Tian 1994

7/23/2019 Tian 1994

http://slidepdf.com/reader/full/tian-1994 1/7

Development

of Analytical Design

Equation

for Gas Pipelines

Shifeng Tian SPE, and Michael A Adewumi SPE, Pennsylvania State U.

Summary

Based on mass and momentum balance, a rigorous analytical equation is derived for compressible fluid flow in pipelines. This

equation gives a functional relat ionship between flow rate, inlet pressure, and outlet pressure. It is very useful in design calculations where

any

of

these variables need to be estimated

if

the others are given. The equation can be used for any pipeline topology and configuration,

including size and orientation. A number

of

problems

of

engineering importance are studied with this equation. They include bottomhole

pressure (BHP) calculations in gas wells, gas injection calculations, and long-distance gas pipeline design calculations. The excellent agree

ment between predicted results and field data, with this equation for this wide variety of problems and conditions, demonstrate the efficacy

of

this equation for engineering applications. Simple computer programs, in both

FORTRAN

and BASIC are developed to handle these ap

plications. The BASIC program can be run on any programmable calculator with 3 kilobytes of memory.

Introduction

The problem

of

compressible fluid flow through pipelines and con

duits has been studied by many investigators. In the natural gas in

dustry, the problems

of

interest fall into two categories: gas pipeline

flow calculations and gas-well calculations. Because of the differ

ing sets

of

assumptions usually invoked, these two problems have

been treated as almost mutual ly exclusive in the literature. For pipe

lines, the most commonly used equations for these calculations are

the Weymouth equation and the Panhandle equations. For BHP pre

diction in gas wells, the most popular methods are those developed

by Sukkar and Cornell

l

and Cullender and Smith.

2

Basically, the Weymouth and Panhandle equations are derived

for gas flow in horizontal and slightly incl ined pipelines. For slight

ly inclined pipes, the elevation change is accounted for by simply

adding the static head of

the gas column to the pressure difference

calculation. While this may be adequate for small elevation

changes, as obtained in gas pipelines, it is inadequate in gas wells

where the pipe is either vertical or nearly vertical. The reason is that,

in this case, the gravity term is sufficiently significant to affect fluid

velocity and hence the friction term.

Sukkar and Cornell

l

and Cullender and Smith2 developed nu

merical procedures for integrating the energy equation to calculate

BHP in flowing gas wells. Azi

z

3

presented a numerical algorithm

for computing flowing BHP for gas wells. His method improves the

convergence rate in the iterative procedure usually involved. The

common denominator in all these investigations is that kinetic ener

gy was neglected. By assuming that temperature and compressibili

ty are constant throughout the pipeline and by neglecting the kinetic

energy term, an analyt ical expression can be derived from the funda

mental energy equation. The resulting expression is well known.

4

-

6

However, no such expression is published for the case where the ki

netic energy term is included.

Young

4

conducted a comprehensive analysis

of

the comparative

magnitude

of

the error introduced into the gas-well flow calcula

tions as a result

of

the various assumptions usually made. He con

cluded that, although the kinetic energy term is negligible in many

cases, the error introduced is more significant than that arising from

setting temperature and compressibility factor to some constant av

erage values. He concluded that a numerical algorithm must be used

when the kinetic energy term is significant enough that it cannot be

neglected. His study further shows that, for gas wells, this condition

will arise when the well is less than 4,000 ft deep and wellhead pres

sure is below 100 psia. This would seem to apply to shallow-well,

low-pressure systems common, for example, in the Appalachian ba

sin

ofthe

U.S. Young also concluded that including the kinetic ener

gy term is important when the pressure traverse in the gas well is de

sired, even for wellhead flowing pressure as high as 500 psia.

This paper presents an analytical equation derived from the fun

damental differential equation describing compressible fluid flow

in pipes without neglecting the kinetic energy term. The derivation

process retains the assumption

of

constant temperature and com-

Copyright 994 Society of Petroleum Engineers

100

pressibility factor. This new equation is equally suitable for use in

pipeline calculations and gas-well calculations. Furthermore, be

cause this equation eliminates the need for numerical integration

quadrature (common in previous techniques), it is better suited for

estimating any

ofthe

commonly sought variables in either case. Ex

ample calculations are presented to demonstrate the versatility of

this equation. The excellent agreement between the results pre

dicted with this equation and field data, which include BHP calcula

tions for gas wells and long-distance gas pipeline transportation,

shows that this equation is accurate for any of these applications.

Development of the

Analytical Equation

For a pipeline with constant cross-sectional area, the ID continuity

equation for gas flow is

d pv)/dx = O (1)

Eq.

I

implies that the product

of

the gas density and gas velocity,

pv is constant along the pipeline. Thus,

pv =

Povo

=

w/A

_ (2)

where w=gas mass flow rate and is constant throughout the pipeline,

A=constant cross-sectional area of the pipeline, and the subscript 0

indicates standard conditions.

Ifwe letx

denote flow direction (along the pipeline axis), the ID

form of the energy equation for gas flow can be written as

d(pv2) dp fpv2 .

= -

dx -2 f -pg sm

a

(3)

where .f=friction factor and a=angle of pipeline elevation. Fig. 1

shows the signs of a for various pipeline configurations. The fric

tion factor can be calculated with Chen s5 equation:

1 {e d 5.0452 [ I e)1.I098 5.8506]}

Ii = 2 log 3.7065 ~ log 2.8257

d

+

~ : 9 8

where N

Re

= pvd//1

and, by definition,

_ (4)

(5)

p = Mgp/zRT _

(6)

Eqs. 4 through 6 constitute the basic governing equations for gas

flow in conduits. The differential equation (Eq. 3) is usually inte

grated with numerical methods. However, by some mathematical

analysis, an analytical solution ofthe equation can be obtained. Let

us consider a short interval of the pipeline. Because pv is constant

and the gas viscosity, /1 can be assumed to be constant within this

short interval, the Reynolds number,

NRe

is constant. With these, an

examination

ofEq.

4 should reveal that the friction factor,J, is also

constant throughout the pipe segment under consideration, regard

less of the fluid flow regime (laminar or turbulent).

SPE Production Facilities, May 1994

7/23/2019 Tian 1994


P2

P

____

cx=-9cf

q

q

a = + 90°

P

1

P

2

Fig.

1-Different

pipe and flow configurations.

Substituting Eqs. 2 and 6 into Eq. 3, we obtain

wdv dp

fwv

Mggpsina

Adx

= -

dx -2dA

zRT

(7)

From Eqs. 2 and

6,

the relation between

v

and p is

v

=

wzRT/MgAp. .

(8)

Substituting Eq. 8 into Eq. 7 gives

w

2

zRT dp dp fw2ZRT Mggp sin a

MgA2p

2

dx

=

dx + 2dM

g

A2

p

+ zRT

(9)

The above ordinary differential equation can be integrated to yield

w

2

zRT

MgA2p2

- - - - - . . . -=- - - - - : - - -dp =

Mggp sin a fw2

Z

RT

zRT 2dA2M

g

P

L

f x.

. . . . . . . . . . . (10)

o

Assuming that temperature and compressibility factor are

constant and set equal to some average values, a closed-form in

tegration of Eq.

10

can be obtained. The compressibility factor, z

is a function

of

T and Pro We define the average compressibility fac

tor as the compressibility factor at average pressure and tempera

ture,

P

PI

+

P2

P r P c ~

j

TI + T2

and

Tr

= Tc = ---:yr;;-.

(11)

(12)

A correlation is needed to express compressibility factor as a

function of reduced temperature and reduced pressure. Many meth

ods are available for this estimation. Takacs

6

reviewed these meth

ods. The Dranchuk

et

at 7 method is used in this study because of

its accuracy. This method is based on the Benedict-Webb-Rubin

equation of state and hence has a complex algebraic form. t is labo

rious to use on a calculator; however, once it is coded into a pro

gram, it is very straightforward to use.

The closed-form solution of Eq.

10

follows:

(13)


q

P,

...

_________________ 1

P2

q I

P,.,I 'I

- - ' - - "" ; ; " ' - -11

P,,2

I

I q

p,.,= P, P

2

.

,

- i

i p 2 . 2

I

I q

P 3 . , I I - - ~ - . . . . : ; , - - - . 1 P3.2

FIa = P2,2 P2 = P3.2

Fig.

2-Division

of pipe into computational segments.

This equation reflects the relationship of

PI,

P2 and W. It can be

used to calculate any of them if the other two are known. The rela

tionship between gas mass flow rate and volume flow rate is

w wRTo

qo = Po = Mgpo , (14)

where the volume flow rate, qo is at standard conditions.

Eq. 13 can be used for any pipeline except horizontal cases,

where

a=O

The reason is that Eq.

13

is singular at this point. How

ever, this singularity is removable by applying L Hopital s rule on

Eq. 13. Alternatively, the expression for horizontal pipe can be ob

tained by setting a=O in Eq. 10. For horizontal pipelines, Eq.

10

be

comes

P

w

2

zRT

f

~ d =

fw2ZRT P

PI 2dA2MgP

The solution becomes

(15)

Eqs. 13 and 16 are derived from the fundamental governing dif

ferential equations for fluid flow and hence have wide applications.

These equations provide relationships among flow rate, inlet pres

sure, outlet pressure, and the usual pipeline parameters. They can be

used to estimate any

of

these variables

if

the others are specified.

Simple computer programs based on these analytical solutions have

been developed in both FORTRAN and BASIC to handle these ap

plications. The implicit form of these equations does not allow the

inlet and outlet pressures to be solved explicitly; hence some itera

tive scheme is necessary to solve for them. The Newton-Raphson

method is used in the program to solve the analytical equations for

inlet or outlet pressures, and the compressibility factor is updated af

ter every convergence of the calculated pressure. The calculation

of

gas flow rate from given inlet and outlet pressures does not require

an iterative scheme by itself. The friction factor, which appears in

the expressions, is a function of flow rate (by virtue of its depen

dence on Reynolds number) and hence iteration on friction factor is

necessary when solving for flow rate. The program has the opt ions

of calculating the given pipeline as one piece or dividing the pipe

line into any specified number

of

shorter segments for more accu

rate results. The

BASIC

program has been successfully tested on a

TI-74 BASICALC calculator. Any programmable calculator with at

least a 3K memory capacity is capable of running this program.

Analysis of

the

Analytical Equation

Eq. 13 (or Eq. 16 for horizontal pipelines) provides a functional rela

tionship between gas flow rate and inlet and outlet pressures. Eqs.

13 and

16

are based on fundamental fluid flow equations, so they

can be applied for a wide variety of problems. In deriving Eq. 13 or

16, we assumed that temperature and compressibility factor are

constant. For a very short piece of pipeline, this assumption should

be valid and thus the equations should

be

accurate. On the other

hand, we can always divide a long pipeline into small segments for

101

7/23/2019 Tian 1994


TABLE 1 EFFECT

OF AVERAGE z-FACTOR ON PREDICTED PRESSURE

Parameters and Conditions

Pipe length, miles

Pipe diameter, in.

Pipe roughness, in.

1,000

30

0.0006

Inlet pressure, psia

Gas flow rate, MMscf/O

Temperature, • F

2,500

600

50

Gas Composition

CH4

=75.57

C2HS= 11.22

C3H8=

7.78

i C4

= 0.78

n C4

= 1.71

i-C

s

= 0.28

n-C

s

=0.31

Hyp1= 0.41

Hyp2=0.33

Hyp3=0.28

C02

=0.32

N2

. = 1.01

Number of Segments 2

Calculated Outlet Pressure, pSia 1,561.7

1,548.4

stepwise calculations to retain the same level of accuracy. Fig. 2

shows the procedure for calculating outlet pressure from a given in

let pressure and gas flow rate.

Comprehensive testing

of

this analytical equation demonstrated

that the assumption of constant compressibility factor does not

cause significant error and that, for most engineering applications,

we can obtain very accurate results without dividing the pipeline

into shorter segments. Table 1 shows the results of one test.

As Fig. 2 indicates, a long pipeline can be calculated as one piece

or in several pieces. The length, angle, roughness, and temperature

of each segment may differ. For instance,

if

different segments of he

pipe are

of

different age and internal conditions, it may be judicious

to assign different values

of

roughness to them. From the inlet pres

sure and flow rate, the outlet pressure

of

the first segment can be cal

culated by Eq. 13 (or Eq. 16). The outlet pressure

of

the first pipe

segment is the inlet pressure of the second segment and can be used

to calculate the outlet pressure from this segment. Following this

procedure, we can obtain the outlet pressure of the whole pipeline.

This procedure can also be used for inverse problems where the out

let pressure is known and evaluationof he inlet pressure is required.

For instance, given the required deliverability of he pipeline and the

delivery pressure specified in a contract, we may need to design the

compressor station needed to achieve these requirements. This

would call for an estimate

of

the required compressor outlet pres

sure.

Table 1 gives an example application of these equations. All the

input parameters are listed. The 1,000-mile-long pipeline is hori

zontal. The inlet pressure is set to 2,500 psia; gas flow rate is 600

MMscflD. The outlet pressures listed in Table 1 are calculated by

dividing the total length into different numbers of shorter segments.

From the table, we can see that, as the number of segments into

which the total length is divided increases, the outlet pressure con

verges to 1,543.4 psia. The difference between the calculated outlet

pressure taking the total length as a single segment and the con

verged outlet pressure is only 18.3 psia (1.2 ), even though the

pipeline is 1,000 miles long. I f calculated with only two 500-mile

long segments, the error is less than 0.35 . This demonstrates that

the assumption of a constant compressibility factor, calculated at the

average pressure and temperature of the pipeline, will cause very

small errors. Nevertheless, we can always divide a long pipeline into

several segments to obtain better results. Results of all other exam

ples presented in this paper are obtained without subdividing the

pipe into segments because this does not introduce any significant

error into the predictions.

All the previously developed analytical equations neglected the

kinetic energy term. Young

4

studied the effect

of

neglecting the ki

netic energy by using a numerical method. His results show that, for

most engineering applications, the kinetic energy is not important.

In some special cases, however, the deviation caused by neglecting

the kinetic energy could be significant. Our study arrives at the same

conclusion. In high-pressure gas transportation, the relative error

caused by neglecting the kinetic energy is very small. In low-pres

sure cases, especially when the calculated pressure is below 100

psia, the relative error could be very significant. Two examples

where the contribution

of

the kinetic energy term was found to be

significant are presented. The first case is a 3,000-ft-long, 4-in.

diameter (roughness of 0.0006 in.) pipeline upwardly inclined at 1°

102

5 20 50

1,544.2

10

1,543.6 1,543.5 1,543.4

100

1,543.4

angle. The transported gas has a specific gravity of 0.75, a viscosity

of

0.018 cp, and a critical temperature and pressure 411

OR

and 661

psia, respectively. The specified gas flow rate is 10 MMscflD; the

inlet pressure is 200 psia; and the average temperature of 85°F. The

calculated delivery pressures, with and without the kinetic energy

term, are 39 and 50 psia, respectively an error

of

28 .

The second example is a gas well. All the parameters are the

same as for the first problem except that the BHP is specified as 600

psia, the depth is 8,000 ft, the production rate is 17 MMscflD, and

the average temperature is 100°F. The well is assumed to be vertical

(i.e., inclination angle of 90°). The predicted wellhead pressures,

with and without the kinetic energy term, are 63 and 90 psia, respec

tively, reflecting a 43 error. As we can see, neglecting the kinetic

energy term would cause significant error in some cases. It is advis

able to use the equations presented in this paper when the signifi

cance of the kinetic energy term is unknown. Such cases would be

prevalent in the analysis of complex network of pipelines where the

results are not clear a priori

Applications of the

Analytical

Equation

Handling Hilly Terrain. Although the example in Table 1 is for a

horizontal pipeline, the same procedure applies to inclined and ver

tical pipes. If there is any change in size, angle, or roughness along

a pipeline, we must divide the pipeline into segments in such a way

that ensures uniform parameters for each segment. Fig. 3 shows an

example in which the pipeline changes its inclination angle at

17,000 and 35,800 ft. In this case, we need to divide the pipeline

into at least three segments, each with a constant angle.

We

can also

subdivide each section into several segments. However, because the

results of the tests conducted show that such subdivision does not

significantly alter the results, no subdivision is done here. The com

position of the gas used in this example is the same as that used in

Table 1. It falls in the two-phase region at the given pressure and

temperature. Therefore, a two-phase pipel ine flow model developed

by Adewumi and Mucharam

8

is also used to test this example. The

pressure profile and the liquid holdup profile obtained from the

7 0 0 . - - - - - - - - - - - ~ - - - - - - - - - - _ r - - - - - - - - - - _

Pipeline Prorlle

• • • • • Pressure Profile

- - - - - 2

Analytical Equalion)

5 0 0 ~ ~ ~ ~ ~ ~ ~

Distance (ft)

Fig. 3 Predicted pressure profile

for

undulating terrain.

SPE Production Facilities, May

1994

7/23/2019 Tian 1994


1 W O r - - - - - ~ - - - - - - _ r - - - - - - _ - - - - - - ~ - - - - _

900

- - - - - - - - q=8.7722MMSCF/D

- - - q = 12 0 MMSCF/D

-

Injection Pressure: 800 0 psia

Tubing

Diameter:

3 0 inches

Roughness: 0 0006 inches

Gas Specific Gravity: 0 75

Average Temperature: 100 0 F

-

Well Depth (ft)

[2J

,

'--. ,IJ

Fig. 4-Pressure profile

for

gas-injection wells.

two-phase model are also shown in Fig. 3 It is interesting to note

that the pressure profile obtained from the analytical equation is

very close to that calculated from the two-phase model. This indi

cates that having a small amount of liquid in the system does not sig

nificantly affect the pressure loss in the pipeline.

Profile of Design Variables Along

the

Pipeline. Eqs. 13 and 16

provide an explicit relationship between pipe length, L and other

variables. Hence, they are very handy in generating the profile

of

any given design variable along the pipeline. For example, at a gi ven

inlet pressure and flow rate, the pressure profile along the pipeline

can be easily generated by substituting a differentP2 into Eq. 13 or

16 to calculate the corresponding pipe lengths. In fact, this could

also be used instead of the Newton-Raphson method as the basis for

a trial-and-error method to solve for the inlet or outlet pressure. Be

cause all the constants grouped together need to be calculated only

once, the trial-and-error method should involve a minimum amount

of calculations.

Network

Analysis. Natural gas gathering and distribution usually

involve complex pipeline network. Because of he complexity

of

he

network, an analytical expression for single pipelines, such as the

one presented here, is very useful in handling this analysis expedi

ently. Eqs.

13

and 16 are derived without assuming the flow type and

can therefore be used for both laminar and turbulent flow. Further

more, the effect

of

kinetic energy is included in the equations. In oth

er words, the analytical equation presented here can be applied in a

wider range

of

conditions and to a wider range

of

problems than any

existing analytical equations for compressible fluid pipeline flow.

It is therefore more suitable for pipeline network analysis, where

widely differing situations could be encountered. Ref. 9 gives de

tailed explanation of the application of this equation to network

analysis.

Gas

Injection Calculation. Young

4

pointed out that a discontinuity

can develop when the fundamental energy equation is integrated nu

merically for injection cases. We have examined the injection cases

and found out that this discontinuity is caused by the sign of dp used

in the numerical procedure.

Unlike the uphill and horizontal pipe configurations where both

the friction and the gravity forces oppose flow, in the downhill flow

ing case (pipeline with negative angle), such as for gas injection, the

gravity force helps the flow while friction opposes it. In this case,

these two forces oppose each other. This could result in three types

of

pressure profiles along the pipe, as Fig. 4 shows. In this figure,

Curve I represents the "normal" case where pressure is expected to

decrease with distance. This occurs because the friction force is

greater than the gravity force. If the friction force is just balanced by

the gravity force, the pressure profile will be a horizontal straight

line (Curve 2). In other words, no pressure loss will be experienced

during flow. In this case, the gas velocity and density will also re

main constant along the pipe. If the gravity force is greater than the

friction force, which could happen in the injection case where high

pressure and low flow rate exist, the pressure will increase with the

distance (Curve 3). A pressure increase will cause an increase in gas

density and a decrease in gas velocity. These, in turn, cause a de

crease in frictional force and an increase in gravity force. We can

conclude that, as long as the pipe ID remains constant, the pressure

profile will either increase or decrease monotonically. In the numer

ical integration

of

Eq. 10 for the injection case, the elemental

dummy variable, dp could be positive ornegative. If he wrong sign

TABLE 2-COMPARISON OF PREDICTED BHP WITH FIELD DATA

Field

Well operator

Date of test

Well

Well depth, ft

Total Flow

Rate

(MMcf/D)

15.606

Bigstone

Amoco Canada Petroleum Co. Ltd.

Oct. 10, 1972

Pan American HB,

C-1

10,965

Measured Parameters

Tubing Head

Condensate

Production

P

Rate

(psia)

(B/D)

2,314.5 155 0.0

Tubing size

10),

in.

Reservoir temperature,

0

F

Gas specific gravity

Pseudocritical pressure, psia

Pseudocritical temperature, R

Water/Gas Predicted

Ratio BHP

BHP Pl

(bbl/MMcf) (psia)

(psia)

5.21 3,295.5

3,312.7

TABLE 3-COMPARISON

OF PREDICTED BHP WITH FIELD DATA

Field

Well operator

Date of test

Well

Well depth, ft

Total Flow

Rate

(MMcf/D)

17.359

Bigstone

Amoco Canada

Petroleum Co. Ltd.

Oct. 10, 1972

Pan American HB, G-2

11,029

Measured Parameters

Tubing Head

Condensate

Production

Rate

(B/D)

P2

(psia)

2,599.5

T

170

0.0

SP

Production

&

Facilities, May 1994

Tubing size

10),

in.

Reservoir temperature, 0 F



Pseudocritical temperature, 0 R

Water/Gas

Ratio

(bbl/MMcf)

1.3

BHP

(psia)

3,690.4

Predicted

BHP Pl

(psia)

3,707.2

2.992

243

0.702

798.3

409.5

Deviation From

Measured

l

(%)

+0.52

2.992

243

0.702

798.3

409.5

Deviation From

Measured

l

(%)

+0.46

103

7/23/2019 Tian 1994


TABLE

4-COMPARISON OF PREDICTED BHP WITH FIELD DATA

Field

Well operator

Date

of

test

Well

Well depth, ft

Total Flow

Rate

(MMcf/D)

6.527

9.598

12.048

14.277

Bigstone

Amoco Canada Petroleum Co. Ltd.

Sept. 20, 1970

Pan American HB, G-2

11,029

Measured Parameters

Tubing Head

Condensate

Production

P2

Rate

(psia)

.-f£L

(B/D)

3,249.4 132 0.0

3,168.4

140

0.0

3,078.4

153 0.0

2,990.4 158 0.0

is used, a discontinuity could develop during the numerical integra

tion procedure and the result would be incorrect. This problem can

not arise with the analytical equation.

Comparison of Predicted Results With

Field Data

Tables 2

through

8 compare the predicted results from the analyti

cal equation with the field data. The field data are from Refs. 10 and

11 All the pertinent information i s contained in these references, so

no tuning or parameter adjustment was necessary in our comparison.

Tables 2 through 4 show predictions

of

BHP for the gas wells

studied. All the wells are vertical with different depths and sizes.

The temperature used in the calculation is the arithmetic average of

the reservoir temperature and the tubing-head temperature. The

agreement of the predicted and measured BHP's is very good. The

average absolute deviation is less than 1 , with a maximum devi

ation of 3.16 . In fact, the overall accuracy is better than the predic

tion of Abou-Kassem'slO method, which involves more complex

calculations.

Tubing size

10),

in.

Reservoir temperature, • F



Pseudocritical temperature, • R

Water/Gas Predicted

Ratio BHP

BHP Pl

(bbl/MMcf) (psia) (psia)

1.5 4,249.6 4,290.0

1.5 4,205.8 4,221.9

1.5

4,163.2

4,142.4

1.5 4,123.2 4,083.8

2.992

246

0.6997

801.2

410.9

Deviation From

Measured

Pl

( Yo)

+0.95

+0.38

-<l.50

-<l.96

As indicated in the tables, all the wells also produced a small

amount of condensate or water with the gas. This demonstrates that

the analytical equation can handle pressure drop calculations for gas

pipelines bearing small amounts of liquid.

We also considered linear variation of temperature in the well

bore, using the tubing-head temperature and the reservoir tempera

ture as the two endpoints. No significant effect was found in any of

the test runs; in fact, the maximum error reduces only from 3.16

to

3.10 .

Tables 6 through 8 show the calculation

of

pressure loss in some

long-distance gas pipelines. The detailed field test procedure and

measurements are given in Ref. 11 Only the parameters needed as

input data for the analytical equation are listed in the tables. Three

types of calculations are conducted: I) determination of the outlet

pressure given the measured inlet pressure and gas flow rate, (2) es

timation

of

he inlet pressure given the measured outlet pressure and

gas flow rate, and (3) calculation of the gas flow rate given the mea

sured pressures. The predicted results are listed in the last three col-

TABLE

5-COMPARISON OF PREDICTED BHP WITH FIELD DATA

Field

Dunvegan Tubing size

10),

in.

Well operator

Anderson Exploration Ltd. Reservoir temperature, • F

Date of test

April

5,

1971


Well

Dunvegan 6-29


Well depth, ft

4,753

Pseudocritical temperature, •R

Measured Parameters

Tubing Head

Condensate

Total Flow

Production Water/Gas Predicted

Rate

P2

T

Rate Ratio BHP

BHA, P

(MMcf/D)

(psia)

.-f£L

(B/D)

(bbIlMMcf) (psia) (psia)

0.833

1,660.0 47

3.48 0.0 1,928.7 1,905.4

1.606

1,566.0

53

5.22

0.0 1,808.4 1,796.0

2.186

1,473.0

57 10.44 0.0 1,701.8 1,690.2

2.910

1,360.0

58 5.22 0.0 1,567.5 1,566.5

4.291

895.0 68 1.74 0.0 1,103.7 1,068.8

4.043

862.0

69 12.97 0.0 1,048.0 1,026.7

TABLE 6-COMPARISON OF PREDICTED PIPELINE DESIGN VARIABLES WITH FIELD DATA

Test Conditions

Approximate Gas Composition

L miles

d in.

h

ft

10.678

12.25

161.0

CH4 =89.80

C2HS= 3.50

C3HS= 0.51

i-C4 = 0.07

n-C4=0.13

i-C

s

=0.02

n-C

s

=0.02

Cs =0.03

C7 =0.01

N2

= 5.3

CO

2

=

0.2

He = 0.4

2.441

115

0.6402

669.9

367.2

Deviation From

Measured

P

( Yo)

-1.21

-<l.69

-<l.68

-<l.06

-3.16

-2.03

Measured Parameters

Predicted Results

T

II x 10

6

Ex10

6

Test

(lbmlft-sec)

1 524.0 7.957

757

2 518.8 7.883

863

3

519.7

7.866 780

4 518.9 7.831 772

5 518.3 7.802

758

6

518.3

7.787 765

104

P

P2

(psia) (psia)

814.8 749.8

797.7

741.0

769.3

730.5

754.5 720.4

729.8

711.2

716.6 701.7

q

(MMcf/D)

77.11

70.72

57.45

52.99

37.25

32.26

P

(psia)

812.6

795.8

768.4

753.8

729.7

716.6

P2

(psia)

752.1

743.0

731.4

721.1

711.3

701.7

q

(MMcflD)

78.6

72.0

58.2

53.6

37.3

32.2


7/23/2019 Tian 1994


T BLE 7-COMPARISON OF PREDICTED PIPELINE DESIGN V RI BLES WITH FIELD D T

Test Conditions Approximate Gas Composition

L,miles 63.068 CH

4

=76.99 n-C

4

=0.67 C

7

= 0.12

d

in. 19.4375

C2HS=

5.21

i-C

s

=0.08

N2 =13.28

h ft -518.0

C3Ha= 2.85

n-C

s

=0.08

C02=

0.30

i-C

4

= 0.30 C

e

=0.02 O

2

= 0.10

Measured Parameters Predicted Results

T

p.

x 10

6

EX

1

6

Pl P2

Pl

P2

Test

...i'BL

lbm/ft-sec)

(psi

a)

(psia)

(MMcf/D) (psia) (psia)

(MMcf/D)

-

1

523.3

7.517 901 602.7 587.6

51.53 603.1 587.2

51.1

2 523.0 7.512 601 612.1 576.3 72.94 612.9 575.4 72.3

3 522.5

7.494 456

611.1

559.0

86.53 612.6 557.3

85.4

4

522.8

7.392 923

515.5

495.7

50.20

515.9 495.3

49.8

5

522.3

7.379 627 518.7 481.0

66.49 519.6

480.0

65.8

6 528.0 7.892

790

812.7

795.0

68.49

813.0 794.7

68.1

7

530.0

7.894 630 811.3 774.8

88.14

811.7 774.4

87.8

8 531.8 7.896

491

809.0 756.5 102.76 809.9 755.5

102.0

9 532.0 7.866 512 812.6 725.9

124.89 812.6 725.9

124.9

10 532.7 7.864

487

814.0 712.4 133.86 813.9 712.5

133.9

T BLE -COMPARISON OF PREDICTED PIPELINE DESIGN V RI BLES WITH FIELD D T

Test Conditions Approximate Gas Composition

L miles

152.376

CH

4

=76.99

n-C

4

=0.67 C

7

= 0.12

d

in.

19.4375

C2HS= 5.21

i-C

s

=0.08

N2

=13.28

h

ft

-1.198.0

C3Ha= 2.85

n-C

s

=0.08

C02= 0.30

i-C4 = 0.30 Cs =0.02 O2 = 0.10

Measured Parameters

Predicted Results

T

p. x 10

6

EX 1

6

Pl

P

Pl P2

Test

...i'BL

lbm/ft-sec)

(psia) (psia)

(MMcf/D) (psia) (psia) (MMcf/D)

-

1

515.0 7.409 1.067 602.7 563.9

51.45

603.1 563.5 51.3

2

516.0 7.395

825

612.1

513.9

72.88

612.3 513.6

72.8

3 515.0

7.352

636

611.1

463.6

86.48 612.0

462.3

86.2

4 515.0 7.284 1.101

515.5 463.9

50.14 515.9

463.5

50.0

5 516.0 7.270

846

518.7 413.8

66.44

519.2 413.1

66.3

6

525.0

7.847 902 812.7 766.6

68.43 813.2 766.0 68.2

7 525.5 7.808 789 811.3 713.9

88.08

811.4 713.8

88.0

8 526.0 7.765

651

809.0 663.6 102.71 808.2 664.7

103.0

9

526.5 7.694 638 812.6 564.1 124.85 811.3 566.2 125.2

10 526.5 7.662 622 814.0 513.3 133.83 812.5 516.1

134.3

11

526.5 7.605

661

813.7

413.4

145.47 807.8 426.6

146.9

900

14

'

800

Q

'

tJ12

en

CIl

:::E

61

;::I

<11

CIl

-

Il

'

£

t: :::

0

0

80

;::I

CIl

0

'

<1 1

<1 1

:::E

;::I

60

n

C<j

500

<11

:::E

4

900

Fig.

S-Comparison

of predicted pressure and field data. Fig. 6-Comparison

of

predicted

flow

rate and field data.

SPE Production Facilities. May 1994

1 5

7/23/2019 Tian 1994


umns

of

each table. All the calculations are condu cted with the total

length

of

the pipeline taken as a single segment, because the test runs

show that the same results will

be

obtained

if

the total length is di

vided into smaller segments,

even

for the 152-mile pipeline, the

longest pipeline in the field test.

For convenience, the comparison

of

the predicted pressures and

flow rates with the field test data is also shown in Figs. 5 and

6.

The

agreement

of

the predicted results and field data is excellent. These

field tests were conducted with a wide range

of

gas flow rates

and

pressures, as well as different pipe sizes, lengths, and elevations.

This further demonstrates the predictive and descriptive capability

of

the analytical equati on presented for engineering applications.

Conclusions

An

analytical equation for steady gas flow is obtained without elimi

nating any

of

the terms in the fundamental governing differential

equation. The resulting expression gives a relationship among gas

flow rate and inlet and outlet pressures and is very useful in gas pipe

line design

and

gas-well flow calculations. Comprehensive testing

of

this equation was conducted. Case studies include horizontal, in

clined, and vertical pipes (gas wells). The excellent agreement ob

tained between the predicted results and field data, which include

BHP calculations for natural gas wells and long-distance natural gas

pipeline transportation, shows that this equation is very accurate for

any

of

these applications. Simple computer programs in both BASIC

and

FORTRAN

have been developed to handle these applications.

The

BASIC program can

be

run on any programmable calculator.

The

analytical equation would considerably enhance gas pipeline

design in terms

of

both ease of use and accuracy. It would have tre

mendous application in pipe net work analysis where repetitive cal

culations are common.

Nomenclature

A = cross-sectional area

of

pipeline, L

d = pipeline diameter, L

f = friction factor

g =

gravitational acceleration, U t

2

L = pipeline length, L

Mg = gas molecular weight, m

P

=

pressure, mlLt

2

Pc

=

critical pressure, mlLt

2

Pr

= reduced pressure, mlLT2

PI

= inlet pressure, mlLT2

P2

= outlet pressure, mlLT2

qo = gas volumetric flow rate at standard conditions, L3/t

R =

universal gas constant

T = temperature, T

c = critical temperature, T

r

=

reduced temperature, T

v = gas velocity, Ut

w = mass flow rate

of

gas,

mit

x =

axial coordinate

z = gas compressibility factor

a = pipeline angle

f 1

=

gas viscosity, mILt

p

=

gas density, mlL

3

a = pipeline roughness

References

1.

Sukkar,

Y.K.

and Cornell,

D.

, Direct Calculation

of

Bottom-Hole Pres

sures in Natural Gas Wells, Pet. Trans. AIME (1955) 204, 43.

2. Cullender, M.H. and Smith, R.Y.: Practical Solution

of

Gas-Flow

Equations for Wells and Pipelines with Large Temperature Gradients,

Pet. Trans. AIME (1956) 207, 281.

106

3. Aziz, K. : Calculation of Bottom-Hole Pre

ss

ure in Gas Wells, JPT

(July 1967) 897.

4. Young, K.L.: Effect

of

Assumptions Used to Calculate Bottom-Hole

Pressures in Gas Wells, JPT (April 1967) 547: Trans. AIME, 240.

5.

Chen, N.

H.:

An Explicit Equation for Friction Factor in Pipe, Ind.

Eng. Chern. Fundarn . (1979) 296.

6. Takacs, G.: Comparison s Made for Computer Z-Factor Calculations,

Oil Gas

J. (Dec. 20, 1976) 64.

7. Dranchuk, P.M. and Abou-Kassem, J.H. : Calculation

ofZ

factors for

Natural Gases Using Equations

of

State, J. Cdn. Pet. Tech. (July-Sept.

1975) 34.

8. Adewumi, M.A. and Mucharam,

L.:

Compositional Multiphase Hy

drodynamic Modeling

of

Gas/Gas-Condensate Dispersed Flow in Gas

Pipelines, SPEPE (Feb. 1990) 85.

9. Tian,

S.

and Adewumi, M.A.:

A

Simple Algorithm for Analyzing Gas

Pipeline Networks, paper SPE 25475 presented at the 1993 SPE Pro

duction Operations Symposium, Oklahoma City, March 21-23.

10. Abou-Kassem, J.H .: Determination

of

Bottom-Hole Pressure in Flow

ing Gas Wells, MS thesis, U.

of

Alberta, Edmonton, Canada (1975).

11.

Uhl, A.E. et al : Steady Flow in Gas Pipelines, IGT Report No. 10,

American Gas Assn., New York City (1965).

51 Metric Conversion Factors

cp X 1.0 E+OO=

mPa'

s

ft x 3.048* E-{)I

=

m

OF

CF-32)/1.8

=

°C

Ibm

x

4.535 924 E-{)I = kg

mile

x 1.609344* E+OO= km

R CRlI.8) = R

psi x

6.894757 E+OO=

kPa

*Conversion factor is exact.

SPEPF

Original SPE manuscript received for review Oct. 4, 1992. Revised manuscript received Nov.

30 1993. Paper accepted for publication Jan.

20

, 1994. Paper (SPE 24861) first presented

at the 1992 SPE Annual Technical Conference and Exhibition hel d in Washington Oct . 4--7.

Shifeng Tian is a postdoctoral research associate

in

the Petro

leum and Natural Gas Engineering Dept. at Pennsylvania State

U in University Park. He conducts research in air-drilling well

bore hydraulics, multlphase

flow

in pipes, and produced-water

treatment. He holds a BS degree in mechanical engineering and

an MS degree in drilling engineering from China U of Geosci

ences and a PhD degree in petroleum and natural gas engineer

ing from Pennsylvania StateU Michael A.

Adewumi

is associate

professor of petroleum and natural gas engineering at Pennsyl

vania State U His research in terests are multi phase flow in

pipes, fluid dynamics in porous media, phase behavior, and well

bore hydraulics. Adewuml holds a BS degree in petroleum engi

neering from the U

of

Ibadan, Nigeria, and

MS

and PhD degrees

in

gas engineering from the Illinois Inst,

of

Technology.

Tlan

Adewumi


Documents

Tian 1994