Tarek Ahmed. Reservoir engineering handbook

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the turbulent flow condition. In a linear form, Equation 8-17 can be

expressed as:

The laminar flow coefficient a

and inertial-turbulent flow coefficient

can be determined from the linear plot of the above equation as shown

in Figure 8-5.

Having determined the coefficient a

and b

, the gas flow rate can be

determined at any pressure from:

The application of Equation 8-29 is also restricted by the assumptions

listed for the pressure-squared approach. However, the pressure method

is applicable at pressures higher than 3000 psi.

aa bpp

rwf

-+ + -

(

)

(8 - 33)

abQ

rwf

(8 - 32)

548 Reservoir Engineering Handbook

Figure 8-5. Graph of the pressure-method data.

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 548

c. Pseudopressure Quadratic Approach

Equation 8-19 can be written as:

where

The term (a

) in Equation 8-34 represents the pseudopressure

drop due to laminar flow while the term (b

) accounts for the

pseudopressure drop due to inertial-turbulent flow effects.

Equation 8-34 can be linearized by dividing both sides of the equa-

tion by Q

to yield:

The above expression suggests that a plot of versus Q

on a Cartesian scale should yield a straight line with a slope of b

and

intercept of a

as shown in Figure 8-6.

Given the values of a

and b

, the gas flow rate at any p

is calcu-

lated from:

It should be pointed out that the pseudopressure approach is more rig-

orous than either the pressure-squared or pressure-approximation method

and is applicable to all ranges of pressure.

aab

rwf

-+ + -

()yy

(8 - 38)

rwf

abQ

(8 - 37)

1422

(8 - 36)

1422

075=

ln . (8 - 35)

rwf g

aQ bQ-= +

(8 - 34)

Gas Well Performance 549

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 549

In the next section, the back-pressure test is introduced. The material,

however, is intended only to be an introduction. There are several excel-

lent books by the following authors that address transient flow and well

testing in great detail:

• Earlougher (1977)

• Matthews and Russell (1967)

• Lee (1982)

• Canadian Energy Resources Conservation Board (1975).

The Back-Pressure Test

Rawlins and Schellhardt (1936) proposed a method for testing gas

wells by gauging the ability of the well to flow against various back pres-

sures. This type of flow test is commonly referred to as the conventional

deliverability test. The required procedure for conducting this back-pres-

sure test consists of the following steps:

550 Reservoir Engineering Handbook

Figure 8-6. Graph of real gas pseudo-pressure data.

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 550

Step 1. Shut in the gas well sufficiently long for the formation pressure to

equalize at the volumetric average pressure p

–

Step 2. Place the well on production at a constant flow rate Q

for a suf-

ficient time to allow the bottom-hole flowing pressure to stabilize

at p

wf1

, i.e., to reach the pseudosteady state.

Step 3. Repeat Step 2 for several rates and the stabilized bottom-hole

flow pressure is recorded at each corresponding flow rate. If three

or four rates are used, the test may be referred to as a three-point

or four-point flow test.

The rate and pressure history of a typical four-point test is shown in

Figure 8-7. The figure illustrates a normal sequence of rate changes

where the rate is increased during the test. Tests may be also run, howev-

er, using a reverse sequence. Experience indicates that a normal rate

sequence gives better data in most wells.

The most important factor to be considered in performing the conven-

tional deliverability test is the length of the flow periods. It is required

that each rate be maintained sufficiently long for the well to stabilize,

i.e., to reach the pseudosteady state. The stabilization time for a well in

the center of a circular or square drainage area may be estimated from:

Gas Well Performance 551

Figure 8-7. Conventional back-pressure test.

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 551

where t

= stabilization time, hr

f=porosity, fraction

= gas viscosity, cp

= gas saturation, fraction

k = gas effective permeability, md

–

= average reservoir pressure, psia

= drainage radius, ft

The application of the back-pressure test data to determine the coeffi-

cients of any of the empirical flow equations is illustrated in the follow-

ing example.

Example 8-2

A gas well was tested using a three-point conventional deliverability

test. Data recorded during the test are given below:

, psia y

, psi

/cp Q

, Mscf/day

–

= 1952 316 ¥ 10

1700 245 ¥ 10

2624.6

1500 191 ¥ 10

4154.7

1300 141 ¥ 10

5425.1

Figure 8-8 shows the gas pseudopressure y as a function of pressure.

Generate the current IPR by using the following methods.

a. Simplified back-pressure equation

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

i. Pressure-squared approach, Equation 8-29

ii. Pressure-approach, Equation 8-33

iii. Pseudopressure approach, Equation 8-26

c. Compare results of the calculation.

gge

1200

(8 - 39)

552 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 552

Solution

a. Back-Pressure Equation:

Step 1. Prepare the following table:

, psi

¥ 10

–

- p

), psi

¥ 10

, Mscf/day

–

= 1952 3810 0 0

1700 2890 920 2624.6

1500 2250 1560 4154.7

1300 1690 2120 5425.1

Step 2. Plot (p

–

- p

) versus Q

on a log-log scale as shown in Figure

8-9. Draw the best straight line through the points.

Step 3. Using any two points on the straight line, calculate the expo-

nent n from Equation 8-22, as

n =

log( ) log( )

4000 1800

1500 600

087

Gas Well Performance 553

5.00E+07

0.00E+00

1.00E+08

1.50E+08

2.00E+08

2.50E+08

3.00E+08

3.50E+08

500 1000 1500 2000 2500

Pressure

Real Gas Potential

Figure 8-8. Real gas potential vs. pressure.

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 553

Step 4. Determine the performance coefficient C from Equation 8-23

by using the coordinate of any point on the straight line, or:

Step 5. The back-pressure equation is then expressed as:

= 0.0169 (3,810,000 - p

)

0.87

Step 6. Generate the IPR data by assuming various values of p

and

calculate the corresponding Q

, Mscf/day

1952 0

1800 1720

1600 3406

1000 6891

500 8465

0 8980 = AOF = (Q

)

max

C Mscf psi==

1800

600 000

0 0169

087

(, )

554 Reservoir Engineering Handbook

Figure 8-9. Back-pressure curve.

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 554

b. LIT Method

i. Pressure-squared method

Step 1. Construct the following table:

–

- p

), psi

¥ 10

, Mscf/day (p

–

- p

)/Q

–

= 1952 0 0 —

1700 920 2624.6 351

1500 1560 4154.7 375

1300 2120 5425.1 391

Step 2. Plot (p

–

- p

)/Q

versus Q

on a Cartesian scale and draw the

best straight line as shown in Figure 8-10.

Step 3. Determine the intercept and the slope of the straight line to give:

intercept a = 318

slope b = 0.01333

Step 4. The quadratic form of the pressure-squared approach can be

expressed as:

(3,810,000 - p

) = 318 Q

+ 0.01333 Q

Gas Well Performance 555

320.00

300.00

340.00

360.00

380.00

400.00

1000 2000 3000 4000 60005000

Flow Rate

Differential Pressure Over Flow Rate

Figure 8-10. Pressure-squared method.

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 555

Step 5. Construct the IPR data by assuming various values of p

and

solving for Q

by using Equation 8-28.

–

- p

), psi

¥ 10

, Mscf/day

–

= 1952 0 0

1800 570 1675

1600 1250 3436

1000 2810 6862

500 3560 8304

0 3810 8763 = AOF = (Q

)

max

ii. Pressure-approximation method

Step 1. Construct the following table:

–

- p

, Mscf/day (p

–

- p

)/Q

–

= 1952 0 0 —

1700 252 262.6 0.090

1500 452 4154.7 0.109

1300 652 5425.1 0.120

Step 2. Plot (p

–

- p

)/Q

versus Q

on a Cartesian scale as shown in

Figure 8-11.

Draw the best straight line and determine the intercept and

slope as:

intercept a

= 0.06

slope b

= 1.111 ¥ 10

-5

Step 3. The quadratic form of the pressure-approximation method is

then given by:

(1952 - p

) = 0.06 Q

+ 1.111 (10

-5

) Q

Step 4. Generate the IPR data by applying Equation 8-33:

–

- p

, Mscf/day

1952 0 0

1800 152 1879

1600 352 3543

1000 952 6942

500 1452 9046

0 1952 10827

556 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 556

iii. Pseudopressure approach

Step 1. Construct the following table:

y, psi

/cp (y

–

-y

, Mscf/day (y

–

-y

)/Q

–

= 1952 316 ¥ 10

00 —

1700 245 ¥ 10

71 ¥ 10

262.6 27.05 ¥ 10

1500 191 ¥ 10

125 ¥ 10

4154.7 30.09 ¥ 10

1300 141 ¥ 10

175 ¥ 10

5425.1 32.26 ¥ 10

Step 2. Plot (y

–

-y

)/Q

on a Cartesian scale as shown in Figure 8-12

and determine the intercept a

and slope b

, or:

= 22.28 ¥ 10

= 1.727

Step 3. The quadratic form of the gas pseudopressure method is given by:

(316 ¥ 10

- y

) = 22.28 ¥ 10

+ 1.727 Q

Step 4. Generate the IPR data by assuming various values of p

, i.e.,

, and calculate the corresponding Q

from Equation 8-38.

Gas Well Performance 557

0.02

0.00

0.04

0.06

0.08

0.10

0.12

0.14

1000 2000 3000 4000 60005000

Flow Rate

Differential Pressure Over Flow Rate

Figure 8-11. Pressure-approximation method.

Reservoir Eng Hndbk Ch 08 2001-10-24 11:13 Page 557