Tian 1994
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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).
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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)
SPE Production Facilities, May 1994
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
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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.
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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
Gas specific gravity
Pseudocritical pressure, psia
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
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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
Gas specific gravity
Pseudocritical pressure, psia
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
Gas specific gravity
Well
Dunvegan 6-29
Pseudocritical pressure, psia
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
SPE Production Facilities, May 1994
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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
http://slidepdf.com/reader/full/tian-1994 7/7
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
SPE Production Facilities, May 1994