Margaris D Etal_2001_validation Of Existing Models For The Prediction Of The Two-phase Flow Behavior In Oil-natural Gas Pipelines

ASME - GREEK SECTION, First Nat. Conf. on Recent Advances in Mech. Eng., September 17-20, 2001, Patras, Greece Proceedings of First Nat. Conf. on Recent Advances in Mech. Eng. September 17-20, 2001, Patras, Greece

ANG1/P103 VALIDATION OF EXISTING MODELS FOR THE PREDICTION OF THE TWO-PHASE FLOW BEHAVIOR IN OIL-NATURAL GAS PIPELINES

Dionissios P. Margaris Assist.Prof., Dr.-Ing. University of Patras Mechanical Engineering and Aeronautics Department Fluid Mechanics Laboratory 26500 Patras, Hellas. Tel. & Fax: 061 997202

Dimitrios G. Papanikas Prof., Dr.-Ing. University of Patras Mechanical Engineering and Aeronautics Department Fluid Mechanics Laboratory 26500 Patras, Hellas. Tel. & Fax: 061 997201

Andronicos E. Filios Dr.-Ing. University of Patras Mechanical Engineering and Aeronautics Department Fluid Mechanics Laboratory 26500 Patras, Hellas. Tel. & Fax: 061 997202

email: [email protected]

email: [email protected]

email: [email protected]

fraction of the pipe volume occupied by the liquid phase, and finally of the pressure drop. The complexity of two-phase flow, and more specifically that of gas-liquid flow, has rendered disadvantageous the use of Computational Fluid Dynamics methods, whilst it has led to the use of simplified prediction models, which, however, need to be evaluated before their application. After a laborious search, an assessment of a number of such models suitable for use in hydrocarbons transport pipelines is made in the framework of a CEC supported research [11]. Some of the results gained are presented here. Of the different types of two-phase flow, gas-liquid flows are the most complex, since they combine the characteristics of a deformable interface and the compressibility of the gas phase. For given conditions of two-phase flow in a pipe, the gas-liquid interfacial distribution can take any of a great number of possible forms. Several factors tend to limit the range of possibilities, these being the flow conditions and the physical properties of the phases. A wide variety of flow regimes have been defined in the literature; this results partly from the subjective nature of flow regime definitions and partly from the variety of names being to the same flow regime. The most commonly occurring flow patterns in a horizontal or a vertical pipe are shown in Fig. 1 and 2 respectively. Annular and stratified air-water and steamwater flows are the subject of this presentation.

ABSTRACT Pressure drop and liquid holdup along with the flow pattern recognition are the most important characteristics for the design and analysis of two-phase Oil/Natural Gas Systems and are predicted by means of a computational system based on formulations of two-phase flow models, usually employed for industrial applications. The results gained during the validation phase of the generated code are compared with experimental data, making apparent that the prediction certainty of some established and widely applied formulations is restricted in narrow ranges of pressure, flow rate and quality parameters. KEYWORDS Two-phase flow, Oil-natural gas, Pressure drop, Liquid holdup, Annular flow, Stratified flow. INTRODUCTION Two-phase flow often appears in the production and transportation systems of hydrocarbon fluids (oil/natural gas) and affects considerably their operation. The prediction of the flow behavior of two-phase mixtures is particularly necessary in the design procedure of oil/natural gas pipe systems, especially in the calculation of liquid holdup, this being the

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ASME - GREEK SECTION, First Nat. Conf. on Recent Advances in Mech. Eng., September 17-20, 2001, Patras, Greece

In any case, while predicting two-phase pressure drop, the determination of the flow pattern is a very important step. Different flow patterns lead to very different prevailing conditions in the pipe and therefore to different pressure drops.

Bubble Flow

Slug or Plug Flow

Churn Flow

Annular Flow

Wispy Annular Flow

Figure 2. Flow patterns in vertical gas-liquid flow. PHYSICOMATHEMATICAL MODELING FOR TWO-PHASE FLOW In most applications of stratified flow, where the liquid is assumed to lie on the lower part of the pipe due to the gravitational force, an angle γ is also defined, as shown in Fig. 3, as well as the liquid height hL. Moreover, a pipe carrying a two-phase mixture is characterized by its length L, its diameter d and its angle a with respect to the horizontal plane. If we assume that the velocities of each phase are constant, in any given cross-section within the zone occupied by the phase, it is possible to write separated flow equations for the conservation of mass and momentum for each phase. The continuity equation of the liquid film of thickness δ in a pipe of sectional area A has the form: ρ L (1 − ε G )u L A + δz

∂ [ρ L (1 − ε G )u L A]+ m&e δz − ∂z

∂ ρ L (1 − ε G )u L A + [ρ L (1 − ε G ) Aδz ] = 0 ∂t

(1)

where ρ is the density, ε the void fraction, u the superficial &e the rate of conversion of liquid to gas per unit velocity, m length, z the axial distance, and the subscripts L and G denote the liquid and gas phase respectively. For a pipe perimeter P, &e is given by the equation: m

Figure 1. Flow patterns in horizontal gas-liquid flow. On the other hand, predictions cannot be easily ascertained through experimental data due to the flow complexity. The reported experimental measurements are usually limited to specific flow cases, and it is difficult to be used for the production of generalized conclusions for twophase flow phenomena. Despite the above-mentioned, experimental measurements are necessary for the validation of computated results. In the present study, experimental data concerning water-air or steam-water flows were used, after being compiled in the literature ([2], [4], [6], [10]).

P &e = q& m h LG

(2)

where hLG is the latent heat of vaporization, and q&is the heat flux from the surface. For the liquid and the gas phases respectively, Eq.(1) becomes:

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ASME - GREEK SECTION, First Nat. Conf. on Recent Advances in Mech. Eng., September 17-20, 2001, Patras, Greece

∂ [ρ L (1 −ε G )A]+ ∂ [ρ L u L (1 − ε G )A] = −m&e ∂z ∂t ∂ ∂ (ρ G ε G A ) + (ρ G ε G u G A ) = m&e ∂t ∂z

(3)

SG

(4)

AG

∂ (ρ TP A ) + ∂ (m&A ) = 0 ∂t ∂z

γ AL

(5)

(6)

UG

In a similar way we can write the momentum balance equation for the pipe–element of Fig. 3 for separated flow (12): τ P ∂ ∂p − g ρ TP sin a − o = [ρ L (1 − ε G )u L + ρ G ε G u G ] + A ∂z ∂t

[

1 ∂ 2 ρ L A(1 − ε G )u 2L + ρ G A ε G u G A ∂z

hL

τWG

& = ρ L u L (1 − E G ) + ρ G E G u G m

+

D

SL

and remembering that:

−

hG

Si

The general separated flow continuity equation represents the overall balance for the two-phase mixture of density ρTP, and can be derived by adding Eq.(3) and (4):

]

Gas τi

UL

τi Liquid

τWL Figure 3. Modeling of horizontal stratified flow. PRESSURE DROP PREDICTION

(7)

For steady flow, which is examined in the present work, the overall momentum balance can be rewritten in terms of &G / m &) : &, Eq.(5), and quality x (x = m mass flux m

where το is the wall shear stress. The energy equation of the separated flow can be derived in a similar way. The numerical solution of the resulting system of equations, for a more accurate analysis of two-phase flow, is the subject of another work phase [11].

−

∂p τ o P d  (1 − x )2 x2  &2 = +m +   + gρ TP sin α (8) ∂z dz  ρ L (1 − ε G ) ρ G ε G  A

The three terms on the right-hand side, represent the three components of pressure gradient, that is, the frictional, accelerational and gravitational pressure drops respectively. The analytical solution of Eq.(8), combined with Eq.(6), it is not the subject of the present paper. On the other hand, a great number of predictive models have been published internationally, in an attempt to provide solutions in connection with experimental data in order to produce a convenient computational tool for industrial applications. Some of the most widely used and reliable ones have been selected, coded and introduced into an integrated computational system. The results produced have been analyzed against a set of experimental data and are presented below. LIQUID HOLDUP ESTIMATION Liquid hold up, that is, the fraction on the pipe volume occupied by the liquid phase, is usually a prerequisite for the two-phase pressure drop calculation. Some of the most well-

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ASME - GREEK SECTION, First Nat. Conf. on Recent Advances in Mech. Eng., September 17-20, 2001, Patras, Greece

known and widely used models for the calculation of liquid holdup are evaluated below.

1.0

The Flanigan model Flanigan [9] takes into account only the influence of volume flow rate and pipe diameter on the liquid holdup. The relationship giving liquid holdup is the following:

0.8

HL

1

Liquid Holdup

3

(9)

1 + 0,3264 u 1G,006

)

1: Tests 2: Premoli 3: Flanigan 4: Muckherjee - Brill

0.4 1 2 3 4

0.0 0

50

100

150

200

250

300

3

Gas Flow Rate (m /h) Figure 4. Liquid holdup for air-water flow in vertical pipes. The Taitel-Dukler model This model enables the calculation of liquid holdup (or void fraction) when the flow pattern is stratified, using the following equation [9]: 1 2  ε G = cos −1 2 h L − 1 − 2 h L − 1 1 − 2 h L − 1  (11) π 

The Mukherjee-Brill model The model of Mukherjee and Brill [5] was based on experiments performed with a system consisting of gas phase (air) and liquid phase (kerosene and lube oil) in a pipe of diameter 3.8 cm and inclination with respect to the horizontal varied between α=0ο and 90ο. The temperature varied between t = –7.8 to 55.56 οC. The following general equation for liquid holdup calculation resulted from the experimental measurements:

(

0.6

0.2

where uG is the gas superficial velocity. The calculation of liquid holdup in this way is simpler, but of more limited accuracy than the other models. The Flanigan model is based on experiments performed with inclined pipes and upwards flow with pipe diameter d=16 in. The influence of pipe inclination on HL is considered negligible. Results obtained by the application of the Flanigan model are compared with experimental data in Fig. 4. As is apparent, the predictions of the model are of moderate accuracy, but are slightly better for very low or very high gas flow rates.

C5   N GV  H L = exp  C1 + C 2 sin a + C 3 sin 2 a + C 4 N 2L C6   N LV  

QL=0.97 m /h d =4.54 cm

(

) (

)

(

)

where εG is the void fraction, and h L the dimensionless liquid height (given by a complex implicit equation not presented here). Results from the application of the Taitel-Dukler model, compared with experimental measurements are presented in Fig. 5. As is apparent, this model has remarkable agreement with the experimental data for high pressures (75 bar) and high gas flow rates, while for smaller flow rates or for higher pressures (120 bar) the disagreement of its predictions with the experimental data becomes considerable, and therefore it should be used with special care.

(10)

where NL is the liquid viscosity number, NGV the gas velocity number, NLV the liquid velocity number, and C1-C6 resulted from the experimental correlations. The model was found to be less accurate when the value of liquid holdup is less than 10% or greater than 90% (in most cases these values of liquid holdup occur in the stratified and annular glow regime). Results obtained from the application of Mukherjee-Brill model, are compared with experimental data in Fig. 4, where its relatively poor agreement with the latter is also displayed. In addition to that, the Mukherjee-Brill model was the one with the less accurate predictions among the three (Premoli and Flanigan being the remaining two) applied to the same flow conditions.

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ASME - GREEK SECTION, First Nat. Conf. on Recent Advances in Mech. Eng., September 17-20, 2001, Patras, Greece 1.0

the whole it gave much better results than the other two models compared (Flanigan and Mukherjee-Brill).

p=75 bar d=18 cm

hL calculated

2

G=1015 kg/m s Quality=0.01-0.2

0.8

PRESSURE DROP ESTIMATION The Lockhart-Martinelli flow model This model uses frictional multipliers and does not examine the influence of surface tension on the pressure drop [1]. In addition to that, it is inadequate in representing a wide range of two-phase flow pressure gradient. Despite its deficiencies, the model has been widely used in the Oil and Natural Gas Industry. Results produced in medium pressure (20 bar), as well as in high pressure (60 bar), and comparison of them with the results produced by other similar models and with experimental data is given in Fig. 6 and 7.

0.6

0.4

0.2

: Dukler

0.2

0.4

0.6

0.8

1.0

hL experimental Figure 5. Comparison between calculated and experimental holdup values for water-steam stratified flow.

Calculated Pressure Gradient

12800

The Premoli model This empirical model[1] is based on a wide range of experimental data and calculates first the ratio S of the mean velocities of the gas and liquid phase according to the equation:   uG y = S = 1 + E1  − y E 2  uL  1 + yE2 

1/ 2

(12)

where y = β/(1-β) with β = QG/(QL+QG), and E1 and E2 are the dimensionless numbers ρ E1 = 1,578 Re − 0,19  L  ρG

  

2 1

p=20 bar d=25 mm

4800 1: Lockhart - Martinelli 2: Friedel 3: Beggs - Brill 800 800

4800

8800

12800

Experimental Pressure Gradient (13)

  

Figure 6. Comparison between calculated and experimental pressure drop for steam-water flow. The Friedel model Evaluation and comparison of about 25,000 data points with some of the existing correlations produced the Friedel [1] model:

−0,08

(14)

where Re is the Reynolds number and We the Weber number. Then, the void fraction can be calculated by: εG =

8800

0,22

ρ E 2 = 0,0273 We Re − 0,51  L  ρG

3

QG uG = SQ L + Q G S u L + u G

Φ 2LO = E +

(15)

3,24 F H Fr0,045 We 0,035

(16)

E, F and H are dimensionless numbers, calculated as a function of gas and liquid densities, gas and liquid friction factors, and gas and liquid viscosities, and Fr and We are the Froude and Weber numbers respectively. Results produced by the application of the Friedel model for specific flow conditions, and comparison of them with experimental data are given in Fig. 6 and 7.

where uG and uL are the gas and liquid superficial velocities and QG and QL the gas and liquid volume flow rates. This model proved to be the most accurate one as shown in Fig. 4 for this specific application case. Its accuracy becomes slightly poorer for the medium gas flow rates, but on

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ASME - GREEK SECTION, First Nat. Conf. on Recent Advances in Mech. Eng., September 17-20, 2001, Patras, Greece

Lockhart - Martinelli 2.10

2000

1.60

Pressure Gradient (Pa/m)

Calc./Exper. Pressure Gradient

the shear stress profile. Results produced by the application of Wallis model in special cases of annular flow have been compared with experimental data, as well as with the results produced by the Dukler-Flanigan model, Fig.8, 9.

Beggs - Brill Friedel

1.10 p=60 bar d=50 mm 0.60 0.57

0.67

0.77

d=12.5 mm annular flow

2

Gc=35.9 kg/m s

1500

Wallis

1000 Duckler - Flanigan 500

0.87

Tests

Quality 0

Figure 7. Pressure gradient deviation vs. quality, for steamwater flow (G=227,5 kg/m2s).

0

10

20

30

40

2

50

Liquid Mass Flux (kg/m s) Figure 8. Pressure gradient vs. liquid mass flux, for air-water vertical flow.

The Beggs-Brill model More advanced than the previously presented model, and probably as frequently used as that, this model consists of three parts: one for the prediction of the two-phase flow pattern prevailing in the pipe, according to the transport properties of the fluids and the flow rates of the two phases, one for the calculation of liquid holdup and one for the calculation of the pressure gradient. The correlation proposed by Beggs and Brill [3] can be used to account for the frictional or gravitational pressure drop. Their model is applicable to downward two-phase flow, such as might occur in offshore gathering lines. A comparison of results produced by this model with experimental data and some other similar models is given in Fig. 6 and 7. The Wallis model This model is semi empirical and predicts two-phase liquid holdup and pressure drop [8]. It is applicable to either horizontal or vertical co current upwards-annular two-phase flow. The flow is considered to be steady, one dimensional, and the various coefficients used in its computational procedure have been obtained by experimental observations in air-water flow. More specifically, the wall friction factor is computed for either laminar or turbulent flow. Subsequently, the void fraction and the frictional pressure gradient for horizontal pipes and computed by use of the semi empirical “annular geometry model”, developed by Wallis, while for vertical pipes the void fraction and the total pressure gradient are computed iteratively using the modified Martinelli correlation which takes into account the effect of curvature of

The Dukler-Flanigan model Dukler [9] developed a model for the prediction of twophase pressure drop, which in combination with the elevation pressure drop correlation generated by Flanigan, gives the total pressure drop occurring in a two-phase flow pipeline. The model is mostly used in the Natural Gas and Oil Industries, in contradiction to some other models developed by experimental results in air-water pipelines. It uses an empirical correlation for the calculation of the frictional component of pressure drop as a function of the single and two-phase friction factor, the two-phase mixture density, the mixture velocity, the pipeline segment length and the pipe internal diameter. The elevation component of the two phase pressure gradient is calculated using an experimental derived relation, as a function of the Flanigan liquid holdup fraction, the pipeline-segment vertical elevation rise and the sum of the elevation rises of all segments. The method gives better results if used by dividing the pipeline into small segments since it takes account of the variation of fluid properties. The total two-phase pressure drop is given as the sum of the frictional and elevation components.

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ASME - GREEK SECTION, First Nat. Conf. on Recent Advances in Mech. Eng., September 17-20, 2001, Patras, Greece 25000

Among the general pressure drop models used, as made apparent in Fig. 6, the Lockhart-Martinelli model, despite the fact that it is the oldest and perhaps the simplest one, was definitely not the one of poorest accuracy. For the specific cases, the Friedel model was generally the most accurate one. Comparing Fig. 6 with Fig. 8 we reach the conclusion that the higher the operating pressure, the poorer the accuracy of the above-mentioned models. This is expected in a way, taking into consideration the fact that all these models were derived from experimental data, and therefore, should not be expected to behave properly far from the ranges of the data used for their generation. As regards the more elevated and specific models compared, the Wallis model, as shown in Fig. 9, behaves in a better fashion for lower operating pressures and lower liquid flow rates. Compared with the Dukler-Flanigan model, the latter displays considerably better behavior with respect to the experimental data, while the Wallis model seems to behave very badly at high liquid flow rates, as shown in Fig. 8, and therefore it should not be used in such cases.

2

Pressure Gradient (Pa/m)

GL=5.3 kg/m s 2

GL=297.1 kg/m s

20000

Wallis

p=2.4 bar d=31.8 mm

15000

10000

Wallis

Tests 5000

0

Tests

0

50

100

150

200

250

300

2

Gas Mass Flux (kg/m s) Figure 9. Pressure gradient vs. gas mass flux, in vertical annular air-water flow.

OTHER MODELS PRESSURE DROP

FOR

LIQUID

HOLDUP

AND

In order to provide more detailed and accurate models than the ones mentioned above, some investigators have tried to consider thoroughly the physics of specific flow regimes. A great number of such models have been developed in order to be applied in one flow regime each. The results of such specific models are not necessarily better than the ones produced by simpler models, some of which have been presented here. Nevertheless, about ten other formulations for different flow regimes have been selected and are in the process of validation [11], this being necessary before their use for the analysis and design of Industrial Oil and/or Natural Gas Systems. CONCLUSIONS As regards liquid holdup, the comparison of the models’ predictions against experimental data was adequately successful for most of the models used. More specifically, the Premoli model displayed the closest agreement with the experimental data, while the Mukherjee-Brill had of the poorest relative accuracy. The predictions of all models were better for very low or very high gas flow rates. The Dukler model, as stated before, displayed remarkable agreement with experimental data (as shown in Fig. 5) for medium pressure and high flow rates, while for lower flow rates its prediction accuracy became much poorer. For even higher pressures (120 bar) the predictions of the model were far from these expected.

REFERENCES [1] Hetsroni G., (1982), Handbook of Multiphase Systems, Mc Graw-Hill, New York. [2] Owen D.G., Hewitt G.F. (1987), An improved annular two-phase flow model, Proc. of the 3rd Int. Conf. on Multiphase Flow, the Hage, Netherlands. [3] Beggs D.H., (1985), Gas Prediction Operation Publications, Tulsa, USA. [4] Oliemans R.V.A., Pots B.F.M., Trompe N., (1986), Modelling of Annular Dispersed Two-Phase Flow in Vertical Pipes, J. of Multiphase Flow, Pergamon.

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ASME - GREEK SECTION, First Nat. Conf. on Recent Advances in Mech. Eng., September 17-20, 2001, Patras, Greece

[5] Mukherjee H., Brill J.P., (1983), Liquid holdup correlations for inclined two-phase flow, J. of Petroleum Technology , pp. 1003-1008. [6] Chen J.J.J., Spedding P.L., (1984), Holdup in two-phase flow, Int. J. of Multiphase Flow, Vol. 10, No 3, pp. 307-309. [7] Manzano-Ruiz J., Hernander A. et al., (1967), Pressure drop in Steam/Water Flow through largebore horizontal piping, Proc. 3rd Int. Conf. on Multiphase Flow, the Hage, Netherlands. [8] Wallis G.B., (1969), One Dimensional Two-Phase Flow, Mc Graw-Hill, USA. [9] Gas Suppliers Ass., (1987), Engineering Data Book, Vol. I, II, Tulsa, Oklahoma, USA. [10] Kawaji M., Anoda Y., Nakamura H., Tasaka T., (1987), Phase and Velocity Distributions and Holdup in High Pressure Steam/Water Stratified Flow in a large diameter horizontal pipe, Int. J. of Multiphase Flow, Vol. 13, No 2, pp. 145-149. [11] MPF-Project Team, (1989-90), Multiphase Pipe Flow Modelling in Natural Gas and Oil Systems, Project 09052/89 CEC, Directorate of Energy.

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