Tarek Ahmed. Reservoir engineering handbook

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We should emphasize that all of the above derivations are based

on the assumption that no free gas exists from the start of the

flood till abandonment.

Step 9. Calculate the surface water–oil ratio WOR

that corresponds to

each value of f

(as determined in step 2) from Equation 14-28:

Step 10. Calculate the oil and water flow rates from the following

derived relationships:

Introducing the surface water–oil ratio into the above expres-

sion gives:

Solving for Q

gives:

and

where Q

= oil flow rate, STB/day

= water flow rate, STB/day

= water injection rate, bbl/day

Step 11. The preceding calculations as described in steps 1 through 10

can be organized in the following tabulated form:

(

)

WOR 14 - 56

B B WOR

ow s

(

)

14 - 55

i Q B Q WOR B

wooo sw

iQBQB

wooww

WOR

−













928 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 928

(df

/dS

–

inj

WOR

wBT

• S

wBT

DBT

PBT

iBT

0 •••

•••••••••••••

(1 – S

) 1.0 ••••••••100% 0 •

Step 12. Express the results in a graphical form.

Example 14-10

The data of Example 14-7 are reproduced here for convenience:

w 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75

ro 30.23 17.00 9.56 5.38 3.02 1.70 0.96 0.54 0.30 0.17 0.10

w 0.062 0.105 0.173 0.271 0.398 0.541 0.677 0.788 0.869 0.922 0.956

/dS

w 0.670 10.84 1.647 2.275 2.759 2.859 2.519 1.922 1.313 0.831 0.501



= 2.0 cp 

= 1.0 cp

= 1.25 bbl/STB B

= 1.02 bbl/STB

 = 25% h = 20 ft

= 20% S

= 20%

= 900 bbl/day (PV) = 775,779 bbl

= 496,449 STB E

= 100%

Predict the waterflood performance to abandonment at a WOR

of 45

STB/STB.

Solution

Step 1. Plot f

vs. S

as shown in Figure 14-30 and construct the

tangent to the curve. Extrapolate the tangent to f

=1.0 and

determine:

= S

wBT

= 0.596

= f

wBT

= 0.780

(df

/dS

)

swf

= 1.973

iBT

= 1/1.973 = 0.507

= 0.707

wBT

Principles of Waterflooding 929

Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 929

Figure 14-30. Fractional flow curve for Example 14-10.

Step 2. Calculate E

DBT

by using Equation 14-50:

DBT

−

0 707 0 20

1020

0 634

930 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 930

Step 3. Calculate (N

)

by applying Equation 14-52:

Step 4. Calculate cumulative water injected at breakthrough from Equa-

tion 14-42:

Step 5. Calculate the time to breakthrough:

Step 6. Calculate WOR

exactly at breakthrough by applying Equation

14-28:

Step 7. Describe the recovery performance to breakthrough in the fol-

lowing tabulated form:

t, days W

inj

= 900 t WOR

= Q

WOR

00 0 0 0 00

100.0 90,000 72,000 720 0 0 0

200.0 180,000 144,000 720 0 0 0

300.0 270,000 216,000 720 0 0 0

400.0 360,000 288,000 720 0 0 0

436.88 393,198 314,780 720 4.34 3125 0

Step 8. Following the computational procedure as outlined for recov-

ery performance after breakthrough, construct the following

table:

inj

WOR STB STB

−









125

102

078

434

. /

t days

393 198

900

436 88

W bbl

iBT

(

)

=775 779 0 507 393198,. ,

N STB

(

)

(

)

=496 499 0 634 314 780, .,

Principles of Waterflooding 931

Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 931

/dS

inj

t, days W

WOR

0.598 0.784 1.948 0.513 0.709 0.636 315,773 397,975 442 82,202 4.45 155 690

0.600 0.788 1.922 0.520 0.710 0.638 316,766 403,405 448 86,639 4.56 153 698

0.700 0.922 0.831 1.203 0.794 0.743 368,899 933,262 1,037 564,363 14.49 56 814

0.800 0.974 0.293 3.407 0.889 0.861 427,486 2,643,079 2,937 2,215,593 45.91 19 859

Step 9. Express graphically results of the calculations as a set of perfor-

mance curves, as shown in Figure 14-31.

Figure 14-31. Performance curves of Example 14-10.

II. AREAL SWEEP EFFICIENCY

The areal sweep efficiency E

is defined as the fraction of the total

flood pattern that is contacted by the displacing fluid. It increases steadily

with injection from zero at the start of the flood until breakthrough

occurs, after which E

continues to increase at a slower rate.

The areal sweep efficiency depends basically on the following three

main factors:

1. Mobility ratio M

2. Flood pattern

3. Cumulative water injected W

inj

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Mobility Ratio

In general, the mobility of any fluid λ is defined as the ratio of the

effective permeability of the fluid to the fluid viscosity, i.e.:

where λ

, λ

= mobility of oil, water, and gas, respectively

, k

= effective permeability to oil, water, and gas,

respectively

, k

= relative permeability to oil, water, and gas, respectively

k = absolute permeability

The fluid mobility as defined mathematically by the above three relation-

ships indicates that λ is a strong function of the fluid saturation. The

mobility ratio M is defined as the mobility of the displacing fluid to the

mobility of the displaced fluid, or:

For waterflooding then:

Substituting for λ:

Simplifying gives:

(

)

14 - 60

displacing

displaced

µµ

kkk

(

)

14 - 59

µµ

kkk

(

)

14 - 58

µµ

kkk

(

)

14 - 57

Principles of Waterflooding 933

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Muskat (1946) points out that in calculating M by applying Equation 14-60,

the following concepts must be employed in determining k

and k

• Relative permeability of oil k

. Because the displaced oil is moving

ahead of the water front in the noninvaded portion of the pattern, as

shown schematically in Figure 14-32, k

must be evaluated at the ini-

tial water saturation S

Figure 14-32. Oil and water mobilities to breakthrough.

• Relative permeability of water k

. The displacing water will form a

water bank that is characterized by an average water saturation of

in the swept area. This average saturation will remain constant until

breakthrough, after which the average water saturation will continue to

increase (as denoted by ). The mobility ratio, therefore, can be

expressed more explicitly under two different stages of the flood:

From the start to breakthrough:

where = relative permeability of water at

wBT

rw wBT

ro wi w

(

)

14 - 61

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Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 934

@ S

= relative permeability of oil at S

The above relationship indicates that the mobility ratio will remain

constant from the start of the flood until breakthrough occurs.

After breakthrough:

Equation 14-62 indicates that the mobility of the water krw/µw will

increase after breakthrough due to the continuous increase in the aver-

age water saturation . This will result in a proportional increase in

the mobility ratio M after breakthrough, as shown in Figure 14-33.

Figure 14-33. Mobility ratio versus time relationship.

In general, if no further designation is applied, the term mobility ratio

refers to the mobility ratio before breakthrough.

Flood Patterns

In designing a waterflood project, it is common practice to locate

injection and producing wells in a regular geometric pattern so that a

rw w

ro wi w

(

)

14 - 62

Principles of Waterflooding 935

Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 935

symmetrical and interconnected network is formed. As shown previously

in Figure 14-9, regular flood patterns include these:

• Direct line drive

• Staggered line drive

• Five spot

• Seven spot

• Nine spot

By far the most used pattern is the five spot and, therefore, most of the

discussion in the remainder of the chapter will focus on this pattern.

Craig et al. (1955) performed experimental studies on the influence of

fluid mobilities on the areal sweep efficiency resulting from water or gas

injection. Craig and his co-investigators used horizontal laboratory mod-

els representing a quadrant of five spot patterns. Areal sweep efficiencies

were determined from x-ray shadowgraphs taken during various stages

of the displacement as illustrated in Figure 14-34. Two mobility ratios,

1.43 and 0.4, were used in the study.

Figure 14-34. X-ray shadowgraphs of flood progress. (Permission to publish by

the Society of Petroleum Engineers.)

936 Reservoir Engineering Handbook

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Figure 14-34 shows that at the start of the flood, the water front takes on a

cylindrical form around the injection point (well). As a result of the continu-

ous injection, pressure distribution and corresponding streamlines are devel-

oped between the injection and production wells. However, various stream-

lines have different lengths with the shortest streamline being the direct line

between the injector and producer. The pressure gradient along this line is the

highest that causes the injection fluid to flow faster along the shortest stream-

line than the other lines. The water front gradually begins to deform from the

cylindrical form and cusp into the production well as water breakthrough

occurs. The effect of the mobility ratio on the areal sweep efficiency is

apparent by examining Figure 14-34. This figure shows that at breakthrough,

only 65% of the flood pattern area has been contacted (swept) by the injec-

tion fluid with a mobility ratio of 1.43 and 82.8% when the mobility ratio is

0.4. This contacted fraction when water breakthrough occurs is defined as

the areal sweep efficiency at breakthrough, as denoted by E

ABT

. In gener-

al, lower mobility ratios would increase the areal sweep efficiency and

higher mobility ratios would decrease the E

. Figure 14-34 also shows

that with continued injection after breakthrough, the areal sweep efficiency

continues to increase until it eventually reaches 100%.

Cumulative Water Injected

Continued injection after breakthrough can result in substantial increases

in recovery, especially in the case of an adverse mobility ratio. The work of

Craig et al. (1955) has shown that significant quantities of oil may be swept

by water after breakthrough. It should be pointed out that the higher the

mobility ratio, the more important is the “after-breakthrough” production.

Areal Sweep Prediction Methods

Methods of predicting the areal sweep efficiency are essentially divid-

ed into the following three phases of the flood:

• Before breakthrough

• At breakthrough

• After breakthrough

Phase 1: Areal Sweep Efficiency Before Breakthrough

The areal sweep efficiency before breakthrough is simply proportional

to the volume of water injected and is given by:

Principles of Waterflooding 937

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