk Ch 14 2001-10-25 17:38 Page 1022

9. Using Stiles’ and the Dykstra-Parsons methods, calculate the vertical

sweep efficiency as a function of producing water–oil ratio, given:

Layer h, ft k, md

1 2 5.0

2 2 7.0

3 2 11.0

4 2 4.0

5 2 14.0

6 2 21.0

7 2 68.0

8 2 13.0



= 8.0 cp



= 0.9 cp

M = 1.58

= 0.45

= 0.08

φ = 15%

= 1.2 bbl/STB

= bbl/STB

Show your results graphically.

10. An oil reservoir is characterized by the following six-layer system:

Layer h, ft k, md 

1 10 50 0.20

2 10 40 0.16

3 5 35 0.12

4 5 30 0.12

5 5 25 0.10

6 5 20 0.10

The first layer is identified as the base layer with the following rela-

tive permeability data:

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.75 0.8 0.9 0.10

00000.04 0.11 0.20 0.30 0.36 0.44 0.68 1.00

1.00 1.00 1.00 1.00 0.94 0.80 0.44 0.16 0.045 0 0 0

Principles of Waterflooding 1023

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The other rock and fluid properties are given below:

= 0.65

= 0.30

= 0.05

= 1.5 cp

= 0.8 cp

= 1.2 bbl/STB

= 1.01 bbl/STB

= 12 MMSTB

Constant (P

inj

– P

) = 950 psi

Wellbore radius = 0.3 ft

a. Generate the performance curves for the base layer.

b. Generate the composite (overall) performance curves for the reser-

voir.

REFERENCES

1. Buckley, S., and Leverett, M., “Mechanism of Fluid Displacement in

Sands,” Trans. AIME, 1962, Vol. 146, p. 107.

2. Caudle, B., and Witte, M., “Production Potential Changes During Sweep-out

in a Five-Spot System,” Trans. AIME, 1959, Vol. 216, pp. 446–448.

3. Cole, F., Reservoir Engineering Manual. Houston, TX: Gulf Publishing

Company, 1969.

4. Craft, B., and Hawkins, M., Applied Petroleum Reservoir Engineering.

Englewood Cliffs, NJ: Prentice Hall, 1959.

5. Craig, Jr., F. F., The Reservoir Engineering Aspects of Waterflooding. Dal-

las: Society of Petroleum Engineers of AIME, 1971.

6. Craig, F., Geffen, T., and Morse, R., “Oil Recovery Performance of Pattern

Gas or Water Injection Operations from Model Tests,” JPT, Jan. 1955, pp.

7–15, Trans. AIME, p. 204.

7. Dake, L., Fundamentals of Reservoir Engineering. Amsterdam: Elsevier Sci-

entific Publishing Company, 1978.

8. Deppe, J., “Injection Rates—The Effect of Mobility Ratio, Areal Sweep, and

Pattern,” SPEJ, June 1961, pp. 81–91.

9. de Souza, A., and Brigham, W., “A Study on Dykstra–Parsons Curves,” TR29.

Palo Alto, CA: Stanford University Petroleum Research Institute, 1981.

1024 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 14 2001-10-25 17:38 Page 1024

10. Dyes, A., Caudle, B., and Erickson, R., “Oil Production After Breakthrough

as Influenced by Mobility Ratio,” JPT, April 1954, pp. 27–32; Trans. AIME,

p. 201.

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

Flood,” in Secondary Recovery of Oil in the United States, 2nd ed. Washing-

ton, DC: American Petroleum Institute, 1950, pp. 160–174.

12. Fassihi, M., “New Correlations for Calculation of Vertical Coverage and

Areal Sweep Efficiency,” SPERE, Nov. 1986, pp. 604–606.

13. Felsenthal, M., Cobb, T., and Heur, G, “A Comparison of Waterflooding

Evaluation Methods,” SPE Paper 332 presented at the SPE 5th Biennial Sec-

ondary Recovery Symposium, Wichita Falls, TX, May 7, 1962.

14. Johnson, C., “Prediction of Oil Recovery by Waterflood—A Simplified

Graphical Treatment of the Dykstra-Parsons Method,” Trans. AIME, 1956,

Vol. 207, pp. 345–346.

15. Khelil, C., “A Correlation of Optimum Free Gas Saturation,” SPE Paper

01983, 1983.

16. Landrum, B., and Crawford, P., “Effect of Directional Permeability on

Sweep Efficiency and Production Capacity,” JPT, Nov. 1960, pp. 67–71.

17. Miller, M., and Lents, M., “Performance of Bodcaw Reservoir, Cotton Val-

ley Field Cycling Project, New Methods of Predicting Gas-Condensate

Reservoirs,” SPEJ, Sep. 1966, p. 239.

18. Muskat, M., Flow of Homogeneous Fluids Through Porous Systems. Ann

Arbor, MI: J. W. Edwards, 1946.

19. Muskat, M., “The Theory of Nine-Spot Flooding Networks,” Prod. Monthly,

March 1948, Vol. 13, No. 3, p. 14.

20. Stiles, W., “Use of Permeability Distribution in Waterflood Calculations,”

Trans. AIME, 1949, Vol. 186, p. 9.

21. Terwilliger, P., Wilsey, L., Hall, H., Bridges, P., and Morse, R., “An Experi-

mental and Theoretical Investigation of Gravity Drainage Performance,”

Trans. AIME, 1951, Vol. 192, pp. 285–296.

22. Thomas, C. E., Mahoney, C. F., and Winter, G. W., Petroleum Engineering

Handbook. Dallas: Society of Petroleum Engineers, 1989.

23. Welge, H., “A Simplified Method for Computing Oil Recovery By Gas or

Water Drive,” Trans. AIME, 1952, pp. 91–98.

24. Willhite, G. P., Waterflooding. Dallas: Society of Petroleum Engineers,

1986.

Principles of Waterflooding 1025

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1026

A phase is defined as that part of a system that is uniform in physi-

cal and chemical properties, homogeneous in composition, and sepa-

rated from other coexisting phases by definite boundary surfaces. The

most important phases occurring in petroleum production are the

hydrocarbon liquid phase and the gas phase. Water is also commonly

present as an additional liquid phase. These can coexist in equilibrium

when the variables describing change in the entire system remain con-

stant with time and position. The chief variables that determine the

state of equilibrium are system temperature, system pressure, and

composition.

The conditions under which these different phases can exist are a mat-

ter of considerable practical importance in designing surface separation

facilities and developing compositional models. These types of calcula-

tions are based on the concept of equilibrium ratios.

VAPOR PRESSURE

A system that contains only one component is considered the simplest

type of hydrocarbon system. The word component refers to the number

of molecular or atomic species present in the substance. A single-

component system is composed entirely of one kind of atom or mole-

cule. We often use the word pure to describe a single-component

system. The qualitative understanding of the relationship that exists

CHAPTER 15

VAPOR–LIQUID PHASE

EQUILIBRIA

Reservoir Eng Hndbk Ch 15 2001-10-25 17:41 Page 1026

between temperature T, pressure p, and volume V of pure components

can provide an excellent basis for understanding the phase behavior of

complex hydrocarbon mixtures.

Consider a closed evacuated container that has been partially filled

with a pure component in the liquid state. The molecules of the liquid

are in constant motion with different velocities. When one of these mol-

ecules reaches the liquid surface, it may possess sufficient kinetic ener-

gy to overcome the attractive forces in the liquid and pass into the vapor

spaces above. As the number of molecules in the vapor phase increases,

the rate of return to the liquid phase also increases. A state of equilibri-

um is eventually reached when the number of molecules leaving and

returning is equal. The molecules in the vapor phase obviously exert a

pressure on the wall of the container and this pressure is defined as the

vapor pressure, p

. As the temperature of the liquid increases, the aver-

age molecular velocity increases with a larger number of molecules pos-

sessing sufficient energy to enter the vapor phase. As a result, the vapor

pressure of a pure component in the liquid state increases with increas-

ing temperature.

A method that is particularly convenient for expressing the vapor pres-

sure of pure substances as a function of temperature is shown in Figure

15-1. The chart, known as the Cox chart, uses a logarithmic scale for the

vapor pressure and an entirely arbitrary scale for the temperature in °F.

The vapor pressure curve for any particular component, as shown in Fig-

ure 15-1, can be defined as the dividing line between the area where

vapor and liquid exists. If the system pressure exists at its vapor pressure,

two phases can coexist in equilibrium. Systems represented by points

located below that vapor pressure curve are composed only of the vapor

phase. Similarly, points above the curve represent systems that exist in

the liquid phase. These statements can be conveniently summarized by

the following expressions:

• p < p

→ system is entirely in the vapor phase

• p > p

→ system is entirely in the liquid phase

• p = p

→ vapor and liquid coexist in equilibrium

where p is the pressure exerted on the pure component. Note that the

above expressions are valid only if the system temperature T is below the

critical temperature T

of the substance.

Vapor–Liquid Phase Equilibria 1027

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