This article presents a one-dimensional numerical model for vertical upward multiphase flow dynamics in a pipeline. A quasi-steady-state condition is used for the gas phase as well as liquid and gas momentum equations. A second-order polynomial for homogeneous flows and a sixth-order polynomial for separated flows are derived to determine the two-phase flow dynamics, assuming that the gas flow mass is conserved. The polynomials are formulated based on the homogenous and separate flows' momentum equation and the homogeneous flows' rise velocity equation and their zeros are the flow actual liquid holdup. The modeling polynomial approach enables the study of the polynomial liquid holdup zeros existence and uniqueness and as a result the design of a stable numerical model in terms of its outputs. The one-dimensional solution of the flow for the case of slug and bubble flow is proven to exist and to be unique when the ratio of the pipe node length to the time step is inferior to a specific limit. For the annular flow case, constraints on the inlet gas superficial velocity and liquid to gas density ratio show that the existence is ensured while the uniqueness may be violated. Simulations of inlet pressure under transient condition are provided to illustrate the numerical model predictions. The model steady-state results are validated against experimental measurements and previously developed and validated multiphase flow mechanistic model.

References

References
1.
Brennen
,
C. E.
,
2005
,
Fundamentals of Multiphase Flows
,
Cambridge University Press
,
Pasadena, CA
.
2.
Taitel
,
Y.
, and
Dukler
,
A. E.
,
1976
, “
A Model for Prediction of Flow Regime Transitions in Horizontal and Near Horizontal Gas-Liquid Flow
,”
Am. Inst. Chem. Eng.
,
22
(
1
), pp.
47
55
.
3.
Taitel
,
Y. S. O.
, and
Brill
,
J. P.
,
1989
, “
Simplified Transient Solution and Simulation of Two-Phase Flow in Pipelines
,”
Chem. Eng. Sci.
,
44
(
6
), pp.
1353
1359
.
4.
Xiao
,
J. J.
,
Shoham
,
O.
, and
Brill
,
J. P.
,
1990
, “
A Comprehensive Mechanistic Model for Two-Phase Flow in Pipelines
,”
SPE
Annual Technical Conference and Exhibition
,
Society of Petroleum Engineers
, New Orleans, LA, Sept. 23–26, pp.
167
180
.
5.
Lord Rayleigh,
1913
, “
On the Motion of a Viscous Fluid
,”
Philos. Mag.
,
6
(
154
), p.
776
.
6.
Lee
,
H.
,
2013
, “
Hydrodynamics Model for Gas–Liquid Stratified Flow in Horizontal Pipes Using Minimum Dissipated Energy Concept
,”
J. Pet. Sci. Eng.
,
108
, pp.
336
341
.
7.
Petalas
,
N.
, and
Aziz
,
K.
,
2000
, “
A Mechanistic Model for Multiphase Flow in Pipelines
,”
J. Can. Pet. Technol.
,
39
(
6
), pp.
43
55
.
8.
Jones
,
A. V.
, and
Prosperetti
,
A.
,
1985
, “
On the Suitability of First-Order Differential Models for Two-Phase Flow Prediction
,”
Int. J. Multiphase Flow
,
11
, pp.
133
148
.
9.
Beggs
,
H. D.
, and
Brill
,
J. P.
,
1973
, “
A Study of Two-Phase Flow in Inclined Pipes
,”
J. Pet. Technol.
,
25
(
5
), pp.
607
617
.
10.
Harmathy
,
T. Z.
,
1960
, “
Velocity of Large Drops and Bubbles in Media of Infinite or Restricted Extent
,”
Am. Inst. Chem. Eng.
,
6
(
2
), pp.
281
288
.
11.
Nicklin
,
D. J.
,
Wilkes
,
J. O.
, and
Davidson
,
J. F.
,
1962
, “
Two-Phase Flow in Vertical Tubes
,”
Trans. Inst. Chem. Eng.
,
42
, pp.
61
68
.
12.
Hasan
,
A. R.
, and
Kabir
,
C. S.
,
1988
, “
A Study of Multiphase Flow Behaviour in Vertical Wells
,”
Soc. Pet. Eng.
,
3
(
2
), pp.
263
274
.
13.
Barnea
,
D.
,
1987
, “
A Unified Model for Predicting Flow-Pattern Transitions for the Whole Range of Pipe Inclinations
,”
Int. J. Multiphase Flow
,
13
(
1
), pp.
1
12
.
14.
Taitel
,
Y.
,
Barnea
,
D.
, and
Dukler
,
A. E.
,
1980
, “
Modelling Flow Pattern Transitions for Steady Upward Gas–Liquid Flow in Vertical Tubes
,”
Am. Inst. Chem. Eng.
,
26
(
3
), pp.
345
354
.
15.
Varquez
,
M.
, and
Beggs
,
H.
,
1980
, “
Correlations for Fluid Physical Property Prediction
,”
J. Pet. Technol.
,
32
(
6
), pp.
968
970
.
16.
Dindoruk
,
B.
, and
Christman
,
P. G.
,
2004
, “
PVT Properties and Viscosity Correlations for Gulf of Mexico Oils
,”
Soc. Pet. Eng.
,
7
(
6
), pp.
427
437
.
17.
Sanjari
,
E.
,
Lay
,
E. N.
, and
Peymani
,
M.
,
2011
, “
An Accurate Empirical Correlation for Predicting Natural Gas Viscosity
,”
J. Nat. Gas Chem.
,
20
(
6
), pp.
654
658
.
18.
Dranchik
,
P. M.
, and
Abou-Kassem
,
J. H.
,
1975
, “
Calculation of z-Factors for Natural Gases Using Equations of State
,”
J. Can. Pet.
,
14
(
3
), pp.
34
36
.
19.
Londono
,
F. E.
,
Archer
,
R. A.
, and
Blasingame
,
T. A.
,
2002
, “
Simplified Correlations for Hydrocarbon Gas Viscosity and Gas Density-Validation and Correlation of Behaviour Using a Large Scale Database
,”
SPE
Gas Technology Symposium, Society of Petroleum Engineers
, Calgary, Alberta, Canada, Paper No. SPE-75721-MS.
20.
Moody
,
L. F.
,
1947
, “
An Approximation Formula for Pipe Friction Factors
,”
Trans ASME
,
69
, pp.
1005
1006
.
21.
Barnea
,
D.
, and
Taitel
,
Y.
,
1993
, “
A Model for Slug Length Distribution in Gas-Liquid Slug Flow
,”
Int. J. Multiphase Flow
,
19
(
5
), pp.
829
838
.
22.
Wallis
,
G. B.
,
1969
,
One-Dimensional Two-Phase Flow
,
McGraw-Hill
,
New York
.
23.
Yuan
,
G.
,
2011
, “
Liquid Loading of Gas Wells
,” M.S. thesis, The University of Tulsa, Tulsa, OK.
24.
Fang
,
X.
,
Xu
,
Y.
, and
Zhou
,
Z.
,
2011
, “
New Correlations of Single-Phase Friction Factor for Turbulent Pipe Flow and Evaluation of Existing Single-Phase
,”
Nucl. Eng. Des.
,
241
(
3
), pp.
897
902
.
25.
Florence
,
M.
,
1941
, “
Descartes' Rule of Signs
,” Stanford University, Stanford, CA.
26.
Alves
,
G. E.
,
1954
, “
Co-Current Liquid-Gas Flow in a Pipe-Line Contactor
,”
Chem. Eng. Prog.
,
50
, pp.
449
456
.
27.
Mukherjee
,
H.
,
1979
, “
An Experimental Study of Inclined Two-Phase Flow
,” Ph.D. thesis, The University of Tulsa, Tulsa, OK.
28.
Griffith
,
P.
,
Lau
,
C. W.
,
Hon
,
P. C.
, and
Pearson
,
J. F.
,
1973
, “
Two Phase Pressure Drop in Inclined and Vertical Pipes
,” Heat Transfer Laboratory, Cambridge, MA,
Report No. 81
.
29.
Nguyen
,
V. T.
,
1975
, “
Two-Phase, Gas–Liquid Co-Current Flow: An Investigation of Holdup, Pressure Drop and Flow Pattern in a Pipe at Various Inclinations
,”
Ph.D. thesis
, University of Auckland, Auckland, New Zealand.
30.
Govier
,
G. W.
,
Radford
,
B. A.
, and
Dunn
,
J. S. C.
,
1957
, “
The Upward Vertical Flow of Air—Water Mixtures I. Effect of Air and Water Rates on Flow Pattern, Holdup and Pressure Drop
,”
Can. J. Chem. Eng.
,
35
, pp.
58
70
.
31.
Woldesemayat
,
M. A.
, and
Ghajar
,
A. J.
,
2007
, “
Comparison of Void Fraction Correlations for Different Flow Patterns in Horizontal and Upward Inclined Pipes
,”
Int. J. Multiphase Flow
,
33
(
4
), pp.
347
370
.
You do not currently have access to this content.