Tarek Ahmed. Reservoir engineering handbook

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m = 0

= 0

= R

Imposing the above conditions on the MBE reduces the equation to the

following simplified form:

with

∆p = p

− p

where p

= initial reservoir pressure

p = current reservoir pressure

Hawkins (1955) introduced the oil compressibility c

into the MBE to

further simplify the equation. The oil compressed is defined in Chapter

2 by:

rearranging, gives:

− B

= c

∆p

Combining the above expression with Equation 12-26 gives:

The denominator of the above equation can be written as:

cB p B

Sc c

ooi oi

wi w f

−













∆∆

(12 - 27)

ooi

−

∆

BB B

Sc c

ooi oi

wi w f

−+

−













()

∆

(12 - 26)

798 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 12 2001-10-25 16:04 Page 798

Since there are only two fluids in the reservoir, i.e., oil and water, then:

+ S

= 1

Equation 12-28 can then be expressed as:

The term between the two brackets is called the effective compress-

ibility and defined by Hawkins (1955) as:

Combining Equations 12-27, 12-28, and 12-29, the MBE above the

bubble-point pressure becomes:

Equation 12-30 can be expressed as an equation of a straight line by:

Figure 12-6 indicates that the reservoir pressure will decrease linearly

with cumulative reservoir voidage N

Rearranging Equation 12-31 and solving for the cumulative oil pro-

duction N

gives:

NNc













∆ (12 - 32)

NB c

oi e

=−













(12 - 31)

Bc p

Bc P P

(12 - 30)

oi e

oi e i

−

()

∆

Sc S c c

oi o wi w f

−1

(12 - 29)

Sc S c c

oi o wi w f

−













∆

oi o

wi w

−













∆ (12 - 28)

Predicting Oil Reservoir Performance 799

Reservoir Eng Hndbk Ch 12 2001-10-25 16:04 Page 799

The calculation of future reservoir production, therefore, does not

require a trial-and-error procedure, but can be obtained directly from the

above expression.

Example 12-3

The following data are available on a volumetric undersaturated-oil

reservoir:

= 4000 psi p

= 3000 psi N = 85 MMSTB

= 5 × 10

−6

psi

−1

= 15 × 10

−6

psi

−1

= 3 × 10

−6

psi

−1

= 30% B

= 1.40 bbl/STB

Estimate cumulative oil production when the reservoir pressure drops

to 3500 psi. The oil formation volume factor at 3500 psi is 1.414

bbl/STB.

800 Reservoir Engineering Handbook

Figure 12-6. Pressure voidage relationship.

Reservoir Eng Hndbk Ch 12 2001-10-25 16:04 Page 800

Solution

Step 1. Determine the effective compressibility from Equation 12-29.

Step 2. Estimate N

from Equation 12-32.

Saturated-Oil Reservoirs

If the reservoir originally exists at its bubble-point pressure, the reser-

voir is referred to as a saturated-oil reservoir. This is considered as the

second type of the solution-gas-drive-reservoir. As the reservoir pressure

declines below the bubble-point, the gas begins to evolve from solution.

The general MBE may be simplified by assuming that the expansion of

the gas is much greater than the expansion of rock and initial water and,

therefore, can be neglected. For a volumetric and saturated-oil reservoir

with no fluid injection, the MBE can be expressed by:

The above material balance equation contains two unknowns, which are:

• Cumulative oil production N

• Cumulative gas production G

The following reservoir and PVT data must be available in order to

predict the primary recovery performance of a depletion-drive reservoir

in terms of N

and G

a. Initial oil-in-place N

Generally the volumetric estimate of in-place oil is used in calculating

the performance. Where there is sufficient solution-gas-drive history,

NB G NR B

BB R RB

po p ps g

ooi sisg

+−

−+−

()

()()

(12 - 33)

MSTB

=× ×









−

−−

()(.)

()

85 10 23 43 10

1 411

1 400

4000 3500

985 18

psi

×+ ×+×

−

=×

−−−

−−

(.)( ) (.)( )

0 7 15 10 0 3 3 10 5 10

103

23 43 10

666

Predicting Oil Reservoir Performance 801

Reservoir Eng Hndbk Ch 12 2001-10-25 16:04 Page 801

however, this estimate may be checked by calculating a material-bal-

ance estimate.

b. Hydrocarbon PVT data

Since differential gas liberation is assumed to best represent the condi-

tions in the reservoir, differential laboratory PVT data should be used

in reservoir material balance. The flash PVT data are then used to con-

vert from reservoir conditions to stock-tank conditions.

If laboratory data are not available, reasonable estimates may some-

times be obtained from published correlations. If differential data are

not available, the flash data may be used instead; however, this may

result in large errors for high-solubility crude oils.

c. Initial fluid saturations

Initial fluid saturations obtained from a laboratory analysis of core

data are preferred; however, if these are not available, estimates in

some cases may be obtained from a well-log analysis or may be

obtained from other reservoirs in the same or similar formations.

d. Relative permeability data

Generally, laboratory-determined k

and k

data are averaged to

obtain a single representative set for the reservoir. If laboratory data

are not available, estimates in some cases may be obtained from other

reservoirs in the same or similar formations.

Where there is sufficient solution-gas-drive history for the reservoir,

calculate (k

) values versus saturation from Equations 12-15 and

12-17, i.e.:

= (1 − S

) (1 − N

/N) (B

)

= (GOR − R

) (µ

/µ

)

The above results should be compared with the averaged laboratory

relative permeability data. This may indicate a needed adjustment in

the early data and possibly an adjustment in the overall data.

All the techniques that are used to predict the future performance of a

reservoir are based on combining the appropriate MBE with the instanta-

neous GOR using the proper saturation equation. The calculations are

repeated at a series of assumed reservoir pressure drops. These calcula-

tions are usually based on one stock-tank barrel of oil in place at the bub-

802 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 12 2001-10-25 16:04 Page 802

ble-point pressure, i.e., N = 1. This avoids carrying large numbers in the

calculation procedure and permits calculations to be made on the basis of

the fractional recovery of initial oil in place.

There are several widely used techniques that were specifically devel-

oped to predict the performance of solution-gas-drive reservoirs, including:

• Tracy’s method

• Muskat’s method

• Tarner’s method

These methodologies are presented below.

Tracy’s Method

Tracy (1955) suggests that the general material balance equation can

be rearranged and expressed in terms of three functions of PVT vari-

ables. Tracy’s arrangement is given in Chapter 11 by Equation 11-53 and

is repeated here for convenience:

N = N

+ G

+ (W

− W

) Φ

(12 - 34)

where Φ

, Φ

, and Φ

are considered PVT-related properties that are

functions of pressure and defined by:

with

Den B B R R B m B

ooi sisg oi

=−

(

)

+−

(

)

+−













1 (12 - 35)

Den

−BRB

Den

osg

Predicting Oil Reservoir Performance 803

Reservoir Eng Hndbk Ch 12 2001-10-25 16:04 Page 803

For a solution-gas-drive reservoir, Equations 12-34 and 12-35 are

reduced to the following expressions, respectively:

N = N

+ G

(12 - 36)

and

Den = (B

− B

) + (R

− R

) B

(12 - 37)

Tracy’s calculations are performed in series of pressure drops that pro-

ceed from known reservoir condition at the previous reservoir pressure p*

to the new assumed lower pressure p. The calculated results at the new

reservoir pressure become “known” at the next assumed lower pressure.

In progressing from the conditions at any pressure p* to the lower

reservoir pressure p, consider that the incremental oil and gas production

are ∆N

and ∆G

, or:

= N

+∆N

(12 - 38)

= G

+∆G

(12 - 39)

where N

, G

= “known” cumulative oil and gas production at

previous pressure level p*

, G

= “unknown” cumulative oil and gas at new pressure

level p

Replacing N

and G

in Equation 12-36 with those of Equations 12-38

and 12-39 gives:

N = (N

+∆N

) Φ

+ (G

+∆G

) Φ

(12 - 40)

Define the average instantaneous GOR between the two pressure p*

and p by:

The incremental cumulative gas production ∆G

can be approximated

by Equation 12-7 as:

∆G

= (GOR)

avg

∆N

(12 - 42)

()

GOR

GOR GOR

avg

(12 - 41)

804 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 12 2001-10-25 16:04 Page 804

Replacing ∆G

in Equation 12-40 with that of 12-41 gives:

N = [N

+∆N

] Φ

+ [G

+∆N

(GOR)

avg

] Φ

(12 - 43)

If Equation 12-43 is expressed for N = 1, the cumulative oil produc-

tion N

and cumulative gas production G

become fractions of initial oil

in place. Rearranging Equation 12-43 gives:

Equation 12-44 shows that there are essentially two unknowns, the

incremental cumulative oil production ∆N

and the average gas oil ratio

(GOR)

avg

Tracy suggested the following alternative technique for solving Equa-

tion 12-44.

Step 1. Select an average reservoir pressure p.

Step 2. Calculate the values of the PVT functions Φ

and Φ

Step 3. Estimate the GOR at p.

Step 4. Calculate the average instantaneous GOR (GOR)

avg

= (GOR* +

GOR)/2.

Step 5. Calculate the incremental cumulative oil production ∆N

from

Equation 12-44 as:

Step 6. Calculate cumulative oil production N

= N

+ ∆N

∆

ΦΦ

GOR

po pg

o avg g

−+

1( )

()

∆

ΦΦ

GOR

po pg

o avg g

−+

1( )

()

(12 - 44)

Predicting Oil Reservoir Performance 805

Reservoir Eng Hndbk Ch 12 2001-10-25 16:04 Page 805

Step 7. Calculate the oil and gas saturations at selected average reservoir

pressure by using Equations 12-15 and 12-16, as:

= (1 − S

) (1 − N

) (B

)

= 1 − S

− S

Step 8. Obtain relative permeability ratio k

at S

Step 9. Calculate the instantaneous GOR from Equation 12-1.

GOR = R

+ (k

) (µ

/µ

)

Step 10. Compare the estimated GOR in Step 3 with the calculated GOR

in Step 9. If the values are within acceptable tolerance, proceed

to next step. If not within the tolerance, set the estimated GOR

equal to the calculated GOR and repeat the calculations from

Step 3.

Step 11. Calculate the cumulative gas production.

= G

+∆N

(GOR)

avg

Step 12. Since results of the calculations are based on 1 STB of oil ini-

tially in place, a final check on the accuracy of the prediction

should be made on the MBE, or:

+ G

= 1 ± tolerance

Step 13. Repeat from Step 1.

As the calculation progresses, a plot of GOR versus pressure can be

maintained and extrapolated as an aid in estimating GOR at each new

pressure.

Example 12-4

The following PVT data characterize a solution-gas-drive reservoir.

806 Reservoir Engineering Handbook

The example data and solution are given by Economides, M., Hill, A., and Economides,

C., Petroleum Production System, Prentice Hall Petroleum Engineering series, 1994.

Reservoir Eng Hndbk Ch 12 2001-10-25 16:04 Page 806

The relative permeability data are shown in Figure 12-7.

psi bbl/STB bbl/scf scf/STB

4350 1.43 6.9 × 10

−4

840

4150 1.420 7.1 × 10

−4

820

3950 1.395 7.4 × 10

−4

770

3750 1.380 7.8 × 10

−4

730

3550 1.360 8.1 × 10

−4

680

3350 1.345 8.5 × 10

−4

640

Predicting Oil Reservoir Performance 807

Figure 12-7. Relative permeability data for Example 12-3. (After Economides, M.,

et al., Petroleum Production Systems, Prentice Hall Petroleum Engineers Series, 1994.)

Reservoir Eng Hndbk Ch 12 2001-10-25 16:04 Page 807