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QC320
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18 1977
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YW.
TWO PHASE PRESSURE DROP IN
INCLINED AND VERTICAL PIPES
Peter Griffith Chun Woon Lau
Pou Cheong Hon
John F. Pearson
Report Number DSR 80063-81
Mobil Oil Company
/00'
Heat Transfer Laboratory
Department of Mechanical Engineering
___Mssachusetts Institute of Technology
Cambridge, Massachusetts 02139
August, 1973
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TECHNICAL REPORT NUMBER 80063-?11
1
TWO PHASE PRESSURE DROP IN
INCLINED AND VERTICAL
PIPES
Peter Griffith
Chun Woon Lau
Pou Cheong Hon
John F. Pearson
Sponsored as a grant in aid by the Mobil Oil Company
DSR 80063
August, 1973.
HEAT TRANSFER LABORATORY
DEPARTMENT OF MECHANICAL ENGINEERING
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
CAMBRIDGE, MASSACHUSETTS 02139
2
ABSTRACT
A method of calculating the pressure drop in inclined and vertical
oil-gas wells is proposed.
The data used to establish the method is from
a variety of sources but is largely from air and water flowing in
systems close to one atmosphere in pressure and in pipes from 1 to 2
inches in diameter.
are included.
All inclinations from vertical to almost horizontal
The method proposed is used to calculate the pressure
distribution in ten oil and gas wells.
The predictions for the overall
pressure drop are good to + 10% for these wells.
INTRODUCTION
The purpose of this work is to provide a simple, physically based
calculation method for determining the pressure distribution in oil and
gas wells.
The equations to be proposed are, by now, well established but
have not been systematically applied to this problem.
The body of this report constitutes a succinct presentation of the
proposed method with a brief discussion of its characteristics and
limitations.
A sample of the various kinds of data used to establish
the correlation is also included in the report.
The appendicies in the
report contain the bulk of the data.
Also included is Appendix D which shows how well this method works
on actual oil well data.
THE REQUIREMENTS FOR THE PRESSURE DROP CALCULATION METHOD
In general the two phase drop in a pipe is a sum of three terms:
a gravity, a friction and a momentum pressure drop.
The momentum
pressure drop is negligible for oil and gas wells because they are
so long.
For this application, then, we can say
AP= AP, +Ap
m
The friction term is empirical for single phase flow and remains empirical
for two phase flow.
For this application it is proposed to evaluate this
term fram an extension of the Thom friction pressure drop calculation method
as described in Wallis (1).
The gravity term cannot be evaluated using any of the established
overall pressure drop calculations and techniques.
This is because they all
relate the void fraction to the quality and properties but leave out
the important effect of pipe diameter on void for vertical or inclined
pipes.
This effect is important because there is an optimum diameter for
any given flow rate, and one would certainly want to choose a pipe of the
optimum size (2) for the well.
This optimum arises from the fact that
at fixed flow rates for two phases the gravity pressure drop increases with
increases in pipe diameter while the friction pressure drop generally
decreases.
See Figure (1).
Neither the Thom, Martinelli or any other
common method shows this optimum.
wells however.
It
is
quite important for oil and gas
5
AP
. Qtota I ....WqrdVity
D
Two phase pressure gradient as a function of pipe diameter at
Figure 1
fixed flow rate for the two phases, in inclined or vertical pipes showing
how the gravity and friction pressure drop contributions change when the
pipe diameter is altered.
4
BUBBLY AND SLUG FLOW VOID AND PRESSURE DROP IN INCLINED PIPES
All flow regimes are found in inclined pipes.
we will consider the bubbly and slug flow regimes.
In this section,
Whatever the flow
regime, however, the void fraction is to be substituted into equation
(2) to determine the gravity pressure drop.
L sinG
+g,(o4I
oc)
0P.
~
(2)
Though one can easily see bubbly flow in inclined pipes (3), the
void fraction range in which bubbly flow can be detected by void measurements
is minute.
The reason is
the bubbles rapidly migrate to the top of the pipe
where they can soon collide and agglomerate into slug-flow bubbles.
at void fractions less than 10%.
scarcely exists.
This occurs
As a result bubbly flow, as such,
Figure (2) (to be explained soon) shows this.
For
vertical pipes, bubbly flow occurs up to a void fraction of about
15%.
Inclined pipes differ significantly from vertical pipes in this
respect because the tilt substantially increases the number of collisions
between bubbles and promotes a rapid transition to slug flow.
For
simplicity in this calculation procedure it is suggested that the
existence of bubbly flow be ignored as it occurs .in such a small region
that its unique contribution to the overall pressure drop for a typical
oil is not very significant.
Bubbly flows will be treated as slug flows
occuring at the same flow rates for each phase.
As the slug flow occurs almost immediately, the Zuber-Findlay (4)
method of treating slug flow void data is
appropriate.
Figure (2)
sample of the data plotted as suggested in reference (4).
is
a
It is evident
o
BUBBLY FLOW
*
TRANSITION FLOW
A
SLUG FLOW
00
0
A.o
0O
0
w n5
*
0
Vb
a
&Is4
vbu-I.ISV +1.50 FT/SEC.
4
5
FT/SEC
Figure 2: Zuber-Findley plot for a 1.9 I.D. pipe inclined 30* from the
vertical at room temperature and pressure.
Reference(3) .
8
on Figure (2)
(and the other curves of Appendix A) that the region where
bubbly flow can be detected is
negligible.
For the data of Figure (2),
the Zuber-Findlay constants are
C
0
= 1.15
VO = K2 VrgD
which is
to be substituted into the equation
--
The C
is
C.
( .j~-)+
.
IT
V0
-
(3)
3
found to be independent of inclination and everything else, as
shown on Figure on Figure (3)
(data of Reference
(5 and 6),
also
Appendix B).
V
0
should be a function of all the variables which can affect
the bubble rise velocity in vertical pipes.
and a weak diameter dependence are evident.
using the data shown on Figure 4.
their significance is
lost in
In fact only the tilt
V
should be calculated
Though more variables might be important,
the scatter of the data.
The wall shear stress can be either negative or positive in slug
flow.
The reason is the liquid flowing by a bubble runs down while the
liquid in the slugs moves up with the mixture velocity.
The wall shear
stress is usually small, however, when the liquid is flowing down.
This
can be seen on Figure (5) (Ref. 3) where the total pressure drop is
sometimes a little more and sometimes a little less than the gravity
pressure drop.
For this reason it is recoimnended that the Thom friction
multipliers be used to calculate the friction pressure drop.
enr=
4f1P(4)
That is,
A-Lau-D=1.90"
o-Beggs-D=", 1,5"
1.4
1.2
Co
U
0
-0
0
1.0
.8
.6
.4
.2
O1
n*
30*
600
90
Figure 3: C -versus angle of inclination showing the lack of any
angle or diameter effect. References (3) and (4). 0 is measured
from the horizontal.
1.0
o-Lau (1)
&-Singh and Griffith (6)
.8
K2
.6
.4
.2
0L
O"
30"
60*
90*f
Figure 4: K2 versus angle of inclination for substitution in
equation (1).
Reference (2). 0 is measured from the horizontal.
1.0
0.9
0.8
1i-
+
7E
W0.
(L
+t
O.
L
.
+
to
z- 0.
=-
dP
dPi
dI
d
:
WO.
Cr 0
0-
.1
0.I
0.5
0.3
0.4
VOID FRACTION, DIMENSIONLESS
0.6
Figure 5: Pressure gradient versus void fraction for vertical pipes.
Air and water in a 1.90 inch ID pipe at room temperature. Reference (3).
11
The multiplier is to be obtained from Figure (6) (ref. 7).
are explained in the next section.
These curves
The friction factor is to be evaluated
assuming pure liquid is flowing in the pipe at the mixture mass velocity.
.
,
-'
pir1
22'
" 14
6
0
6
/
II
/
ii
/
,/
~I
/j
/
5
/~
II
'I
//
1/
/
/
/
/
*1
/
/
/
/
/
ANNULAR FLOW VOID AND PRESSURE DIROP IN INCLINED PIPES
Compared to slug flow, the contribution of gravity to the pressure
drop is less significant in annular flow.
For this reason, a simple
method for calculating the annular flow gravity pressure drop is all that
is needed.
It is proposed to use the velocity ratio curves from Thom
(7) in order to calculate the void fraction.
Figure 7 shows the
appropriate curve while equation (5) is to be used to evaluate the void.
Equation (5) is derived from the continuity equation as applied to
each phase.
:1
(5)
V,
Typically one can calculate that the void fraction is
80% in the annular flow region.
greater than
This method for calculating void is
not very precise but, because of the reduced importance of the gravity
term in
the annular flow region,
purposes.
If
the accuracy is
adequate for those
one tries to calculate the velocity ratio from the measured
void fraction, however, the scatter is very poor when using equation (5).
The friction term is quite significant in the annular flow
region.
For this it is proposed that the Thom calculation method, already
mentioned, be used.
As presented the Thom friction multiplier curves only
apply to steam and water at pressures above 200 psia.
For our purpose,
they must be extended in two ways.
Instead of pressure (for water and
steam)
it
as the independent variable,
ratio can be used.
will be assumed that the density
On Figure 6 the pressure variable was replaced by the
ratio of densities for that pressure.
apply to any pair of fluids.
These curves are then assumed to
That is, the effect of viscosity, surface
tension or any other property variations on the two phase pressure drop
will be assumed to be negligible.
in addition, to extend the Thom
curves to lower pressures it will be assumed that one can use the
Martinelli curves (taken for air and water at one atmosphere) and that
the turbulent-turbulent friction multiplier line should be used to
make the extrapolation.
The altered Thom curves are as shown on Figure
(6).
Thom was chosen for the basic multiplier because the steam-water
data used to develop them was taken at the appropriate density ratios and
pipe sizes.
Unfortunately, the bulk of the Thom data is in the slug
flow region which is not entirely desirable for this application.
In
spite of this no systematic deviation in the Thom data is evident in the
high quality region where annular flow is expected to occur.
A comparison of the combined void and friction pressure drop
calculation
methods is
shown on Figures (8),
(9) and (10).
The
data in this case is from reference (6) while the curves are drawn
from the methods suggested in Reference (5), which are the methods put
forward here.
A discussion of these curves is appropriate.
Figure (8) shows how the calculated and measured void fractions
compare using the slug flow equation, equation (3).
The actual
void is generally a little greater than calculated and deviates
increasingly at voids greater than 80%.
The low void deviation
is felt to be a result of gas entrained in the liquid.
Generally
the data of reference (6) are taken at a higher velocity than those
used to establishe the correctness of equation (3), so more gas entrainment
is present.
At high voids (greater than 80%) the deviation is due to
the onset of annular flow.
a void greater than 1/C
point.
There is no way equation (3) can show
so this kind of deviation must occur at some
The boundary between the slug and annular flow regions is
subject of the next section.
the
Appendix B shows the void and pressure
15
drop data of (6) reduced in this way by Hon (5).
The pressure drop curves shown here show some scatter but over
the bulk of the data a systematic deviation is not apparent.
From
experimenter to experimenter, however, deviation does occur.
The pressure
drop data of Reference (6) shows very little systematic deviation.
of Reference
(8) shows some systematic deviation with the measured
pressure drops typically 20% greater than calculated.
ShoA
That
all the data of reference (8)).
(Appendix C
The reason is not clear, though
perhaps the higher general velocity level of these data are responsible.
I
1~
-
-4.--I--Il
~---~1
V9
vi?
.-7*1
I
Li..
1
[.~ii
I
t~T
I
~
*
Iii
I
*
-VI
1
S
t
t--
I
~
.1
10
~
p
100
-t
~
I ~
~~IlL
i
Pf
1000
Pg
Velocity ratio as a function of density ratio.
Figure 7:
Reference (7).
1.0
O-Annular
A-Slug
I
Ca
3S
.8
cP
Slug
a
Annular
I
01
0
.2
.4
.6
MEASURED
VOID
Figure 8: Measured and calculated void fraction for 35* inclination
from the vertical for data of Reference (6), Reference (3). The
systematic deviation at high void is evident. The remaining data is
in Appendix B.
I
1.0
.96
08j
.92
.88
.84
.80
.80
.84
.88
.92
.96
MEASURED
VOID
1.0
Figure 9
Plot of calculated versus measured void in the annular flow
region for the data of Beggs (3), adapted from Reference (2). See
Appendix B for the rest of the data.
1.0
0=350
a-Slug
&~-Annular
.1a
01
.0
0~q
1.0
0.1
MEASURED
( kE
Figure 10: Plot of calculated versus measured total pressure drop for
Reference (5). See Appendix B
both flow regines. Data of Beggs (3).
for the rest of the data.
THE SLUG FLOW ANNULAR FLOW REGIME TRANSITION
The choice of the means of distinguishing between the slug and annular
flow regimes forced us to consider all the factors which govern both
types of flow.
It is instructive to review these thoughts again even
though they are not necessary for calculating when this transition
occurs.
The primary purpose in distinguishing between slug and annular
flow is to allow one to choose the right expression for calculating
the void fraction.
The obvious thing to do is to equate the appropriate
slug and annular flow expressions for void fraction and choose the
transition void as the one where they both coincide.
This doesn't
work for the following reasons.
The ideal expression for calculating any quantity for that flow
regime will not work very well on the fringes of that regime.
Therefore,
the solution for the commom void at transition is not very satisfactory
because the input equations are inadequate.
The desired conmom
solution is sometimes non-existent and sometimes multivalued.
separate transition criterion is needed.
A
Such a transition criterion
should include effects of diameter for low velocities and gas
to liquid density ratios for various pressures.
presents just such a transition criterion.
just as it is given in Reference (9).
work.)
Reference (9)
It is to be used here
(Reference (1) summarizes this
The criterion is, annular flow exists if
&*>1.5
arV1d
Jfp
>(7+ o 6 )j
(4)
or if
j4
I~n
j*
>
These criterion can be plotted as shown on Figure (11)
slug and annular regimes.
I
£
showing the
8
9
ig
6
Annular
FlowI
5 00
4
2
2
06
0
Slug Flow
1
2
.3
5
4
Jf
Figure 11:
regimes.
Plot of transition criterion showing the slug and annular
DISCUSSION
The methods proposed here are extraordinarily simple so it is appropriate
to discuss the significance of the variables which have been left out.
These variables are as follows:
1)
Pipe roughness
2)
Viscosity for liquid and gas
3)
Entrainment effects
The effect of pipe roughness in two phase flow is
complex.
Generally
one finds that in bubbly and slug flow, roughness effects are about
the same as for single phase flow.
In annular flow, the presence
of the liquid phase can either increase or decrease the friction drop.
However, the effect is not usually large.
Gas or liquid viscosity are of little
is
so large that laminar flow is
possible.
impossibility for the gas phase but is
consequence unless the viscosity
This is
conceivable for the liquid.
error under these -conditions would be considerable.
whether the flow is
laminar in
a practical
One determines
the bubbly or slug flow regimes by
looking at a Reynolds Number based on mixture velocity,
and viscosity.
The
liquid density
In annular flow the film is almost always effectively
turbulent.
Bubble entrainment is important in that it decreases the effective
liquid density and increases the effective liquid flow rate.
In annular
flow droplet entrainment changes the effective gas phase density too.
Entrainment is
correlation.
probably the most significant factor left out of this
If this correlation turns out to be inadequate in this
form this is probably the place where additional work would yield the
greatest benefits.
22
There is data frcn other sources which has not been analyzed here.
Principal among this is the data of reference (10).
This data was no
good because the pressure level at which it was taken was not reported.
If atmospheric pressure was assumed, absurd answers were obtained for
sane of the runs.
23
LIST OF SYMBOLS
C
Dimensionless coefficient in slug-flow bubble rise equation.
-
0
D
-
Pipe diameter in feet for instance.
f
-
friction factor defined in equation (2) dimensionless.
G
-
Mass velocity in lb/sec ft
g
-
Acceleration of gravity equal to 32.2 ft/sec
2
for instance.
2
2
g,- Gravitational constant; equal to 32.2 ft ibm/sec lb
f*
j
Liquid superficial velocity in ft/sec based on total volume
flow rate and pipe cross-sectional area.
-
1/2
*
f-
Dimensionless liquid velocity equal to if divided by [gDp f-P )/Pf]
Gas superficial velocity in ft/sec based on total volume flow rate
and pipe cross sectional area.
-
*
g -Dimensionless
gas velocity equal to
j / (gD(p -p P
1/2
L - Pipe length ft.
AP
Pressure drop in lb/ft
-
AP
-Gravity
9
for instance.
Friction pressure drop in lb/ft
f3
AP
2
pressure drop in lb/ft
2
for instance.
for instance.
3
- Liquid volume flow rate in ft /sec for instance.
3
Q - Gas volume flow rate in ft. /sec for instance.
9
r - Friction multiplier, dimensionless, from Figure 7 for substitution
in equation 2.
V
V
b
4
mixture velocity in ft./sec.
-
V
f
-
True bubble rise velocity in ft./sec.
= True liquid velocity in ft/sec. for instance.
V
True gas velocity in ft/sec for instance.
g
V
=
o
Bubble rise velocity in stagnant liquid.
in ft/sec.
See equation (1) and Fig. 5,
24
x - Quality, weight fraction of gas flowing.
a- Void fraction, dimensionless.
Pf- Density in
ibm/ft
3
for liquid.
3
p - Density in lbm/ft for gas.
g
0- Angle of tilt measured from the horizontal.
REFERENCES
(1)
Wallis, G.B., "On Dimensional Two Phase Flow", McGraw-Hill, 1969, p.58.
(2)
Singh, G. and P. Griffith, "Determination of the Pressure Drop Optimum
Pipe Size for a Two-Phase Slug Flow in an Inclined Pipe" ASME Paper
No. 70-Pet-15.
(3)
Lau, Chun Woon, "Bubbly and Slug Flow Pressure Drop in an Inclined
Pipe. SB Thesis in Mechanical Engineering, MIT, June 1972.
(copies
available from Professor Griffith)
(4)
Zuber, N. and J.A. Findlay, "Average Volumetric Concentration in TwoPhase Flow Systems" J. of Heat Transfer, Trans. ASME Vol. 87, Series
C, No. 4, p. 4 5 3 - 4 6 8 .
(5)
Hon, Pon Cheong, "Recommended Methods for Determining the Pressure
Drop in Two phase Flow in Inclined Pipes in the Slug and Annular
Flow Regimes. SB Thesis in ME, MIT June 1973.
(6)
Beggs, H.D., "An Experimental STudy of Two-Phase Flow in Inclined
Pipes" Ph.D Thesis,Department of Petrolieum Engineering, Univ. of
Tulsa, 1972.
(7)
Thom, J.R.S., "Prediction of Pressure Drop During Forced Circulation
Boiling of Water" Int. J. Heat and Mass Transfer, Vol. 7, pp.
709-724, 1964.
(8)
Sevigny, R. Jr., "An Investigation of Isothermal CoCurrent, Two-Fluid,
Two-Phase Flow in an Inclined Tube, Ph.D Thesis, Department of Chemical
Engineering, University of Rochester, 1962.
(9)
Haberstroh, R.D. and P. Griffith, "The Transition from the Annular
to the Slug Flow Regime in Two-Phase Flow,"
Technical Report 5003-28
(See Wallis (1) above too)
Department of Mechanical Engineering, MIT 1964.
(10)
Aynsley, E., "The Pressure Drop of a Two-Phase Air-Water Mixture in
an Inclined Pipe. Ph.D Thesis, Department of Chemical Engineering,
University of New Castely-upon-Tyne 1970.
APPENDICIES
Appendix A - Curves and tabulated void fraction data from Lau (3) for the
density of an air and water mixture flowing upward in an inclined pipe.
Appendix B - Calculated and measured pressure drop data from Beggs (6) for
air and water in an inclined pipe.
Appendix C - Curves and tabulated total pressure drop data for air and
water in an inclined pipe from Sevigney (8).
Appendix D - Oil and gas data from Mobil Oil company.
from Dr.Aziz Odeh.
Personal Communication
-
. i- -!'-
APPENDIX A
Appendix (A) contains Lau's data (3) for void fraction used in the
determination of the Constants C
o
and V .
o
Both V
The tabulated data is given on the later pages.
b
and V are in feet per sec.
The apparatus consisted of
a 3 foot calming section, with a plexiglass viewing point, and an eight foot
section which could be isolated by quick closing valves to measure void.
The flow at the top was exhausted to atmosphere.
averaging were used to measure the pressure drop.
inside diameter. The
Transducers with electric
The pipe was 1.90 inches
temperatures reported at the top of the tabulated
data are the temperatures at the flow measuring points.
The volume flow
rates are calculated at the mid-point of the test section for the mixture
temperatures and pressure.
o
BUBBLY FLOW
FLOW
TRANSITION
9A
SLUG FLOW
8-
A
A A
7-
.
-
Vb
6-
*0
w
(00
O
-
o
2A
Fgo
A
A
o
I
tp
A
4-
2-
A
3L
A0
A
p
w
a
0
a
13
U
0
%
IV 0
Vb 1.2 V + 0.70 FT/SEC
2-
01
2
3
4
5
6
7
A
Fig. A1
Vb versus V for vertical pipe.
Air and water in a 1. 9 inch
ID pipe at room temperature and pressure.
8
29-
10
EBUCELY FLOW
o
FLOW
9 TRANSIT 1I
9-
SLUG FLOW
A
U
C
A
8-
00
7A6
AA
A
5-
,
A0
o
,*
0
30
0
-
0
Vb
2-
T/E
.5
1.5V+
0
Ve V 15u +175 FT/piSEC
2b
AA
w
j
Pig. A2
-
i a
Vb versus
V
A
4
5
FT/SEC
v for pipe inclined at
6
400 from vertical.
7
Air and
water in a 1.9 inch ID pipe at room temperature and pressure.
o
BUBBLY FLOW
a
TRANSITION FL OW
A
SLUG FLOW
5
W
4A
r*
o on
-1. 15V + 1.50 FT/11SEC.
3-"Vb
1
2
2A
Fig. A3
3
V=
4
--
5
6
7
T /SEC
Vb versus V for pipe inclined at 30* from vertical.
Air and
water in a 1.9 inch ID pipe at room temperature and pressure.
10-
BUBBLY FLOV
9
TRANSITION FLOW
A
SLUG FLOW
8-
0
--
o
0
<
C
co 0
ooa
a
0
o .
0
0
a
1.0
1.2V +
Vb
FT/SEC
2-
VFig. A 4
4
3
2
OF
A
5
6
7
FT/SZ3EC
V9 versus V for pipe inclined at 20* from vertical.
Air and
water in a 1.9 inch ID pipe at room temperature and pressure.
8
*
BUBBLY FLOW
M
TRANSITION
A
SLUG FLOW
FLOW
A
0-
0
A.
0
U)
LL
0
Vb
l.
*
I
2
V
0.0
F/SE
.
3
4
5
FT/SEC
V -
6
7
8
A
Fig.
A5
Vb versus V for vertical pipe.
Air and water in a 1.9 inch
ID pipe at room temperature and pressure.
UIll
o
BUBBLY FLOW
*
TRANSITION FLOW
I&
SLUG FLOW
7[
0
*
5 -
0
0
EU
0
a
4*
0
Vb- 1.2V + 1.30 FT/SEC
2
U
1
1
1
I
2
3
0
V Fig.
A6
i
Q +Q
A
4
5
6
7
FT/SEC
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50
APPENDIX B
Appendix B contains a comparison of the measured and calculated void
and total pressure drop using the methods put forward in the text.
Data of Reference 6 as reduced in Reference 5.
-
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(6) data the plot of calculated
This section shows for Beggs's
dP
dL
dP
using the recommended methods v.s. the measured (
calculated
dP
(-)
and the measured
dLT
dL
IT
is the total pressure drop.
(
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dL
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T
.
are in psi per foot.
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63
APPENDIX
Appendix (C
calculated
measured
measured
)
P )
b u it
drop.
shows for Sevignv's (8] data the plot of
dP
dLT
dP
dL
C
using the recommended methods v.s.
. Both the calculated
T
are in psi per foot.
Tabulated data is also included.
dP
dL
the
and the
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dL
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These pages
contain
Sevigny's
(8] data sheets.
D
= 0.8245
0
is in degree.
Gf
is in lb. mass ner foot sauare per second.
G
is in lb. mass per foot square per second.
P
is in psia.
T
is in degree F.
PDT
is calculated
inch.
dP
in psi per foot.
dL
PDM
is measured
dP
dL
FLOW
,T
in psi per foot.
,T
refers to flow regime. S indicates slug flow regime.
A indicates annular flow regime.
S 62L 11i
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Appendix D
"COMPARISONS WITH OIL WELL DATA"
Appendix D contains the application of the method of the text to
actual oil well data.
This Appendix includes:
1)
A discription of the various problems encountered in dealing with
crude oil-natural gas systems.
2)
A detailed referenced outline of each calculation of the correlation
for the first step of a one hundred increment iteration.
3)
Well data and pressure drop predictions for ten wells.
4)
A plot of calculated
calculations.
5)
A discussion of the method as applied to this data.
AP
versus measured
AP
for the above
Two phase flow of crude oil and natural gas in high pressure pipelines is much more complicated than the flow of air and water near
atmospheric pressure.
The great disparity is due largely to the solu-
bility of natural gas in crude oil and the deviation of natural gas
from the normal ideal gas model.
In this appendix an outline of these
complications appears,along with a detailed description of the method of
the above text as applied to oil well data.
Included are data from ten
wells as supplied by the Mobil Oil Corp. and the pressure drop calculations
for each well.
The major consideration in using oil-natural gas data is
the des-
cription of gas solubility in crude oil as a function of pressure and
(All of the correlations used are grouped together in Frick
temperature.
(D-1),
The most suitable
though each will be individually referenced.)
correlation found for this physical situation was Lasater's
pressure correlation (D-2).
ratio
In this correlation R
,
bubble point
the gas-oil solution
is found as a direct function of pressure, temperature, and oil-gas
Knowing this ratio, one can determine the amount of gas in
properties.
solution and, therefore, from continuity, the gaseous phase flow rates.
The next major consideration is the deviation of natural gas from the
ideal gas model.
This factor is
deviation.
P
r
and T
r
The compressibility factor,
,
Z , accounts for this
a function of the reduced pressure and temperature,
the ratios of the actual temperature and pressure to the
pseudocritical temperature and pressure.
The tables and plots for calculation
are found in Katz (D-3).
Two other factors arise from the gas-oil solubility.
The first
these is the change of oil viscosity with gas-oil solution ratio and
of
temperature.
The effects of
R
on viscosity are outlined in Chew and
Connally (D-4) and the temperature effects in Beal (D-5).
the change in oil volume as a function of
is
mation factor, B
Rs .
, is correlated as a function of
List of Symbols
A
crossectional area of Pipe, ft2
B
volume formation factor, dimensionless
C
constant from Eq. 3, dimensionless
D
diameter of pipe, ft
f
friction factor, dimensionless
G
2
mixed mass velocity, lbm/sec-ft
g
acceleration due to gravity, ft/sec2
0
0
gravitational constant, ft-lbm/sec -lbf
if
jf
g
j *
j9
K
2
oil superficial velocity, ft/sec
dimensionless oil superficial velocity
gas superficial velocity, ft/sec
dimensionless gas superficial velocity
constant from Eq. 3,
dimensionless
Mf
oil mass flow rate, lbm/sec
M
gas mass flow rate, lbm/sec
P
flowing pressure, psi
P
r
reduced pressure, psi
Qf
oil volumetric flow rate, ft 3/sec
Q
gas volumetric flow rate, ft 3 /sec
The other effect
This oil volume forR
a
in
Frick.
R
produced gas-oil ratio, ft
/B
R
solution gas-oil ratio, ft
/B
Re
Reynold's number, dimensionless
r
friction multiplier from Eq. 4
Tr
reduced temperature, dimensionless
Vb
bubble velocity, ft/sec
x
quality, dimensionless
Z
compressibility factor, dimensionless
AL
incremental depth, ft
dP
dLf
dP
pressure gradient due to friction, psi/ft
pressure gradient due to gravity, psi/f t
g
dP
( )
t
total pressure gradient, psi/ft
a
void fraction, dimensionless
y
specific gravity of gas, dimensionless
y
specific gravity of oil, dimensionless
Pots
viscosity of gas-saturated oil, centipoise
Pf
oil density, lbm/f t3
p
gas density, lbm/f t
3
angle of inclination from horizontal, degrees
Detailed Outline of Pressure Drop Calculations
Following is a detailed outline of the first increment of the 100
step pressure drop calculation of well M-3 from the Mobil data.
(See data sheet for remaining properties)
Oil, gas and well properties
Reservoir Pressure - 2655 psig
Reservoir Temperature - 262 F.
(Average angle of inclination - 80
from the horizontal)
Wellhead Temperature - 141 F.
Step 1)
P dT
dL
reservoir
F
ft
-141-262
8865-0
AL
88
T T
R
Z
-
-
ft
2655 psig
F/ft
-.0136
- 88.65 ft
reservoir
+
+
dT
dL- L - 262 -
530 scf/B
(Reference
.860
(Reference (A-3))
y
-
B
- 1.35
.0136(88.6)
F - 261 F
(A-2))
1.56 x 10-5 lb/sec/ft
2
(References (A-4,
(Reference (A-1,
p.
A-5))
19-25))
(The following flow equations are taken from Orkiszevski (D-6).
The
constants are unit conversion factors.)
Qf
6.49 x 10-5 Q B
Qg-
3.27 x 10
go
M-
Q (4.05
M - 8.85 x 10
g
ZQ
x 10-5 y
-
.0498 ft3 /sec
s
(T+
P
460)
+ 8.85 x 10~
-
.01122. ft
/sec
y R ) - 2.203 lb/sec
Q y (R-R ) - .1124 lb/sec
o g
a
Friction Pressure Deep Calculation
Mf+
1)
G
2)
Re
M
I
A
-
2
-
47.44 lb/sec-ft
2-3;,4-Otl
5
-
-
Uos
3)
.046
-
f
(Re)'
M
4)
x
5)
P
-
H +M
g
f
-
-
4.8%
-
3
44.2 lb/f t
-
3
10.02 lb/ft
M
pg
gQg
!f
p
4.416
-
(Figure (6))
- 1.3
6)
(4PL
44f
dt)f
) 2g~p
)L
f
-
x10
psi/ft
(Equation (4))
Calculation of Gravity Pressure Drop
1)
if
-
j
--
1.205 ft/sec
Q
g
jf *
-
A
.
j
.2299 ft/sec
P
.5
fgD(p- p )
-
.4095
(List of Symbols)
)
<
(Equation (5))
1.5
yj
-
j *
< yj --------
g
(List of Symbols)
.0439
-
gD (P fPg)
g
jf*
2)
(
-
j *
g
0.9
+
(Equation (5))
- 1.44
.6j
slug flow
for slug flow
3)
Q
Vb
C
-
4)
A
+)
A
(
0
b
a
(Equation (3))
AVb
-
K 2 (gD)*5
+
80
,K
2
.44
2.2
-8.52%
(Pi
dP)
dL
- a)
p (a)) JL sine
+
-
.340 psi/f t
(Equation (2))
fgo
Calculation of Pressure for Next Step
()
(
-
+
t
P2
(
f
-P
-
.341 psi/ft
g
(P )AL
1
-
2628 psi
t
Discussion
Although the data used represented a varied selection of oil well
conditions,
pipeline.
none of the data resulted in annular flow in any part of the
Therefore,
of conditions,
though the gravity term was used in a wide variety
the friction term was never tested in a demanding way.
In analyzing the error, several plots were made of error versus
various parameters.
fraction,
These included gas velocity, oil velocity, and void
the three parameters to which the correlation is most sensitive.
The plots resulted in only random scatter with no systematic deviation
whatsoever.
Programming, coefficient,
viscosity, or entrainment effect
error would have produced a marked systematic deviation not observable
from the preceding calculations.
relation to error, it
been present.
is
Since no variable seemed to have any
was concluded that some external error must have
Two such areas seem to be evident and sensitive.
The first
The second is error in the
faulty pressure instrument calibration.
correlation relating the pressure and gas-oil ratio.
Such an error could
produce the systematic percentage deviations within a single set of data
(observed in the calculations).
depending on the particular data.
However,
the error could be plus or minus
Therefore,
this seems to be the most
likely area for investigation in the future.
The other fact became obvious from these calculations.
case the pipe was grossly oversized.
diameter was,
in general,
In all but one
The weight of pipe using the optimum
half that using the nominal diameter.
The use of
optimally sized pipe would result in up to a 50% savings in cost of pipe
alone.
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(0o
SudXX/C, p:>in sof
Iner: /D-am
rtp r
i
erin/n.f-
rer i /oi
3
2
81psigj
.
4,-
te,.c e
,Q"
e
0"/=o*
r
1
'*I-k*k
A-/dL r~ -_Pi,~ G*
ERc'A-1
L)&V//7L~7
II/CAL.S
/enNricb/o-r
//
M-.
l/erk1en
ip/h , -/.
1"~A3
S.
Dep//
APsm,
0/
LI
Pressure
(Psi)
3000
I~t~Th
-. ~-
.:r-. 7-
2777
t#~'-17
L4~L&
~
-.
-
-
~:z~~i1 ~#45i!-
-I
I
.I~p~f4zA:t
_-Measured
- -Calculated
-7
-.
'~2:~I
IrtL
1000
1
-7
2000
A
_4
_
-
0
4000
6000
s9A;<
J7.
e~A.; sc6/ 8
A,,,
Ar
'"/'"&'"'*>
s7o.
s~r/
I
,
-___
/y 0 fu rin,&k
o~
R4*f
"
-
rae,,t D;..,
Z,/,* T 3-7.4
i/ s/
rexrrvoir
T~lgS'/z.
,
_
10,000
,C~/6ie
/
_
0. 3
P/daq
Gas/o/
sc(44
_
Depth (ft.)
a/a~i
Gas/
_
8000
,
MIse
__
1
2000
Aes:(
Gas ,
--Af
IT-
J
I44Seb-
_
/k /aX/
e,/ .X_ . t4 0
I
$
se,,-,,1,,,
a7'
.99 S-
I
fd. p /0
sy
'=
-
1S I|
inc
-9
87
r/
/t/L
/)EV/IA
R c'/vl
-
CL~tA/ A
/P//AD CE~
-
7A
IVCA/-LS
7-L)
P/n/fin/,r /0.
e., dw It
7
/7&o
0
Z03
4_ /.:: 1. q
S49. o
999
.
G
~3419
3 7. q9
*
£,-959
S70
43 . 7
757
6415
-
743
/169
47'30
/40
Ias'/./.
5.: er
Pressure
(psi)
3000
1~1~:t
z:
4z:
-
11 tc-:4fr4~4~7V1EI
--:4';'-1 ->1
~I~T71Hi7TTh
-I
-~
-
:
-..-
hzxiz
-
-c-N
L~.t
Measured
-Calculated
2000
---
-A
1000
4
IT
-
0
2000
Ois,
ITTT1
U:
-~
i-c-
:-~--
8000
Depth (ft.)
6000
4000
~i::TPzt~ ~'~W
t-
mSC3/daq
6~?s/~~ ?Li;d
Gas/0,/
/~AC) Sd!
a
,,
l/
t~~~weigs:a
p/iS&se,-/
~ ;Del 6
14sces,/
ge citad
oo/
.77
FL
-~
+>A~
-4~3>~
~4Z3~-1
e
//S'/.ze.
-
n
f-Ie
B
___________
-6443.
sC4/ckk ,
4r
)?.
?
O/,c.
ir:orr o,/ =
.:o.p4
*
'
P
.(r
ii
. G e,
e$-7
/64/2*
-.-
0..
-
ga:A/
10,00C0
Aftv/L
L OW^
r/ - P//)sC
De 1// AT'
)CROA-
U/er/ic /
Z'ep/'r iCI
Prss "'e ./97
-32.3
./036.).
A.~
, /4.7
4fLit
£4s7'
3.16 1
'06 /I
40101
DArA,
il.C-S
-LD
C,464U
-
..
.10,1
Ib
970/
/
'/75^-7..
-/28/5
3/ b
I"
Pressure
(psi)
3000
7,
a
Measured
-
-Calculated
Ad
-
-J
2000
777
j
1000
0
2000
(AZ .rgar
Fg/w ,ec1 es.
BQ
T00
4000
)
/0A
.'
.. s:
,
,
C,--
Gas/Ztige;</3Miio sef/2.8
n
Ga/,
-/ai, /?-Pr-er
/i
Gniy4s
/c/
Su9/>/.
/wn
S7c.
6Sc-s
s:
Ire k *;t
6,,wi/pct
**'6a
car
le/e
0,000"U
8000
Depth (ft.)
:
.
a/ag
/tVa&e,
Gas, MseC6/day
6000
' 30/*g,f
s (4),r=
, e4p.
etre.reD-.orr e/
*
I
=-.
o
.
t
(3Y~'.
i
/
i
[)EV/ AT 6b
=ROiv4
Teo
1n//1/eia
A4
//o.
l1el-445
.
FY owu
/er l/e /
Dep/ A /7.
1 p/h , I/.
W147A
.C-
M/LT /~ -///)
Pes
i
er we
"-FAv
7e
ng
re ,g
.
97 g
47
i
335
)
M
./4
4 4.#
9.7.0-.7
.
4 35
9, 7f
/97 7
47,7
1
-44WX
/7
(79 -7
17671. s'
717 1
/ 56
/532
11
/733
9 78
/ 775-
A'599'
ree4
Pressure
(psi)
3000
Measured
S .]
-
-- Calculated
-
t
-
.t
2000
1000
0200
0
4000
2000
0
(At' STE~k 4 ,.~~4)
g/
Gf9
/ic,
&./,a,
eir/SPIr>
-erA/irs:
s
G4;ye
o,eeoa/
scf/ B/8
scv/
D:
1 7(27
.
,4
~ 4'*C Gas /r
7akin/
5'or,
.
8000
Depth (ft.)
Se~
/fnk
$ /r
6000
-- ---
404
0
Ass, MScIda
Gas/,4;/
G&s /'/
800
-
/&/er,
.w
-
o.r ea
= /)
evo
-c
t. A .er,
it
37 .
3
-e1&
i
-
f.
Go
A p=/.35
10,00C0
A/t' /T/
-
DCvi
pEAZvi
L/e,,/ifir1eJiac//o
L/er-!c.*/
MendsaiedA
e/,
,
,lw
0
/'Lcs LO/ZA ri4
/,~ //sG
14/CA.Ls
7"'*
rCI
F~2o4'/1!
4/4</x-ATED0
r-ess are ,/>g
1,934,
/
/9>oD
q-7~
Iq79
C,77
f44,
I740
10
7
6 11/
7 q7f
/
132S~
3(C
276'9
A)
&
3///
33
33.33.7
7/
877 g370
/5~
3 7Z-
.
Pressure
(psi)
3000
2000
7-7-
Measured
1
1000
200000LI
600
8000cpLL - - - 8000
Depth (ft.)
6000
4000
2000
0
7i
t Res: Y
G-/s ,
3
sef/
s/
d./,
Gs//
-_7;,,_>g___
-0
0
4 /ar/
But/.~ponfof
.
7
Ta/ra
7~h
D-me
c2/
If
s,/
,
T
cp.
rereOPe
9O -
'
n
10
,i
at
I)
i
C*
3sV(.,.raA.*
.1x
~.
t
u
/
__
uvL
9L 4i/,
r s:/Myre/
s o, 'fudo-cedC
F/aiGnw;ysj
o/s as/
1l0,0cC
0
_____AO
______
£;d /?ic
-
~o
MI-cfday
Gas/L,y
- -
I
Ii
t''
2500
2000
15001e
a
S1000
500-
0
0
50
1000
1500
MEASURED
2000
2500
6P
Figure D : Calculated AP vs. measured AP for the Mobil data.
92
References to Appendix D
D-1)
Frick, T. C., and Taylor, R. W., Petroleum Production Handbook
-
Volume II, Reservoir Engineering, McGraw-Hill, New York (1962).
D-2)
Lasater, J. A., "Bubble Point Pressure Correlation", Trans. AIME,
213, 379 (1958).
D-3)
Katz, D. L., et al., Handbook of Natural Gas Engineering,
McGraw-Hill, New York (1959).
D-4)
Chew, J. N.,
and Connally, C. A.,
"A Viscosity Correlation for Gas-
saturated Crude Oils", Trans. AIME,
D-5)
Beal,
It's
D-6)
C.,
216,
"The Viscosity of Air, Water,
23 (1954).
Natural Gas, Crude Oil and
Associated Gases at Oil Field Temperatures and Pressures",
Trans. AIME,
165,
Orkiszewski,
J., "Predicting Two-Phase Pressure Drops in
94 (1946).
Pipe", Trans. AIME, 240, 829 (1967).
Vertical