Tarek Ahmed. Reservoir engineering handbook

Подождите немного. Документ загружается.

where k = permeability, md

f=porosity, fraction

The term F Q

can be included in all the compressible gas flow equa-

tions in the same way as the skin factor. This non-Darcy term is interpret-

ed as being a rate-dependent skin. The modification of the gas flow equa-

tions to account for the turbulent flow condition is given below:

Unsteady-State Radial Flow

The gas flow equation for an unsteady-state flow is given by Equation

6-147 and can be modified to include the additional drop in the real gas

potential as:

Equation 6-158 is commonly written in a more convenient form as:

where the term DQ

is interpreted as the rate dependent skin factor.

The coefficient D is called the inertial or turbulent flow factor and

given by:

The true skin factor s which reflects the formation damage or stimula-

tion is usually combined with the non-Darcy rate dependent skin and

labeled as the apparent or total skin factor:

s¢=s + DQ

(6-161)

Fkh

1422

(6 -160)

mp mp

sDQ

iwf

itiw

() ( )

log . . .

¥-++

1637

323 087 087

(6 -159)

mp mp

iwf

itiw

( ) ( ) log . .-=

1637

323 087

(6 -158)

428 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 06b 2001-10-24 10:48 Page 428

Equation 6-162 can be expressed in the pressure-squared approxima-

tion form as:

where Q

= gas flow rate, Mscf/day

t = time, hr

k = permeability, md

= gas viscosity as evaluated at p

, cp

Semisteady-State Flow

Equations 6-150 and 6-151 can be modified to account for the non-

Darcy flow as follows:

or in terms of the pressure-squared approach:

kh p p

sDQ

rwf

-++

()

ln .

1422 0 75m

(6 -165)

kh m p m p

sDQ

rwf

-++

[() ( )]

ln .1422 0 75

(6 -164)

QTz

iwf

itiw

1637

323 087-=

log . . ' (6 -163)

mp mp

iwf

itiw

() ( )

log . . '

¥-+

1637

323 087

(6 -162)

Fundamentals of Reservoir Fluid Flow 429

Reservoir Eng Hndbk Ch 06b 2001-10-24 10:48 Page 429

where the coefficient D is defined as:

Steady-State Flow

Similar to the above modification procedure, Equations 6-44 and 6-45

can be expressed as:

where D is defined by Equation 6-166.

Example 6-19

A gas well has an estimated wellbore damage radius of 2 feet and an

estimated reduced permeability of 30 md. The formation has a perme-

ability and porosity of 55 md and 12%. The well is producing at a rate of

20 MMscf/day with a gas gravity of 0.6. The following additional data is

available:

= 0.25 h = 20¢ T = 140°F m

= 0.013 cp

Calculate the apparent skin factor.

Solution

Step 1. Calculate skin factor from Equation 6-143

s =-

025

1 732ln

kh p p

sDQ

ewf

-++

()

ln .

1422 0 5m

(6 -168)

khmp mp

sDQ

iwf

-++

[() ( )]

ln .1422 0 5

(6 -167)

Fkh

1422

(6 -166)

430 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 06b 2001-10-24 10:48 Page 430

Step 2. Calculate the turbulence parameter b by applying Equation 6-155:

b=1.88 (10)

-10

(55)

-1.47

(0.12)

-0.53

= 159.904 ¥ 10

Step 3. Calculate non-Darcy flow coefficient from Equation 6-156:

Step 4. Calculate the coefficient D from Equation 6-160:

Step 5. Estimate the apparent skin factor by applying Equation 6-161:

s¢=1.732 + (1.805 ¥ 10

-4

) (20,000) = 5.342

PRINCIPLE OF SUPERPOSITION

The solutions to the radial diffusivity equation as presented earlier in

this chapter appear to be applicable only for describing the pressure distri-

bution in an infinite reservoir that was caused by a constant production

from a single well. Since real reservoir systems usually have several wells

that are operating at varying rates, a more generalized approach is needed

to study the fluid flow behavior during the unsteady state flow period.

The principle of superposition is a powerful concept that can be applied

to remove the restrictions that have been imposed on various forms of

solution to the transient flow equation. Mathematically the superposition

theorem states that any sum of individual solutions to the diffusivity equa-

tion is also a solution to that equation. This concept can be applied to

account for the following effects on the transient flow solution:

• Effects of multiple wells

• Effects of rate change

• Effects of the boundary

• Effects of pressure change

Slider (1976) presented an excellent review and discussion of the prac-

tical applications of the principle of superposition in solving a wide vari-

ety of unsteady-state flow problems.

D ==¥

( . )( )( )

()()

014 55 20

1422 600

1 805 10

F =¥

3 1612 10

159 904 10 600 0 6

0 013 20 0 25

014

.()(.)

(. )( ) (. )

Fundamentals of Reservoir Fluid Flow 431

Reservoir Eng Hndbk Ch 06b 2001-10-24 10:48 Page 431

Effects of Multiple Wells

Frequently, it is desired to account for the effects of more that one well

on the pressure at some point in the reservoir. The superposition concept

states that the total pressure drop at any point in the reservoir is the sum of

the pressure changes at that point caused by flow in each of the wells in the

reservoir. In other words, we simply superimpose one effect upon the other.

Consider Figure 6-28 which shows three wells that are producing at

different flow rates from an infinite acting reservoir, i.e., unsteady-state

flow reservoir. The principle of superposition shows that the total pres-

sure drop observed at any well, e.g., Well 1, is:

(Dp)

total drop at well 1

= (Dp)

drop due to well 1

+ (Dp)

drop due to well 2

+ (Dp)

drop due to well 3

The pressure drop at well 1 due to its own production is given by the

log-approximation to the E

-function solution presented by Equation

6-146, or:

()()

log . .

pp p

well

ooo

-= =

162 6

323 087

432 Reservoir Engineering Handbook

Figure 6-28. Well layout for Example 6-20.

Reservoir Eng Hndbk Ch 06b 2001-10-24 10:48 Page 432

where t = time, hrs.

s = skin factor

k = permeability, md

= oil flow rate from well 1

The pressure drop at well 1 due to production at Wells 2 and 3 must be

written in terms of the E

-function solution as expressed by Equation 6-78.

The log-approximation cannot be used because we are calculating the

pressure at a large distance r from the well, i.e., the argument x > 0.01, or:

where Q

, Q

, and Q

refer to the respective producing rates of Wells

1, 2, and 3.

The above computational approach can be used to calculate the pres-

sure at Wells 2 and 3. Further, it can be extended to include any number

of wells flowing under the unsteady-state flow condition. It should also

be noted that if the point of interest is an operating well, the skin factor s

must be included for that well only.

Example 6-20

Assume that the three wells as shown in Figure 6-28 are producing

under a transient flow condition for 15 hours. The following additional

data is available:

= 100 STB/day h = 20¢

= 160 STB/day f=15%

= 200 STB/day k = 40 md

= 4500 psi r

= 0.25¢

= 1.20 bbl/STB m

= 2.0 cp

= 20 ¥ 10

-6

psi

-1

= 400¢

(s)

well 1

=-0.5 r

= 700¢

()

log . .

i wf totalat well

ooo

162 6

323 087

70 6

948

70 6

948

mm cr

(6 -169)

Fundamentals of Reservoir Fluid Flow 433

Reservoir Eng Hndbk Ch 06b 2001-10-24 10:48 Page 433

If the three wells are producing at a constant flow rate, calculate the

sand face flowing pressure at Well 1.

Solution

Step 1. Calculate the pressure drop at Well 1 caused by its own produc-

tion by using Equation 6-146.

Step 2. Calculate the pressure drop at Well 1 due to the production from

Well 2.

Step 3. Calculate pressure drop due to production from Well 3.

Step 4. Calculate total pressure drop at Well 1.

()

( . )( )( . )( )

()()

()(.)(.)(

)

()

()()

(.) (. )

( . )( . ) .

psi

due to well

70 6 200 1 2 2

40 20

948 0 15 2 0 20 10 700

40 15

42 36 4 645

42 36 1 84 10 0 08

=--

[]

=¥=

()

( . )( )( . )( )

()()

()(.)(.)( )()

()()

.[ (. )]

(. )(.) .

psi

due to well

70 6 160 1 2 2

40 20

948 0 15 2 0 20 10 400

40 15

33 888 1 5168

33 888 0 13 4 41

=--

(p)

(162.6) (100) (1.2) (2.0)

(40) (20)

log

(40) (15)

(0.15) (2) (20 10 ) (0.25)

3.23 0.87 (0)

270.2 psi

well 1

434 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 06b 2001-10-24 10:48 Page 434

(Dp)

total at well 1

= 270.2 + 4.41 + 0.08 = 274.69 psi

Step 5. Calculate p

at Well 1.

= 4500 - 274.69 = 4225.31 psi

Effects of Variable Flow Rates

All of the mathematical expressions presented previously in this chap-

ter require that the wells produce at constant rate during the transient

flow periods. Practically all wells produce at varying rates and, therefore,

it is important that we be able to predict the pressure behavior when rate

changes. For this purpose, the concept of superposition states, “Every

flow rate change in a well will result in a pressure response which is

independent of the pressure responses caused by other previous rate

changes.” Accordingly, the total pressure drop that has occurred at any

time is the summation of pressure changes causes separately by each net

flow rate change.

Consider the case of a shut-in well, i.e., Q = 0, that was then allowed

to produce at a series of constant rates for the different time periods

shown in Figure 6-29. To calculate the total pressure drop at the sand

face at time t

, the composite solution is obtained by adding the individ-

ual constant-rate solutions at the specified rate-time sequence, or:

(Dp)

total

= (Dp)

due to (Q

- 0)

+ (Dp)

due to (Q

- Q

)

+ (Dp)

due to (Q

- Q

)

+ (Dp)

due to (Q

- Q

)

The above expression indicates that there are four contributions to the

total pressure drop resulting from the four individual flow rates.

The first contribution results from increasing the rate from 0 to Q

and

is in effect over the entire time period t

, thus:

It is essential to notice the change in the rate, i.e., (new rate - old rate),

that is used in the above equation. It is the change in the rate that causes

the pressure disturbance. Further, it should be noted that the “time” in the

equation represents the total elapsed time since the change in the rate has

been in effect.

()

.( )

log . .Dp

162 6 0

323 087

Fundamentals of Reservoir Fluid Flow 435

Reservoir Eng Hndbk Ch 06b 2001-10-24 10:48 Page 435

Second contribution results from decreasing the rate from Q

to Q

, thus:

Using the same concept, the contributions from Q

to Q

and from Q

to Q

can be computed as:

()

.( )

log

()

QQB

kt t

162 6

323 087

436 Reservoir Engineering Handbook

Figure 6-29. Production and pressure history of a well.

Reservoir Eng Hndbk Ch 06b 2001-10-24 10:48 Page 436

The above approach can be extended to model a well with several rate

changes. Note, however, the above approach is valid only if the well is

flowing under the unsteady-state flow condition for the total time elapsed

since the well began to flow at its initial rate.

Example 6-21

Figure 6-29 shows the rate history of a well that is producing under

transient flow condition for 15 hours. Given the following data:

= 5000 psi h = 20¢

= 1.1 bbl/STB f=15%

= 2.5 cp r

= 0.3¢

= 20 ¥ 10

-6

psi

-1

s = 0

k = 40 md

Calculate the sand face pressure after 15 hours.

Solution

Step 1. Calculate the pressure drop due to the first flow rate for the entire

flow period.

()

( . )( )( . )( . )

()()

log

()()

(. )(.)( )(.)

psi

162 6 100 0 1 1 2 5

40 20

40 15

015252010 03

3 23 0 319 6

()

.( )

log

()

QQB

kt t

162 6

323 087

()

.( )

log

()

QQB

kt t

162 6

323 087

Fundamentals of Reservoir Fluid Flow 437

Reservoir Eng Hndbk Ch 06b 2001-10-24 10:48 Page 437