A Mechanistic Model for Two-Phase

Annular-Mist Flow in Vertical Pipes

The physical phenomena associated with two-phase annular-mist S. C. Yao and N. D. Sylvester
flow in vertical pipes are considered. A new mechanistic model is devel- College of Engineering and Applied
oped to predict liquid entrainment, liquid film thickness, in situ veloci- Sciences
ties, and pressure drop. The model is compared to field data and found University of Tulsa
to be superior to the three commonly used empirical correlations. Tulsa, OK 74104

Introduction (1 980) have developed a relation which predicts that annular-

Gas wells that produce liquids can experience liquid loading, mist flow exists if
which increases the pressure drop and thus restricts reservoir
drawdown and reduces production. Liquid loading occurs when
the flowing gas velocity is insufficient to surface the liquids that
are produced with the gas or condensate in the tubing as the
pressure and temperature decrease. As liquids accumulate in This flow regime transition for annular-mist flow is based on the
the lower portion of the tubing, the back pressure on the forma- balance between gravity and drag forces acting on spherical
tion increases, decreasing the production rate. The back pres- droplets. Although this equation has been criticized by Chen
sure on the formation may reach a critical pressure at which the and Spedding (1983), Sylvester (1 984) has shown it to be the
well dies. All gas wells that produce liquids will experience liq- most realistic of the available flow pattern map models, espe-
uid loading with reservoir depletion. Liquid loading problems cially for high-pressure gas-condensate systems.
may be eliminated by reducing the tubing diameter and/or the During annular-mist flow a fraction of the flowing liquid will
surface pressure. Both these techniques increase the gas velocity be entrained or dispersed in the gas stream while the rest is con-
in the well. The prediction and analysis of liquid loading require fined to a thin liquid film on the pipe wall. Wallis (1969) pro-
multiphase flow equations for determining the flow regime and posed an expression for the fraction of the liquid entrained, FE,
tubing pressure as a function of gas rate, gas liquid ratio, well where FE,as shown below, is dependent upon the superficial gas
diameter, well depth, surface pressure, and fluid properties. velocity, gas viscosity, gas density, liquid density, and interfacial
The purpose of this paper is to formulate and develop a new tension.
mechanistic model for two-phase annular-mist flow in vertical
pipes that permits calculation of liquid entrainment, average liq-
uid film thickness, in situ velocities, and pressure drop,
where
Mathematical Model ,
The mechanistic model for vertical annular-mist flow is for-
mulated based on the assumption that the flow is fully developed (3)
and stable. It is further assumed that at any given location in the
tubing, the fraction of entrained liquid is uniformly dispersed as
This expression predicts that the liquid fraction entrained
mist in the gas core. The rest of the liquid resides in a film of
increases as the superficial gas velocity, gas viscosity, or gas den-
uniform thickness, 6, around the pipe wall. The idealized flow
sity increases or as the liquid density or interfacial tension
configuration is shown in Figure 1.
decreases. Knowing the fraction of the liquid entrained permits
When gas and liquid flow concurrently upward a t high gas
the actual mixture density of the gas core to be calculated from
rate and low liquid rate, annular-mist flow occurs. Taitel et al.
the relation

Correspondence concerning this paper should be addressed to N. D. Sylvester. (4)

1008 June 1987 Vol. 33, No. 6 AIChE Journal

 The friction factor is determined from the modified Zigrang-
-
.F Sylvester ( 1 982) equation

-LIQUID FILM OF
THICKNESS a

where the mixture Reynolds number is given by

the mixture viscosity by

Pm - PL + (1 -x)Wg (12)

and e/ D is the relative effective roughness for annular-mist flow.

This roughness is taken to be the ratio of the time-averaged
thickness of the annular liquid film to the pipe diameter. Hen-
stock and Hanratty (1976) determined this ratio to be

6 6.59F 6
D = (1 + 1,400F)''' = D
where

Figure 1. Idealized flow configuration of vertical annular- {[0.707(Re,)0.5]2.5+ [0.0379(Reg)0.9]2.5)oA

mist flow model. F- (14)
(Reg)0~9(~g/rL)(PL/P,)~.~

where the in situ liquid holdup, A, is given by and

The existence of the annular liquid film on the pipe wall

reduces the actual pipe cross-sectional area, which causes the
gas velocity to be greater than the superficial gas velocity. In
addition, since the liquid film surface is wavy it creates a rough The total pressure gradient consists of an elevation term
surface over which the gas core must flow. The roughness (dp/dh)€,a friction term (dp/dh)/,and an acceleration term
increases the loss due to frictional pressure. (dpldh),, and can be written in the form
The pressure gradient due to friction may be written

The elevation term is given by

wheref, is the friction factor and V,,,is the mixture velocity given
by

The friction term was given by Eq. 6, and the acceleration term
where MLG,the mixture mass flow rate in the gas core, is given is
by

Computer Program
and A,, the cross-sectional area of the gas core, is given by Since this new model requires an iterative trial-and-error cal-
culation, a computer program was developed to calculate sur-
r(D - 26)2 _- -r d 2 face pressure given bottomhole conditions, and vice versa. The
A, = (9)
4 4 computer program requires the input data listed in Table 1 . In

AIChE Journal June 1987 Vol. 33, NO.6 1009

 Table 1. Required Input Data Table 3. Summary of Calculational Procedure

Well depth, H Bottomhole temp., TB Calculate the following at bottomhole temperature and pressure
Tubing dia., D Surface temp., TsuR 1. From fluid property correlations calculate solution gas-oil ratio,
Tubing roughness:, c Flowing bottomhole press., Pwf oil formation volume factor, water formation volume factor, oil-
Liquid rate, QL Flowing surface press., PsuR gas interfacial tension, gas compressibilityfactor
Gas rate, Q. 2. Calculate gas and liquid densities and interfacial tension
3. Calculate gas and liquid volumetric flow rates
*Tubing roughness is needed only if all liquid is entrained and/or a single-phase 4. Calculate total gas and liquid mass flow rates
flow pressure calculation is to be made. 5 . Calculate superficialgas velocity
6. Compare superficialgas velocity with Eq. 1 to determine flow re-
gime
addition, the fluid property information shown in Table 2 must 7. If annular-mist flow exists, select number of length increments
be known as a function of temperature and pressure. The calcu- and calculate length increment, A H
lational procedure is summarized in Table 3. The computer pro- 8. Estimate pressure drop A P corresponding to calculated A H
gram consists of a main program and fourteen subprograms. 9. Calculate average temperature
10. Calculate average pressure
Comparison with Field Data 11. Calculate the following at average temperature and pressure
a . From fluid property correlations recalculate fluid property
The effectiveness of the new model was evaluated by compar- values in steps 1 and 2
ing its predictions with field data. It is important to note that the b . Gas and liquid volumetric flow rates
new model or any model may be limited by the accuracy of the c . Superficial gas and liquid velocities
PVT correlations used. d . Fraction of liquid entrained
e . Mass flow rate of liquid entrained
Water-gas data were taken from Camacho (1970) and Rein- f . Mixture mass flow in gas core
icke and Remer (1984). Five cases from Reinicke and Remer g . Volumetric flow rate of entrained liquid
were not included in the analysis because the inclination from h . Liquid holdup
the vertical was significantly more than 10". Although the new i . Effective average liquid film thickness
model was developed for vertical flow, 70 field data points with j . Actual area and gas velocity
k . Mixture viscosity
inclinations less than 10" were included in the analysis. The new 1 . Mixture density
model was also tested against 93 gas-oil data points (Govier and m. Mixture velocity
Fogarasi, 1975). n . Mixture Reynolds number
A summary of the statistical results is shown in Table 4. All 0 . Liquid and gas Reynolds numbers
p . Pseudorelativeroughness
cases tested converged except for one oil-gas data point. There q . Two-phasefriction factor
were eight data points for water-gas flow and four for oil-gas r . Friction gradient
flow in which the pressure drop was significantly overpredicted. s . Elevation gradient
All these had very high gas velocities (>8 m/s). A possible t . Total pressure gradient
explanation for this may be that the Henstock and Hanratty 12. Compute A P from the relation A P = (dp/dz)(Az)
(1976) correlation overpredicts the effective relative roughness 13. If calculated and estimated A P values are not within selected tol-
at high gas velocity. If this occurs, there is a spiral effect that erance, take calculated A P as a new estimate and repeat steps
10-12 until convergence is obtained
overpredicts pressure drop and gas velocity. The Henstock and 14. Compute the next pressure and temperature increment:
P,+,= P i - A P
Table 2. Required Fluid Property Data

Liquid density, p L
Gas density, p g 15. Repeat steps 8-14 until total number of increments is reached
Liquid viscosity, pL
Gas viscosity, p8
Liquid-gasinterfacial tension. u

Table 4. Comparison of Empirical Correlations and New Model

~-
No. of Cases
Correlations No. of Avg. Abs. Avg. Std. Corr. that Failed
or Model Cases % Diff. 76 Diff. Dev. Coeff. to Converee
Water-Gas Flow Data
Present work 119 - 2.48 13.27 21.79 0.994 0
Hagedorn & Brown (1965) 118 9.21 13.86 16.66 0.9927 1
Dun & Ross (1963) 119 25.05 27.60 24.50 0.9856 0
Beggs & Brill (1 973) 110 22.94 25.29 18.00 0.9901 9
Oil-Gas Flow Date
Present work 92 2.12 8.37 13.47 0.973 1
Hagedorn & Brown (1 965) 93 8.45 10.20 9.93 0.9949 0
Dun & Ros (1 963) 93 -3.30 14.83 19.24 0.9837 0
Beggs and Brill (1973) 76 20.04 20.16 13.78 0.9680 17

1010 June 1987 Vof. 33, No. 6 AIChE Journal

Hanratty correlation is based on water-air flow data obtained
over a limited experimental range (Re, 5 260,000). The field o GOVIER AND FOGARASI (1975)
data had calculated gas Reynolds numbers (Re,) significantly 2500
higher than 260,000. In fact, almost all Re, calculated were over
1,000,000. This extrapolation of the Henstock-Hanratty corre-
lation could have an effect on the accuracy of the effective rela-
tive roughness prediction at high gas velocity.
n
There were fifteen points for water-gas flow and two points 9 1500
for the oil-gas flow data that according to the Taitel et al. (1980)
flow regime criteria were not in annular-mist flow a t the bot-
3
tomhole conditions. In addition, there were twelve points for
water-gas flow that did not attain annular-mist flow over the
entire tubing length. For these cases and some of the cases that
achieved annular-mist flow part way up the tubing, most pres-
sure drops were underpredicted. These underpredictions are not
surprising because these cases were in the slug or churn flow ‘
0 500 1000 I500 2000 2500
regimes, which have pressure gradients significantly higher than CALCULATED WELLHEAD PRESSURE

annular-mist flow. Figure 3. Comparison of model prediction with experi-

As shown in Table 4, the new model has a low average percent mental data.
difference, and a low average absolute percentage difference.
Furthermore, the calculated correlation coefficients for both superior to the three commonly used empirical correlations. The
water-gas and oil-gas flow were greater than 0.97, which cer- new model could be improved if better correlations were avail-
tainly suggests that there is a very strong linear relationship able for effective film thickness and liquid entrainment.
between the calculated and field wellhead pressures. Figures 2
and 3 show that the “best” least-squares straight, broken lines Notation
were quite close to the ideal straight lines. A, core cross-sectional area
=
d core diameter
=

Comparison with Available Empirical Correlations D = tubing diameter

fi = two-phase friction factor
Three commonly used empirical correlations were utilized to F = Henstock-Hanratty dimensionless group, Eq. 14
test the predictive accuracy of the new model. These two-phase FE = fraction of liquid entrained
empirical correlations are those of Hagedorn and Brown (1 965), g = acceleration due to gravity
Duns and Ros (1963), and Beggs and Brill (1973). h = depth increment
H = well depth
Table 4 summarizes the statistical results of the comparison MG = gas mass flow rate
of the new model with the three empirical correlations. The new MEN= mass flow rate of entrained liquid
model outperforms each of these correlations. The calculated MLF= mass flow rate of annular liquid film
average percent differences and absolute average percent differ- MLG= mixture mass flow rate in core
M , = total liquid mass flow rate
ences for the new model are consistently lower than the other p = pressure
correlations. PSuR= wellhead pressure
The predictive accuracy of the new model for pressures in Pw, = flowing bottomhole pressure
two-phase annular-mist flow i n vertical pipes is good and is q, = in-situ gas volumetric flow rate
q, = in-siru liquid volumetric flow rate
Q, = volumetric flow rate of liquid
QB= volumetric flow rate of gas
Re, = gas Reynolds number
Re, = liquid Reynolds number
1Ooo-
Re,,, = mixture Reynolds number
TB= bottomhole temperature
6000- TsuR= surface temperature
3
v)
u)
V,,, = in-situ mixture velocity
# 5000- V , = in-situ superficial velocity gas velocity
n V,,. = superficial gas velocity required for annular-mist flon
n
2 4000- 2 = well depth
S
-1
Greek letters
y moo-
n p = dimensionless quantity in Wallis correlation, Eq. 3
A
2000-
6 average effective film thickness
=
c absolute roughness
=
p, = density of gas
1000 - p, = density of liquid
p,,, = density of mixture
I I I I I f I u = liquid-gas interfacial tension
2000 4000 6000 8000
CALCULATEDWELLHEADPRESSURE p, = gas viscosity
p, = liquid viscosity
Figure 2. Comparison of model prediction with experi- p,,, -
mixture viscosity
mental data. X = in-situ liquid core holdup

AIChE Journal June 1987 Vol. 33, No. 6 1011

Literature cited Sylvester, N. D., “Transition to Annular Flow in Vertical Upward Gas-
Liquid Flow,” AIChE J.. 30,700 (1984).
Beggs, H. D., and J. P. Brill, “A Study of Two-Phase Flow in Inclined Taitel, Y.,D. Bornea, and A. E. Dukler, “Modeling Flow Pattern Tran-
Pipes,” J. Per. Technol., 25,607 (1973). sitions for Steady Upward Gas-Liquid Flow in Vertical Tubes,”
Camacho, C. A., “Comparison of Correlations for Predicting Pressure AIChE J.. 26,345 (1980).
Losses in High Gas-Liquid Ratio Vertical Wells,” M.S. Thesis, Univ. Wallis, G. B., One-Dimensional Two-Phase Flow, McGraw-Hill, New
Tulsa ( I 970). York, 393 (1969).
Chen. J. J., and P. L. Spedding, “Transition to Annular Flow in Vertical Zigrang, D., and N. D. Sylvester, “Explicit Approximation to the Solu-
Upward Gas-Liquid Flow,” AIChE J., 29,525 (1983). tion of Colebrook’s Friction Factor Equation,” AZChE J., 28, 514
Duns, H., and N. C. J. Ros, “Vertical Flow of Gas-Liquid Mixtures in (1982).
Wells,” Proc. 6th World Per. Cong.,451 (1963).
Govier, G. W., and M. Fogarasi, “Pressure Drop in Wells Producing
Gas and Condensate.” Canad. J . Pet. Tech., 28 (1975).
Manuscript received Nov. 13, 1986, and revision received Jan. 29, 1987.
Hagedorn. A. R., and K. E. Brown, “Experimental Study of Pressure
Gradients Occurring During Continuous Two-Phase Flow in Small-
Diameter Vertical Conduits,” J. Per. Technol., 17 475 (Apr., 1965).
Henstock, W. H., and T. J. Hanratty, “The Interfacial Drag and the See NAPS document no. 04498 for 6 pages of supplementary materi-
Height of the Wall Layer in Annular Flows,” AIChE J., 22, 990 al. Order from NAPS c/o Microfiche Publications, P.O. Box 3513,
( 1976). Grand Central Station, New York, NY 10163. Remit in advance in
Reinicke. K. M., and R. J. Remer, “Comparison of Measured and Pre- U S . funds only $7.75 for photocopies or $4.00 for microfiche. Outside
dicted Pressure Drops in Tubing for High-Water-Cut Gas Wells,” the U.S. and Canada, add postage of $4.50 for the first 20 pages and
SPE Paper No. 13279, Soc. Pet. Eng. 59th Ann. Meet., Houston, $1.00 for each of 10 pages of material thereafter, $1.50 for microfiche
(Sept., 1984). postage.

1012 June 1987 Vol. 33, No. 6 AIChE Journal