SPE

Society of Petroleum Engineers

SPE 20631
A Comprehensive Mechanistic Model for Two-Phase Flow
in Pipelines
J.J. Xiao, O. Shoham, and J.P. Brill, U. of Tulsa
SPE Members
II
Copyright 1990, Society of Petroleum Engineers, Inc.
This paper was prepared for presentation at the 65th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in New Orleans, LA, September 23-26, 1990.
This paper was selected for presentation by an SPE Program Committee following review of information contained In an abstract submitted by the author(s}. Contents of the paper,
as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect
any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society
ot Petroleum Engineers. Permission to copy is restricted to an abstract ot not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledg-
ment of where and by whom the paper is presented. Write Publications Manager, SPE, P.O. Box 833836, Richardson, TX 75083-3836 U.S.A. Telex, 730989 SPEDAL.
ABSTRACT
A comprehensive mechanistic model has been
developed for gas-liquid two-phase flow in horizontal
and near horizontal pipelines. The model is able first to
detect the existing flow pattern, and then to predict the
flow characteristics, primarily liquid holdup and
pressure drop, for the stratified, intermittent, annular,
or dispersed bubble flow patterns.
A pipeline data bank has been established. The
data bank includes large diameter field data culled from
the A. G. A. database, and laboratory data published in
the literature. Data include both black oil and
compositional fluid systems.
The comprehensive mechanistic model has been
evaluated against the data bank and also compared with
the performance of some of the most commonly used
correlations for two-phase flow in pipelines. The
evaluation, based on the comparison between the
predicted and the measured pressure drops,
demonstrated that the overall performance of the
proposed model is better than that of any of the
correlations, with the least absolute average percent
error and the least standard deviation.
systems. The traditional approach to solve the problem
has been to conduct experiments and develop empirical
correlations. Although these correlations have
contributed significantly to the design of two-phase flow
systems, they did not take into consideration the
physical phenomena.
Since the mid 1970's, significant progress has
been made in this area. Models have been developed to
predict flow patterns. Separate Models have also been
proposed for the prediction of the flow characteristics
for each flow pattern, namely stratified flow,
intermittent flow, annular flow and dispersed bubble
flow. However, up to date, no study has been carried out
to verify the consistency and the applicability of these
models.
The purpose of this study is to develop a
comprehensive mechanistic model for two-phase flow in
pipelines by combining the most recent developments in
this area. The model is then evaluated against a field and
laboratory measurement data bank, and compared with
several commonly used empirical correlations.
FLOW PATTERN PREDICTION MODEL
INTRODUCTION
References and illustrations at end of paper.
pressure
important
Prediction of flow patterns, liquid holdup and
loss for two-phase flow in pipelines is
for designing gas-liquid transportation
167
When gas and liquid flow simultaneously in a
pipe, the two phases can distribute themselves in a
variety of flow configurations or flow patterns,
depending on operational parameters, geometrical
variables as well as physical properties of the two
phases. The existing flow patterns in pipelines have been
classified into four major types: Stratified Flow
(Stratified Smooth and Stratified Wavy), Intermittent
2 A COMPREHENSIVE MECHANISTIC MODEL FOR TWO-PHASE FLOW IN PIPELINES SPE 20631
Flow (Elongated Bubble Flow and Slug Flow), Annular
Flow (Annular Mist Flow and Annular Wavy Flow) and
Dispersed Bubble Flow. These flow patterns are shown in
Fig. 1.
Flow pattern prediction is a central problem in
two-phase flow analysis. The recent trend in this area is
the development of mechanistic models based on the
physical phenomena. The pioneering work is due to
Taitel & Dukler (1976) and Taitel et al. (1980). Later,
Barnea et al. (1982a, 1982b, 1985 and 1987) adopted the
same approach, modified and extended the existing
models to form a unified model for the entire pipe
inclination angles. On the other hand, flow pattern
determination, especially for the onset of slugging, has
been investigated through linear stability theory by
various researchers (Lin & Hanratty 1986, Andritsos
1986 and Wu et al. 1987). Unfortunately, this approach
is mathematically complex and its solution is very
involved for design purposes. Hence, the Taitel & Dukler
(1976) model with some modifications is used in the
present work.
Three major flow pattern transitions are
identified here: The Stratified-Non Stratified transition,
the Intermittent-Annular transition and the
Intermittent-Dispersed Bubble transition. Stratified
flow is further divided into two subregions: Stratified-
Smooth and Stratified-Wavy flow.
Stratified-Non Stratified Transition (S-NS): The
mechanism of wave growth is used for the prediction of
this transition. A finite wave is assumed to exist on the
gas-liquid interface of an equilibrium stratified flow.
Extending the Kelvin-Helmholtz theory to analyze the
stability of finite waves in pipes, Taitel & Dukler
claimed that when the pressure suction force is greater
than the gravity force, waves tend to grow and thus
stratified flow cannot be preserved. Their analysis leads
to the following criterion for this transition:
[ ]
1/2
hL (PL - pg) g COS a Ag
v.+ -0) P . ( ~ ) (1)
This transition is shown as transition A in Fig. 2 for air-
water flow at atmospheric pressure in a 0.05-m diameter
pipe with a inclination angle of -1".
The stratified-slug transition is predicted
satisfactorily by Eq. (1). For the stratified-annular
transition, however, recent experiments conducted by
Lin & Hanratty (1987) showed that the entrainment-
deposition process is dominant for large diameter pipes,
while for small diameter pipes wave-growth is usually
the dominant mechanism. Nevertheless, no generally
168
accepted model based on the entrainment-deposition
mechanism has yet been found in the literature.
Intermittent-Annular Transition <I-A): When waves
are unstable, the flow could change to either intermittent
flow or annular flow, depending on whether there is
enough liquid supply. The proposed critical liquid level
was 0.5 in the Taitel & Dukler (1976) model. Barnea e t
al. (1982a) modified this criterion by taking into
account possible gas void fraction in liquid slug near the
transition. The revised transition is given by:
hL < 0.35 (2)
D
This is shown as transition B in Fig. 2.
Intermittent-Dispersed Bubble Transition a-DB); The
mechanism governing this transition is believed to be the
turbulent process which breaks up bubbles and prevents
bubble coalescence. Barnea et al. (1987) developed a
unified model for the transition to dispersed bubble flow
applicable to all inclination angles. For the condition
considered in this study (-15" ~ a ~ 15") , however, the
original Taitel & Dukler (1976) model is used because of
its simplicity and sufficient accuracy. When the
turbulent force is sufficiently high to overcome buoyant
force, the gas is no longer able to stay at the top of the
pipe, and dispersed bubble flow will occur. The
transition criterion is expressed as:
[
~
1 / 2
VL> 4A
g
g cos a (1 _pg) (3)
Si fL PL
This is shown as transition C in Fig. 2.
Stratified Smooth-Stratified Wavy Transition (SS-SW): In
stratified flow. the gas-liquid interface can be either
smooth or wavy, which gives quite different results for
liquid holdup and pressure drop. Waves may develop due
to either the in terfacial shear or as a result of
instability due to the action of gravity. For waves
induced by "wind" effect, Taitel & Dukler (1976)
proposed the following criterion according to Jeffrey's
theory:
[ ]
1/2
V
g
> 4 ilL (PL - pg) g COS a (4)
s PL pg VL
where, s is a sheltering coefficient. Values ranging from
0.01 to 0.6 have been suggested from theories and
experiments in the literature. Taitel & Dukler (1976)
used a value of 0.01 to match their experimental data. A
recent study by Andritsos (1986) showed that the
criterion given by Eq. (4) with s = 0.01 is not accurate
for gas flow with liquids of high viscosity. They found
SPE 20631 J. J. XIAO. O. SHOHAM ANP J. P. BRILL 3
that a good comparison can be obtained if a value of 0.06
is used. The sheltering coefficient may indeed be a
function of liquid viscosity. In the present work the
value of s = 0.06 is used. This transition is shown as
transition P in Fig. 2. As can be seen, the transition line
is shifted to the left of the line given by the original
Taitel & Pukler model (1976).
For stratified flow in downwardly inclined
pipes, waves can develop under the influence of gravity
even without the presence of interfacial shear. Barnea et
al. (l982a) presented the transition criterion as:
~ > 1 5
1 ~ '""" " , " .. " , (5)
yg hL
In Figure 2, this transition boundary is represented by
Curve E, and is terminated at the transition P where
waves are agitated by interfacial shear.
INDIVIDUAL FLOW PATTERN MODELS
After predicting the actual flow pattern from the
operational conditions, separate models are needed to
calculate liquid holdup and pressure drop for the
predicted flow pattern. These models are developed in
the following section
STRATIFIED FLOW MODEL
In stratified flow. due to gravity, liquid flows in
the bottom portion of the pipe while gas flows in the
upper portion of the pipe, as shown in Fig. 3. Stratified
flow is one of the most dominant flow patterns for two-
phase flow in pipelines, particularly for flow in
downwardly inclined pipes.
Over the years, various theoretical models with
different degrees of complexity have been proposed for
this flow pattern. Significant recent work includes the
Taitel & Pukler (1976) model, the Cheremisinoff (1977)
model and the Shoham & Taitel (1984) two-dimensional
model. However, models considering the liquid phase
velocity profile are neither easy to use nor guarantied to
give better results. This is maybe one of the reason that
the generalized one-dimensional two-fluid model by
Taitel & Pukler (1976) is commonly used. This approach
is adopted in this study.
Using the steady state one-dimensional two-
fluid model approach and neglecting changes of phase
velocities (or liquid level). the momentum equations for
the two fluids reduce to force balances. They can be
written as:
- A g ( : ~ ) - 'ti Si - 'twg Sg - A g pg g sin a =0",, (7)
Under the assumptions of negligible surface
tension and liquid phase hydrostatic pressure gradient,
the pressure gradients in both phases are the same.
Eliminating the pressure gradient from these equations
results in the so called combined momentum equation:
(PL - Pg) g sin a =0 "" (8)
Applying constitutive equations and geometrical
relationships, one can show that Eq. (8) is an implicit
function of hL/P. One problem encountered in solving
Eq. (8) is the multiple roots which occur in some cases
(Baker et al. 1988 and Crowley & Rothe 1988).
Commonly, it is presumed that the smallest value is the
physical one.
After solving this equation for hL/P, the liquid
holdup can be derived from a geometrical relationship:
EL e - sin e " ".. " " " (9)
21t
where
e = 2 cos'} (1 - 2 h
L
) " " " (10)
D.
With the solved liquid holdup, Eq. (6) or (7) can
be used to calculate the pressure gradient. Another way
is to apply both equations by eliminating interfacial
shear, i.e.,
_(d p) = 'twL SL + 't
wg
Sg +
dx A
(:L pL + ;, pg) g sin a .. (11)
Notice that the first term in the right hand side (RHS) of
Eq. (11) represents the frictional pressure gradient. and
the second term represents the gravitational pressure
gradient. Obviously, the accelerational pressure gradient
has been neglected.
Constitutive Equations
- AL ( ~ ) + 'ti Si - 'twL SL - AL PL g sin a =0 .. (6)
dx
169
Shear
wall, gas-wall
friction factors
stress. The shear stresses in liquid-
and interface are evaluated through
as:
4 A COMPREHENSIVE MECHANISTIC MODEL FOR TWO-PHASE FLOW IN PIPELINES SPE 20631
p vt p v
2
p v
2
'twL = fWL-
L
-- 't
wg
= 'ti = (12)
2 2 2
where fwL and f
wg
are obtained as follows.
Hanratty (1987) correlation (D 0.127 m or 5 in) and the
Baker et al. (1988) suggested correlation, i.e.,
for D 0.127 m
If Vsg Vsg,t, then:
f=ll
Re
for Re 2000 (13) .li- =1 (16)
f
wg
_1_ = 3.48 - 4 log (2....&. + .!l....3..5.-)
if D Re if
for Re > 2000 .. (14)
Where E is the pipe wall absolute roughness. Liquid and
gas Reynolds numbers are defined as ReL = PLvLDLlIlL
and Reg = PgvgDg/ll
g
, with hydraulic diameters, DL and
D
g
, given by Eq. (15):
D = 4 A
g
(15)
g (Sg + Si)
Experiments by Kowalski (1987), Andritsos &
Hanratty (1987) and Andreussi & Persen (1987) all show
that the liquid-wall friction factor deviates from the
friction factor of single-phase flow due to the presence
of interfacial waves. However, it is generally agreed that
using new correlations for fL other than the conventional
one does not improve the prediction significantly. In this
study, the effect of fL on the model performance will be
investigated.
Interfacial friction factor. A closure
relationship for the interfacial friction factor is needed
to complete the stratified flow model. In the original
Taitel & Dukler model this friction factor was assumed
to be equal to the gas-wall friction factor.
Underprediction of the pressure gradient due to this
assumption was reported by many later investigators.
Many studies have been focused on improving the
interfacial friction modelling.
An extensive evaluation of available interfacial
friction correlations reveals that current methods for
prediction of the interfacial friction are far from being
satisfactory. It is found out that the correlation
developed by Andritsos & Hanratty (1987) works well
for small diameter pipes but overpredicts the friction
factor when applied to large diameter pipes. A modified
Duns & Ros correlation (Brill & Beggs 1986) used by
Baker et al. (1988), on the other hand, underpredicts
friction factor for small diameter pipes and gives a
correct trend for large diameter pipes. Therefore, it is
recommended to use a combination of the Andritsos &
170
If Vsg > Vsg,t, then:
.li- = 1 + 15 A fh:(V
sg
- 1) (17)
f
wg
'V D vsg,t
where vsg,t is the critical superficial gas velocity for the
transition to wavy regime. From Andritsos & Hanratty
(1987), this velocity can be approximated by:
V
(18)
sg,t -
P
where p is the pressure in Pa (N/m
2
).
for D > 0.127 m
for N
we
Nil 0.005,
Ei =34 cr (19)
pg vr.
for N
we
Nil > 0.005,
170 cr Nv-t
3
Ei = (20)
PgvL
where Ei is the interface absolute roughness. Baker et al.
(1988) proposed that Ei should be bounded between the
pipe wall absolute roughness and 0.25(hL/D). The Weber
number, N
we
, and the liquid viscosity number, Nil' are
defined as:
= pg vr. Ei ................................ (21)
cr
(22)
PL cr Ei
Baker et al. (1988) suggested to replace the
superficial gas velocity in the original Duns and Ros
correlation with the interfacial velocity, Vi. In this
study, VL is substituted for Vi. From Ei and Reg, the
interfacial friction factor is calculated from Eq. (14).
SPE 20631 J. J. XIAO. O. SHOHAM AND J. P. BRILL 5
and considering
an overall liquid
For flow pattern prediction. unfortunately. use
of the above recommended correlations or any other
correlations is questionable. This is due to the fact that
available correlations were usually developed for actual
stratified flow with low hL /D. while for equilibrium
stratified flow hL /D can range from 0.0 to 1.0.
theoretically. At present. a value of fi = 0.0142 is used
for equilibrium stratified flow in flow pattern
determination. This constant value is suggested by
Shoham & Taitel (1984).
INTERMITTENT FLOW MODEL
Intermittent flow is characterized by alternate
flow of liquid and gas (Fig. 4). Plugs or slugs of liquid.
which fill the entire pipe cross sectional area. are
separated by gas pockets. which contain a stratified
liquid layer flowing along the bottom of the pipe. The
mechanism of the flow is that of a fast moving liquid slug
overriding the slow moving liquid film ahead of it. The
liquid in the slug body may be aerated by small bubbles
which are concentrated towards the front of the slug and
at the top of the pipe.
Intermittent flow has been studied by many
investigators. Recently. a consistent approach has been
carried out by Taitel & Barnea (1990). They presented a
general approach to determine the hydrodynamics of the
liquid film of a slug unit using a very detailed one-
dimensional channel flow model. The disadvantage of this
general approach is the requirement of numerical
integration. For practical application. a model which
assumes a uniform liquid level in the film zone is
believed to be sufficient.
With this assumption
incompressible liquid and gas phases.
mass balance over a slug unit gives:
VsL Lu = VL Es L
s
+ vr Er Lr (23)
where. E
s
and Ef are the liquid holdups in slug body and
film zone. respectively. A mass balance can be also
applied at two cross sections relative to a coordinate
system moving at the translational velocity. For the
liquid phase. this results in
(Vt - vd E
s
=(Vt - vr) Er (24)
The total volumetric flow rate is constant at any cross
section in a slug unit. For the slug body and the film
zone cross sections. this implies:
V
s
= VsL + V
sg
= VL Es + Vb (I - Es) (25)
V
s
= vr Er + vg (1 - Er) (26)
171
where V
s
represents the mixture velocity in the slug
body.
The above four equations yield several important
relationships. From Eq. (25). the liquid velocity in the
slug body. vL. is obtained. Then. Eq. (24) is rearranged
to give an expression for the liquid velocity in the film
zone. vf. Using Eq. (26). an expression for the gas
velocity in the film zone. Vg. can be obtained. The average
liquid holdup for a slug unit. EL. is defined as:
EL =E
s
L. + Er Lr (27)
Lu
From Eqs. (23). (24) and (25). a relationship for EL can
be derived:
E
- Vt Es + Vb (I - Es) - vsg (28)
L-
Vt
Since we consider a uniform liquid level along
the film zone. a combined momentum equation. similar to
Eq. (8) in stratified flow. can be obtained for the film
zone:
'tr.L - 'tg [ ( ~ ) + ( ~ i ) (..h. + ~ ) I l +
Ar Ag g Ar Ag ~
(PL - pg) g sin a =0 (29)
Analogously. Eq. (29) is solved for the equilibrium
liquid level. or the liquid holdup in the film zone. Ef.
Then. the liquid and gas velocities and the shear stresses
can be evaluated. Considering the slug length to be
known. the slug unit length can be obtained from Eq. (23)
and L
u
= L
s
+ Lf:
L
u
=L. VL Es - vr Er (30)
VsL - vr Er
The average pressure gradient for intermittent
flow is calculated by using a force balance over a slug
unit:
(
d P) .
- d x = pu g SIll a +
where Pu is the average fluid density of a slug unit:
pu =EL PL + (1 - Ed Pg (32)
6 A COMPREHENSIVE MECHANISTIC MOPEL FOR TWO-PHASE FLOW IN PIPELINES SPE 20631
The first term of the RHS of Eq. (31) is the gravitational
pressure gradient whereas the second term is the
frictional pressure gradient, which results from the
friction loss in the slug body as well as the friction loss
in the film zone.
Constitutive Equations
Shear stress. The shear stresses appearing in
Eq. (29) are calculated in a similar manner as in
stratified flow, Le.,
" PL I vfl Vf
'tf = ~ f ~ . : - : : . . . . : . . . . . . . . . : .
2
'ti =fi Pg I V
g
- Vfl (V
g
- vf) (33)
2
where ff and f
g
are evaluated using Eq. (13) or (14) with
Ref = PLvfDL/f.lL and Reg = PgvgDg/f.lg. Hydraulic
diameters are defined exactly as in stratified flow. A
constant value of fi = 0.0142 is used for the interfacial
friction factor.
The shear stress in the slug body, 1: s, is
calculated as:
'ts = f
s
ps V; (34)
2
where f
s
is obtained from Eq. (13) or (14) using Res =
PsvsD/f.ls. Ps and f.ls are the mixture density and viscosity
in the slug body, respectively:
ps =Es PL + (1 - Es) Pg (35)
J..ls = E
s
J..lL + (1 - Es) J..lg (36)
Correlations for Vt and vb. The correlation
for elongated (Taylor) bubble translational velocity is
based on Bendiksen's recommendation (Bendiksen 1984):
VI = CVs + 0.35 Yg D sin a +
0.54 YgD cos a (37)
where the value of C depends on the liquid velocity
profile in the slug body. C = 1.2 is used for turbulent
flow and C = 2 is used for laminar flow.
The velocity of dispersed bubbles in the slug
body is given by:
Vb = 1.2 Vs +
172
[
0 g (PL - pg)]1/4 0.1 .
1.53 Pt E; SIll a (38)
where E
s
0
.1 is included to account the effect of "bubble
swarm" in the slug body (Ansari 1988).
Liquid holdup in slug Body. The
correlation developed for liquid holdup in the slug body
by Gregory et ai. (1978), given below, is used in this
study.
E
s
= 1 (39)
(
V )1.39
1 + __s_
8.66
The calculated E
s
is bounded between 1.0 and 0.48.
Slug length. For slug length, we use the
correlation developed by Scott (1987):
Ln (Ls) = - 26.6 +
28.5 [Ln (D) + 3.67Jo.
1
(40)
If D < 0.0381 mm (1.5 in), an approximate value of L
s
=
30 D is used.
ANNULAR FLOWMODEL
The liquid phase in annular flow exists in two
forms: a liquid film flowing along the pipe wall; and,
liquid droplets entrained in the gas core (see Fig. 5).
Unlike the vertical flow case, the liquid film in the
horizontal and inclined configurations is not
circumferentially uniform, but is usually thicker at the
bottom than at the top of the pipe.
Early studies for annular flow were summarized
by Hewitt & Hall-Taylor (1970). The classical treatment
for annular flow has been the use of the well-known
triangular relationship between the film flow rate, the
film thickness and the pressure gradient. This treatment
ignores the liquid secondary flow effects,
circumferential variations of the film thickness, and the
deposition and entrainment rates. These phenomena are
important for horizontal and inclined annular flow.
Therefore, two-dimensional models are proposed to
incorporate these mechanisms (James et ai. 1987 and
Laurinat et ai. 1985). Nevertheless, in these models,
complex mathematical formulations are involved, and
numerical methods are often required for the solution.
For vertical annular flow, on the other hand, the one-
dimensional tWO-fluid approach has used by Oliemans e t
ai. (1986) and later by Alves et ai. (1988). Comparing
with field data, Ansari (1988) shows that this approach
gives excellent results.
SPE 20631 J. J. XIAO. O. SHOHAM AND J. P. BRILL 7
In the present work, the two-fluid approach is
extended to fully developed steady state annular flow in
pipelines. For simplicity, an average film thickness is
assumed. In the gas core, the droplets are assumed to
travel at the same velocity as the gas phase. Thus, the gas
core can be treated as a homogeneous fluid. Because of
these assumptions, the treatment of annular flow is
similar to stratified flow, but with a different
geometrical configuration. Here, the two fluids are the
liquid film and the gas core which includes the gas and
the entrained liquid droplets.
Momentum balances on the liquid film and the
gas core yield
_(dP) ='twL SL +
dx A
( ~ PL + ~ pc) g sin a. " .. """.... """".. "" (47)
Clearly, the total pressure gradient is a summation of the
frictional pressure gradient (the first term of the RHS),
and the gravitational pressure gradient (the second term
of the RHS). Again, the accelerational pressure gradient
is neglected.
Constitutive Equations
Shear stress. The shear stresses are defined
as follows:
- Ar ( : ~ ) + 'tj Sj - 'twL SL -
Ar PL g sin a. =0."""".. " .. """"."""""."" (41)
PL vr
'twL =fr--
2
'ti = f; pc eVe - Vf'f ".""""" (48)
2
- Ac (: ~ ) - 'ti Si - Ac pc g sin a. =0 "".. ". (42)
where Pc is the mixture density in the gas core and is
given by:
pc = Ec PL + (1 - Ec) Pg.... """...... """,, ...... (43)
The liquid holdup in the gas core is related to liquid
entrainment fraction, FE, as follows:
E
c
= VsL FE ...... """........ """... """"... (44)
vsg + VsL FE
Eliminating the pressure gradient from these equations
gives the combined momentum equation:
't
w
L!1.. - 'ti Si (_1_ + _1_) +
Ar Ar Ac
(PL - Pc) g sin a. =0.. " .. " " (45)
Similar to the stratified flow case, all the
geometric parameters in Eq. (45) are functions of olD,
the dimensionless average film thickness. Thus, the
combined momentum equation can be solved for this
unknown, from which the liquid holdup can be
calculated:
_ ( 0)2 V
sg
EL - 1 - 1 - 2 - " (46)
I> v sg +VsL FE
The pressure gradient can be evaluated using Eqs. (41)
and (42):
173
where ff is calculated from Eq. (13) or (14) using ReI. =
PI. vfDL/!!L' with the hydraulic diameter defined as DL =
4o(D-o)/D.
Using an overall liquid volumetric flow rate balance for
the film leads to the following relationship for the liquid
film velocity, vf:
Vr = V
s
dl - FE) .. " ... " .. " .. """".... "".. ""... (49)
4 ~ ( 1 - ~ )
Similarly, for the gas core, the mixture velocity is given
by:
_ v
sg
+ Vol FE
Vc - """"" " ",, (50)
(
1
- 2 ~ r
Liquid entrainment and interfacial
friction factor. To complete the annular flow model,
closure relationships for the interfacial friction factor
and the liquid entrainment fraction are needed. Only few
correlations have been developed from experimental data
for horizontal annular flow (Henstock & Hanratty (1976),
Laurinat et ai. (1984. No data are available for inclined
annular flow. Consequently, correlations developed for
vertical annular flow are also considered in this study
(Wallis (1969), Whalley & Hewitt (1978) and Oliemans
et ai. (1986. It is found out that the combination of the
liquid entrainment and interfacial friction correlations
proposed by Oliemans et ai. (1986) gives the best
results. These correlations are given as follows
8 A COMPREHENSIVE MECHANISTIC MODEL FOR TWO-PHASE FLOW IN PIPELINES SPE 20631
1 - FE
FE
lio lil liz li3 li4 lis D
li6
li? liB ~ 9 5
10 PL Pg ilL Ilg (J VsL Vsg g ( 1)
fi = f
c
[1 + 2250 ( ~ ) ] (52)
(Pc eVe ~ vf'f 0)
The 1988 version of the A. G. A. gas-liquid
pipeline data base contains 455 data points (Crowley
1988). Theses data are from measurements in a wide
variety of gas and oil pipelines. Thus, it provides an
appropriate source of data for statistical analysis.
However, many data points contained in this data base
are either almost identical or not reliable. As an
example, there are cases where unrealistically low
pressure drops were reported for very long pipelines.
Data of this nature are discarded. Another concern in the
evaluation process is the accuracy of the fluid physical
property prediction. To reflect the performance of the
model, errors from fluid physical property calculation
should be minimized. For a compositional system
containing free water a possible water-oil emulsion may
occur. Therefore, compositional data containing free
water are not considered. As a consequent of all these
considerations, only 79 data points are selected for this
study, among which 25 data are from compositional
systems.
(53) Re
c
= pc V
c
Dc .
Ilc
where the 13 parameters are regression coefficients. Eq.
(52) is a modification of the original correlation carried
out by Crowley & Rothe (1986). The core friction factor,
f
c
, can be calculated from Eq. (13) or (14) using the
following definition of the Reynolds number
where
Ilc =EeIlL + (1 - Ee) Ilg ...... .. ..... ......... (54)
Dc =D - 2 0 .... .... .. ........ (55)
DISPERSED BImBLEFLOW MODEL
Among the four flow patterns, the model for
dispersed bubble flow is the simplest one. Due to no
slippage between the phases, the pseudo-single phase
model with average properties is suitable for this flow
pattern. The liquid holdup is thus the no-slip liquid
holdup:
Field measurements by Mcleod et al. (1971) are
also included in our data base. These are high quality
data taken in an offshore pipeline of 152.4-mm diameter.
The fluids are modelled as a black-oil system.
Additional laboratory data from Eaton & Brown
(1965) and Payne et aI. (1979) are included. Although
these data were obtained in small diameter pipes, the
operational pressures are very close to field conditions.
STATISTICAL PARAMETERS
The statistical parameters used in this study are
defined in Table 2 and are explained below:
EL = VsL (56)
Vm
Calculation of the pressure gradient calculation
can be carried out as in single phase flow with average
mixture density and velocity:
(
d P) 2 f
m
pm V ~ (57)
- - = + pm g sm a. .
dx D
EVALUATION
The average percentage error, . 1, and the average
error, '4, are measures of the agreement between
predicted and measured data. They indicate the degree of
overprediction (positive values) or underprediction
(negative values). The absolute average percentage error,
.2, and the absolute average error, .5, are considered to
be more important than . 1 and .4, because the negative
and the positive errors do not cancel out. The standard
deviations, .3 and .6, indicate the scatter of the errors
with respect to their corresponding average errors, . 1
and 4.
PIPELINE DATA BASE
The applicability of the proposed comprehensive
mechanistic model is assessed through comparisons with
actual data. For this purpose, a pipeline data bank has
been established. This data base contains a total of 426
field and laboratory data from various sources, as shown
in Table 1.
The first three parameters are more appropriate
to be used for the evaluation of small values, whereas the
rest three are better for large values. In this study, all
the six parameters are considered in the evaluation
RESULTS AND DISCUSSIONS
The evaluation in this study is only carried out
for pressure drops, since most of the cases in the data
base do not contain liquid holdup values. The commonly
174
SPE 20631 J. J. XIAO. O. SHOHAM AND J. P. BRILL 9
NOMENCLATURE
ACKNOWLEDGMENT
correlations for the wide variety of data
contained in the data base.
3. All individual flow pattern models give better
results than any of the empirical correlations.
following the
1. The major uncertainty for the stratified flow
model is the interfacial friction factor. Future
studies should be focused on improving our
understanding of the interfacial shear
phenomena, and developing more accurate
predictive methods.
3. Small diameter laboratory data represent a
large portion of the data base used in this
study. More high quality field data are needed
to further verify the mechanistic model.
2. For annular flow, the correlations for liquid
entrainment and interfacial friction factor are
all developed from vertical annular flow
experiments. More studies are needed for
horizontal and inclined annular flow.
For future studies,
recommendations are made:
Financial support from The University of Tulsa
and the Tulsa University Fluid Flow Projects (TUFFP) to
J. J. Xiao is gratefully acknowledged.
For a flow pattern dependent model such as the
comprehensive model, the evaluation should be also
carried out for each of the individual flow pattern
models. Here, the entire data base is separated into
groups in which all cases have the same dominant flow
pattern (>75% of the total pipe length), namely
stratified, intermittent, annular flow and dispersed
bubble flow. Then, a separate evaluation is conducted for
each flow pattern. The results can be found in Table 4-6
and Figure 7-9. It can be seen that all these models,
particularly the intermittent flow model, perform better
than any of the correlations. No evaluation can be done
for the dispersed bubble flow model because there is no
dispersed bubble flow dominated cases.
used correlations of Beggs and Brill, Mukherjee and
Brill, Dukler and Dukler with the Eaton holdup
correlation ( Brill & Beggs 1986) have also been included
in the evaluation for the purpose of comparison.
The overall evaluation of the comprehensive
mechanistic model using the entire data base is shown in
Table 3. The calculated and measured pressure drops are
also plotted to give an overall picture of the performance
of the model (Fig. 6). The model has negative values for
1 and 4, indicating its underprediction for pressure
drops. All the other statistical parameters of the model
are the smallest, which demonstrates its superior
performance over all the correlations. Of all 426 cases,
there is only one case where the model has a convergent
problem, whereas all correlations have more than five
troublesome cases. In this respect, the comprehensive
mechanistic model is also the best.
The degree of uncertainty in the calculation of
the liquid-wall friction factor for stratified flow has
also been studied. A sensitivity study is undertaken by
varying the value of fL .25% around its calculated value.
The results are reported in Table 7. As shown, except for
some changes in 1 and 4, the other parameters remain
almost the same. This suggests that the performance of
the stratified flow model is generally not sensitive to fL.
CONCLUSIONS AND RECOMMENDATIONS
Based on the results of this study, the following
conclusions have been reached:
1. A comprehensive mechanistic model, which is
capable of predicting two-phase flow pattern,
liquid holdup and pressure drop, has been
formulated.
2. The consistency and applicability of the
comprehensive mechanistic model have been
demonstrated by its overall superior
performance over any of the compared
A
dp/dx
D
E
f
FE
g
h
L
N
N
we
Nfl
p
Re
s
v
pipe cross sectional area or area
occupied by fluid
constant coefficient
differentiation of AL with
respect to hL
pressure gradient
pipe diameter or hydraulic
diameter
liquid holdup
fanning friction factor
liquid entrainment fraction
acceleration of gravity
liquid level
length
number of points
Weber number
liquid viscosity number
pressure
Reynolds number
wetted periphery or sheltering
coefficient
velocity
175
10 A COMPREHENSIVE MECHANISTIC MODEL FOR TWO-PHASE FLOW IN PIPELINES SPE 20631
Greek Letters
a
13
/)
Ap
E
9
JL
P
0"
pipe inclination angle, positive
for upward
regression coefficient
film thickness
pressure drop
roughness or error parameter
angle subtended by interface
viscosity
density
surface tension
summation
shear stress
7.
8.
9.
Barnea, D., Shoham, O. and Taitel, Y.: "Flow Pattern
Transition for Vertical Downward Inclined Two-
Phase Flow; Horizontal to Vertical," Chern. Eng.
Sci. 37, No.5, 735-740 (1982a).
Barnea, D., Shoham, O. and Taitel, Y.: "Flow Pattern
Transition for Vertical Downward Two-Phase
Flow," Chern. Eng. Sci. 37, No.5, 741-744 (1982b).
Barnea, D., Shoham, O. and Taitel, Y.: "Gas-Liquid
Flow Inclined Tubes: Flow Pattern Transitions for
Upward Flow," Chern. Eng. Sci. 40, No.1, 131-136
(1985).
Subscripts
b
10. Barnea, D.: "A Unified Model for Predicting Flow-
Pattern Transitions for the Whole Range of Pipe
Inclinations," Int. J. Multiphase Flow 13, No. I, 1-
bubble 12 (1987).
c core or calculated
f film
g gas phase
interface
L liquid phase
m measured or mixture
s superficial or slug
transition or translational
u slug unit
w wall
REFERENCES
11. Bendiksen, K. H.: "An Experimental Investigation
of the Motion of Long Bubbles in Inclined Tubes,"
Int. J. Multiphase Flow 10, No.4, 467-483 (1984).
12. Brill, J. P. and Beggs, H. D.: "Two-Phase Flow in
futl," Fifth Edition (December, 1986).
13. Cheremisinoff, N. P.: "An Experimental and
Theoretical Investigation of Horizontal Stratified
and Annular Two-Phase Flow with Heat Transfer,"
Ph D. Dissertation, Clarkson College of Technology
(1977).
1.
2.
3.
Alves, I. N., Caetano, E. F., Minami, K. and Shoham,
0.: "Modelling Annular Flow Behavior for Gas
Wells," Presented at the Winter Annual Meeting of
ASME, Chicago (Nov. 27-Dec. 2, 1988).
Andreussi, P. and Persen, L. N.: "Stratified Gas-
Liquid Flow in Downwardly Inclined Pipes," Int. J.
Multiphase Flow 13, No.4, 565-575 (1987).
Andritsos, N.: "Effect of Pipe Diameter and Liquid
viscosity on Horizontal Stratified Flow," Ph. D.
Dissertation, U. of Illinois at Champaign-Urbana
(1986).
14. Crowley, C. J. and Rothe, P. H.: "State of the Art
Report on Multiphase Methods for Gas and Oil
Pipelines," Vol. 3: Theoretical Supplement,
Prepared for Project PR-172-609 of Pipeline
Research Committee, A. G. A. (December, 1986).
15. Crowley, C. J. : "Contents of A. G. A. Data Bank
(August 1988 Release)," Creare (1988).
16. Crowley, C. J. and Rothe, P. H.: "Assessment of
Mechanistic Two-Phase Analysis Method for Gas-
Condensate Pipelines," PSIG Annual Meeting,
Toronto, Ontario, Canada (Oct. 20-21, 1988).
Baker, A., Nielsen, K. and Gabb, A.: "Pressure loss,
Liquid Holdup Calculations developed," Oil & Gas
J., 55-59 (March 14, 1988).
Andritsos, N. and Hanratty, T. J.: "Influence of
Interfacial Waves in Stratified Gas-Liquid Flows,"
AIChE J. 33, No.3, 444-454 (1987).
4.
5.
6.
Ansari, A. M.: "A Comprehensive
Model for Upward Two-Phase Flow,"
The University of Tulsa (1988).
Mechanistic
M. S. Thesis,
176
17. Eaton, B. A. and Brown, K. E.: "The Prediction of
Flow Patterns. Liquid Holdup and Pressure Losses
Occurrini Durini Continuous Two-Phase Flow jn
Horizontal Pipelines," Technical Report, The U. of
Texas (October 1965).
18. Gregory, G. A., Nicolson, M. K. and Aziz, K.:
"Correlation of the Liquid Volume Fraction in the
Slug for Horizontal Gas-Liquid Slug Flow," Int. J.
Multiphase Flow 4, 33-39 (1978).
SPE 20631 J. J. XIAO. O. SHOHAM AND J. P. BRILL 11
19. Henstock, W. H. and Hanratty, T. J.: "The
interfacial Drag and the Height of the Wall layer
in Annular Flow," AIChE J. 22, No.6, 990-999
(1976).
20. Hewitt, G. F. and Hall-Taylor, N. S.: "Annular Two-
Phase Flow," Pergamon Press (1970).
21. James, P. W., Wilkes, N. S., Conkie, W. and Burns,
A.: "Developments in the Modelling of Horizontal
Annular two-Phase Flow," Int. J. Multiphase Flow
13, No.2, 173-198 (1987).
22. Kowalski, J. E.: "Wall and Interfacial Shear Stress
in Stratified Flow in a Horizontal Pipe," AIChE J.
33, No.2, 274-281 (1987).
23. Laurinat, J. E., Hanratty, T. J. and Jepson, W. P.:
"Film thickness Distribution for Gas-Liquid
Annular Flow in a Horizontal Pipe," Int. J.
Multiphase Flow 6, No. 1/2, 179-195 (1985).
24. Laurinat, J. E., Hanratty, T. J. and Dallman, J. C.:
"Pressure Drop and Film Height Measurements for
Annular Gas-Liquid Flow," Int. J. Multiphase Flow
10, No.3, 341-356 (1984).
25. Lin, P. Y. and Hanratty, T. J.: "Prediction of the
Initiation of Slugs with Linear Stability Theory,"
Int. J. Multiphase Flow 12, No. I, 79-98 (1986).
26. Lin, P. Y. and Hanratty, T. J.: "Effect of Pipe
Diameter on Flow Patterns for Air-Water Flow in
Horizontal Pipes," Int. J. Multiphase Flow 13, No.
4, 549-563 (1987).
27. Mcleod, W. R., Rhodes, D. F. and Day, J. J.:
"Radiotracers in Gas-Liquid Transportation
Problems - A Field Case," J. Pet. Tech., 939-947
(August, 1971).
28. Oliemans, R. V. A., Pots, B. F. and Trope, N.:
"Modelling of Annular Dispersed Two-Phase Flow
in Vertical Pipes," Int. J. Multiphase Flow 12, No.
5, 711-732 (1986).
29. Payne, G. A., Palmer, C. M., Brill, J. P. and Beggs,
H. D.: "Evaluation of Inclined-Pipe, Two-Phase
Liquid Holdup and Pressure-Loss Correlations
Using Experimental Data," J. Pet. Tech., 1198-
1208 (September, 1979).
30. Scott, S. L.: "Modelling Slug Growth in Pipelines,"
Ph. D. Dissertation, The University of Tulsa
(1987).
31. Shoham, O. and Taitel, Y.: "Stratified Turbulent-
Turbulent Gas-Liquid Flow in Horizontal and
177
Inclined Pipes," AIChE J. 30, No.3, 377-385
(1984).
32. Taitel, Y. and Dukler, A. E.: "A Model for
Predicting Flow Regime Transitions in Horizontal
and Near Horizontal Gas-Liquid Flow," AIChE J.
22, No. I, 47-55 (1976).
33. Taitel, Y., Bornea, D. and Dukler, A. E.: "Modelling
Flow Pattern Transition for Steady Upward Gas-
Liquid Flow in Vertical Tubes," AIChE J. 26, No.3,
345-354 (1980).
34. Taitel, Y. and Barnea, D.: "A Consistent Approach
for Calculating Pressure Drops in Inclined Slug
Flow," Chern. Eng. Sci. 45, No.5, 1199-1206
(1990).
35. Wallis, G. B.: "One-Dimensional Two-Phase Flow,"
McGraw-Hill (1969).
36. Whalley, P. B. and Hewitt, G. F.: "The Correlation
of Liquid Entrainment Fraction and Entrainment
Rate in Annular Two-Phase Flow," UKAEA Report,
AERE-R9187, Harwell (1978).
37. Wu, H. L., Pots, B. F. M., Hollenberg, J. F. and
Meerhoff, R.: "Flow Pattern Transitions in Two-
Phase Gas/Condensate Flow at High Pressure in An
8-in Horizon tal Pipe," 3rd International
Conference on Multiphase Flow, The Hague,
Netherlands, 13-21 (May 18-20, 1987).
Table 1: Pipeline Data Base Table 2 Table 3
leE 20631
Nominal
Pipe No. of
Data Sources Diameter Data Fluid System
(mm) Points
A.G.A. 76.2 - 79 54-Black Oil System
660.4 25-Compositional
Field Data System
Mcleod ef al. 152.4 12 Black Oil System
Eaton et al. 50.8 139 Natural Gas / Water,
101.6 97 Crude or Distillate
Lab. Data
Payne ef a/. 50.8 99 Natural Gas / Water
Total
No. of Data 426
Points
Definitions of Statistical Parameters
Statistical Definitions Unit
Parameters
!. [ lJ X 100J]
% ]
N l
,
!. ( L I - X 100 I J
%
N
2
( J
L ---x100 -,
%
,
N-1
,
!.I L -
x10' Pa N
,
!. (L I - I )
x10' Pa N
,
VL [( )_4]2
x10' Pa
N-I
Overall Evaluation of
the Comprehensive Mechanistic Model
Using Entire Data Base
Model No. Statistical Parameters
No. or of Data
, I , I , I , I , I ,
Correlation Points x10
4
X10
4
X10
4
(%) (%) (%) (Pa) (Pa) (Pa)
1 This Model 425 -11.7 30.5 50.6 -8.6 12.2 22.0
2 Beggs & Brill 415 10.9 35.0 94.2 9.6 13.2 31.6
3 Muk. & Brill 411 39.4 60.5 128. 16.3 21.3 40.5
4 Dukler 415 32.9 43.0 107. 17.3 18.7 36.7
5 Dukler-Eaton 419 21.5 35.4 89.3 13.8 16.0 33.9
Notes:
....
'"


Calculated Pressure Drop (x 10' Pal
Measured Pressure Drop (xl0
4
Pa)
Table 4 Table 5 Table 6
Evaluation of Stratified Flow Model Using Evaluation of Intermittent Flow Model Using Evaluation of Annular Flow Model Using
Cases with 75% Stratified Flow Cases with 75% Intermittent Flow Cases with 75% Annular Flow
3 I Muk. & Brill
1 I This Model
5 I Dukler-Eaton
2 I Beggs & Brill
Model No. Statistical Parameters
No. or of Data
, I , I , I , I , I ,
Correlation Points XI0
4
X!04 XI0
4
(%) (%) (%) (Pa) (Pa) (Pa)
1 This Model 123 -3.6 39.2 76.5 -17.9 24.0 33.7
2 Beggs & Brill 114 33.3 41.7 163. 21.9 24.7 50.2
3 Muk. & Brill 117 83.0 94.1 215. 35.4 43.3 63.3
4 Dukler 114 49.8 56.5 187 27.8 30.4 53.7
5 Dukler-Eaton 119 29.8 41.8 149 22.1 27.3 54.2
Model No. Statistical Parameters
No. or of Data
] I , I , I , I , I ,
Correlation Points X10
4
X10
4
XI0
4
(%) (%) (%) (Pa) (Pa) (Pa)
1 This Model 121 -18.9 22.7 19.8 -8.7 9.3 12.6
2 Beggs & Brill 129 23.3 31.6 33.0 7.9 9.6 13.9
3 Muk. & Brill 127 27.1 40.0 54.7 4.8 10.1 15.7
4 Dukler 128 28.5 34.4 35.5 10.5 11.4 16.5
5 Dukler-Eaton 129 22.5 29.9 32.4 9.6 10.7 18.2
-9.0 34.9 53.7 6.6 9.0 16.2
16.7 66.9 79.0 16.0 19.3 19.4
23.1 54.4 78.8 17.4 19.0 23.1
Statistical Parameters
32.1 59.2 80.8 21.1 22.5 25.1
I I I I I XI0
4
X\04 XI0
4
(%) (%) (%) (Pa) (Pa) (Pa)
-18.0 34.6 49.1 -3.2 8.1 14.3 89
83
86
86
86
No.
of Data
Points
Dukler
or
Correlation
Model
4
No.
SEE 2063 1
Table 7
=-'lIO "'") J
(SW)
STATIFIED
Sensitivity Analysis of
Liquid-Wall Friction Factor on the
Performance of Stratified Flow Model
Statistical Parameters

=:I- . :]
(EB)
---
JWffiRM,=m

7: o;b;(\
(SL) . 0
.. ' '=:',:
:-:' :0
(AM)]
ANNULAR


(AW)
rD.
0:--:.0
.
0

DISPERSED
.
0
.
(DB) BUBBLE
!-:_o
8.1 14.3
8.0 14.3
8.5 14.4
" I " I " I I I
(%) (%) (%) (Pa) (Pa) (Pa)
-18.0 34.6 49.1 -3.2
-22.1 35.9 48.4 -4.7
-14.4 34.4 49.8 -1.9 89
89
89
No.
of Data
Points
0.75 f L
Friction
Factor
fL
2
No.
Figure 1 - Flow Patterns in Horizontal and Near Horizontal Pipes
(AN)
A
hLlD=O.S;
.....
(1) / .... / ....
__-// /B
B
_c
(SW)
(DB)
c------C-
r----A-----:-...,.--_
" ........ --A
"" ..... ,\.
s=O.06 0.01
\, "
"
, ,
, ,
, ,
'. \
\ \
\ \
, \
, ,
\ ,
\ \
\ \
\
(SS) D \,
.1
.01
.1 10 100
SUPERFICIAL GAS VELOCITY (m/s)
Figure 2 Flow Pattern Map (Air-Water in 5-cm Pipe of -1 Degree Inclination) Figure 3 - Physical Model for Stratified Flow
.---<."'\'1...-- L, ==\
Slug Body
c>.- vb
_VL

Figure 4 - Physical Model For Intermittent Flow Figure 5 - Physical Model for Annular Flow
179
SPE 20631
3000.0
'"
...
2500.0 ,;:.
1'0
~
Cl
2000.0
~
Il::
;J
'"
'"
1500.0
~
<>:
...
Cl
~
1000.0
E;;:
..l ~ 'a
;J
6
U
500.0
0
..l
<
U
0.0
U.O 500.0 1000.0 1500.0 2000.0 2500.0 3000.0
4000.0 3500.0 3000.0 2500.0 2000.0
4000.0
'"
3500.0 ...
,;:.
...
0 3000.0
<>:
Cl
~
2500.0
<>:
;J
'"
'"
'"
2000.0 0
~
<>:
'" ... 0
0
Cl
0 0
0 0
1500.0 0 0
"-l
0
0
0
0
...
<
0",0 00
..l
0
;J ~ o o o
U
..l
<
U
MEASURED PRESSURE DROP (kPa)
MEASURED PRESSURE DROP (kPa)
Figure 6 - Performance of the Comprehensive Model Using Entire Data Bank Figure 7 - Perfoonanee of Stra1ified Flow Model Using Cases with 75% Stratified Flow
4000.0 3500.0 3000.0
o
o
2000.0 2500.0
o 0
o 00
o
o
1500.0
o
o q)0 0
o
CD
o
00
1000.0 500.0
4000.0
'"
3500.0 ...
,;:.
...
0 3000.0
<>:
Cl
"-l
2500.0
<>:
;J
'"
'"
2000.0
~
<>:
...
Cl 1500.0
~
...
<
..l
1000.0
;J
U
..l
<
500.0 0
U
Q
0.0
U.O
3000.0 2500.0 2000.0 1500.0 1000.0
3000.0
'"
...
,;:. 2500.0
...
0
<>:
Cl
2000.0
~ 0
<>:
0
;J
'"
'"
1500.0
"-l
<>:
...
Cl
o "0
~
1000.0
... ",0
-< 0
..l 000
;J
0 IP
U 0
..l
~ o
<
U
MEASURED PRESSURE DROP (kPa) MEASURED PRESSURE DROP (kPa)
Figure 8 - Performance of Intermittent Flow Model Using Cases with 75% Intermittent Flow Figure 9 - Performance of Annular Flow Model Using Cases with 75% Annular Flow
180