Tarek Ahmed. Reservoir engineering handbook

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–

-y

, Mscf/day

1952 316 ¥ 10

1800 270 ¥ 10

46 ¥ 10

1794

1600 215 ¥ 10

101 ¥ 10

3503

1000 100 ¥ 10

216 ¥ 10

6331

500 40 ¥ 10

276 ¥ 10

7574

0 0 316 ¥ 10

8342 = AOF (Q

)

max

c. Compare the gas flow rates as calculated by the four different meth-

ods. Results of the IPR calculation are documented below:

Gas Flow Rate, Mscf/day

Pressure Back-pressure p

-Approach p-Approach y-Approach

19520 0 0 0 0

1800 1720 1675 1879 1811

1600 3406 3436 3543 3554

1000 6891 6862 6942 6460

500 8465 8304 9046 7742

0 8980 8763 10827 8536

6.0% 5.4% 11% —

558 Reservoir Engineering Handbook

27000

26000

28000

29000

30000

31000

32000

33000

1000 2000 3000 4000 60005000

Flow Rate

Differential Pressure Over Flow Rate

Figure 8-12. Pseudopressure method.

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 558

Since the pseudo-pressure analysis is considered more accurate and

rigorous than the other three methods, the accuracy of each of the meth-

ods in predicting the IPR data is compared with that of the y-approach.

Figure 8-13 compares graphically the performance of each method with

that of y-approach. Results indicate that the pressure-squared equation

generated the IPR data with an absolute average error of 5.4% as com-

pared with 6% and 11% for the back-pressure equation and the pressure-

approximation method, respectively.

It should be noted that the pressure-approximation method is limited to

applications for pressures greater than 3000 psi.

Future Inflow Performance Relationships

Once a well has been tested and the appropriate deliverability or

inflow performance equation established, it is essential to predict the IPR

data as a function of average reservoir pressure. The gas viscosity m

and

gas compressibility z-factor are considered the parameters that are sub-

ject to the greatest change as reservoir pressure p

–

changes.

Assume that the current average reservoir pressure is p

–

, with gas vis-

cosity of m

and a compressibility factor of z

. At a selected future aver-

Gas Well Performance 559

500

1000

1500

2000

2500

2000 4000 6000 8000 1200010000

Flow Rate (Mscf/day)

Pressrue (psi)

(Back-pressure)

(Pressure-Squared)

(Pressure)

(Psuedopressure)

Figure 8-13. IPR for all methods.

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 559

age reservoir pressure p

–

, m

and z

represent the corresponding gas

properties. To approximate the effect of reservoir pressure changes, i.e.

from p

–

to p

–

, on the coefficients of the deliverability equation, the fol-

lowing methodology is recommended:

Back-Pressure Equation

The performance coefficient C is considered a pressure-dependent

parameter and adjusted with each change of the reservoir pressure

according to the following expression:

The value of n is considered essentially constant.

LIT Methods

The laminar flow coefficient a and the inertial-turbulent flow coeffi-

cient b of any of the previous LIT methods, i.e., Equations 8-24, 8-29,

and 8-34, are modified according to the following simple relationships:

• Pressure-Squared Method

The coefficients a and b of pressure-squared are modified to account

for the change of the reservoir pressure from p

–

to p

–

by adjusting the

coefficients as follows:

where the subscripts 1 and 2 represent conditions at reservoir pressure

–

to p

–

, respectively.

(8 - 42)

(8 - 41)

(8 - 40)

560 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 560

• Pressure-Approximation Method

where B

is the gas formation volume factor

• Pseudopressure Approach

The coefficients a and b of the pseudo-pressure approach are essen-

tially independent of the reservoir pressure and they can be treated as

constants.

Example 8-3

In addition to the data given in Example 8-2, the following informa-

tion is available:

• (m

z) = 0.01206 at 1952 psi

• (m

z) = 0.01180 at 1700 psi

Using the following methods:

a. Back-pressure method

b. Pressure-squared method

c. Pseudo-pressure method

Generate the IPR data for the well when the reservoir pressure drops

from 1952 to 1700 psi.

Solution

Step 1. Adjust the coefficients a and b of each equation. For the:

• Back-pressure equation:

Using Equation 8-40, adjust C:

(8 - 44)

(8 - 43)

Gas Well Performance 561

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 561

= 0.01727 (1700

- p

)0.87

• Pressure-squared method:

Adjust a and b by applying Equations 8-41 and 8-42

(1700

- p

) = 311.14 Q

+ 0.01304 Q

• Pseudopressure method:

No adjustments are needed.

(245 ¥ 10

) - y

= (22.28 ¥ 10

) Q

+ 1.727 Q

Step 2. Generate the IPR data:

Gas Flow rate Q

, Mscf/day

Back-Pressure p

-Method y-Method

–

= 1700 0 0 0

1600 1092 1017 1229

1000 4987 5019 4755

500 6669 6638 6211

0 7216 7147 7095

Figure 8-14 compares graphically the IPR data as predicted by the

above three methods.

HORIZONTAL GAS WELL PERFORMANCE

Many low permeability gas reservoirs are historically considered to be

noncommercial due to low production rates. Most vertical wells drilled in

tight gas reservoirs are stimulated using hydraulic fracturing and/or

b =

=0 01333

0 01180

0 01206

0 01304.

a =

=318

0 01180

0 01206

311 14

C =

=0 0169

0 01206

0 01180

0 01727.

562 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 562

acidizing treatments to attain economical flow rates. In addition, to

deplete a tight gas reservoir, vertical wells must be drilled at close spacing

to efficiently drain the reservoir. This would require a large number of

vertical wells. In such reservoirs, horizontal wells provide an attractive

alternative to effectively deplete tight gas reservoirs and attain high flow

rates. Joshi (1991) points out those horizontal wells are applicable in both

low-permeability reservoirs as well as in high-permeability reservoirs.

An excellent reference textbook by Sada Joshi (1991) gives a compre-

hensive treatment of horizontal wells performance in oil and gas reser-

voirs.

In calculating the gas flow rate from a horizontal well, Joshi intro-

duced the concept of the effective wellbore radius r¢

into the gas flow

equation. The effective wellbore radius is given by:

with

aLahr

(/)

[(/)[/()]

11 2 2

(8 - 45)

Gas Well Performance 563

Figure 8-14. IPR comparison.

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 563

and

where L = length of the horizontal well, ft

h = thickness, ft

= wellbore radius, ft

= horizontal well drainage radius, ft

a = half the major axis of drainage ellipse, ft

A = drainage area, acres

Methods of calculating the horizontal well drainage area A are present-

ed in Chapter 7 by Equations 7-45 and 7-46.

For a pseudosteady-state flow, Joshi expressed Darcy’s equation of a

laminar flow in the following two familiar forms:

Pressure-Squared Form

where Q

= gas flow rate, Mscf/day

s = skin factor

k = permeability, md

T = temperature, °R

Pseudo-Pressure Form

rwf

()

ln .

1422 0 75

kh p p

Tz rr s

rwf

g avg eh w

[]

()

()ln(/).

1422 0 75m

(8 - 48)

43 560,

(8 - 47)

[]

05 025 2

..(/)

(8 - 46)

564 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 564

Example 8-4

A 2,000-foot-long horizontal gas well is draining an area of approxi-

mately 120 acres. The following data are available:

–

= 2000 psi y

–

= 340 ¥ 10

psi

/cp

= 1200 psi y

= 128 ¥ 10

psi

/cp

avg

= 0.011826 r

= 0.3 ft s = 0.5

h = 20 ft T = 180°F k = 1.5 md

Assuming a pseudosteady-state flow, calculate the gas flow rate by

using the pressure-squared and pseudopressure methods.

Solution

Step 1. Calculate the drainage radius of the horizontal well:

Step 2. Calculate half the major axis of drainage ellipse by using Equa-

tion 8-46:

Step 3. Calculate the effective wellbore radius r¢

from Equation 8-45:

Applying Equation 8-45, gives:

2000

2 1495 8

1 7437

=+ -

a(.)

(/ )

()(.)

203

1 0357

20 2000

a =

2000

05 025

2 1290

2000

1495 8

()( )

rft

(, )( )43 560 120

1290

Gas Well Performance 565

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 565

Step 4. Calculate the flow rate by using the pressure-squared approxima-

tion and y-approach.

• Pressure-squared

• y -Method

For turbulent flow, Darcy’s equation must be modified to account for

the additional pressure caused by the non-Darcy flow by including the

rate-dependent skin factor DQ

. In practice, the back-pressure equation

and the LIT approach are used to calculate the flow rate and construct the

IPR curve for the horizontal well. Multirate tests, i.e., deliverability tests,

must be performed on the horizontal well to determine the coefficients of

the selected flow equation.

PROBLEMS

1. A gas well is producing under a constant bottom-hole flowing pressure

of 1000 psi. The specific gravity of the produced gas is 0.65, given:

= 1500 psi r

= 0.33 ft r

= 1000 ft k = 20 md

h = 20 ft T = 140°F s = 0.40

Mscf day

( . )( )( )( )

()()ln

1 5 20 340 128 10

1422 640

1290

477 54

075 05

9396

Mscf day

( . )( )( )

()()(. )ln

1 5 20 2000 1200

1422 640 0 011826

1290

477 54

075 05

9 594

==rft

1290 200 2

1495 8 1 7437 1 0357

477 54

(/)

.(. )(. )

566 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 566

Calculate the gas flow rate by using:

a. Real gas pseudopressure approach

b. Pressure-squared approximation

2. The following data

were obtained from a back-pressure test on a gas well.

, Mscf/day p

, psi

0 481

4928 456

6479 444

8062 430

9640 415

a. Calculate values of C and n

b. Determine AOF

c. Generate the IPR curves at reservoir pressures of 481 and 300 psi.

3. The following back-pressure test data are available:

, Mscf/day p

, psi

0 5240

1000 4500

1350 4191

2000 3530

2500 2821

Given:

gas gravity = 0.78

porosity = 12%

= 15%

T = 281°F

a. Generate the current IPR curve by using:

i. Simplified back-pressure equation

ii. Laminar-inertial-turbulent (LIT) methods:

• Pressure-squared approach

Gas Well Performance 567

Chi Ikoku, Natural Gas Reservoir Engineering, John Wiley and Sons, 1984.

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 567