Proceedings of the Third (1993) International Offshore and Polar Engineering Conference Singapore, 6-11 June 1993 Copyright © 1993 by The International Society olOffshore and Polar Engineers ISBN ]-880653-05-2 (Set); ISBN ]-880653-07-9 (Vol II)

. A Model for Transient Analysis of Gas-Liquid Slug Flow in Pipelines of an Offshore Well Head

T.N. Wong Nanyang Technological University

Singapore

A. Gilchrist University of Strathc1yde

Glasgow, United Kingdom

Abstract

An approach to model slug flow in pipelines and pipeline-riser systems is presented which is based on a transient Lagrangian formulation. This method is particularly useful for slug flows approaching the end of a pipeline or for predicting the influence of a riser pipe to the production system.

The flow is modelled by considering a series of slug units, each consisting of a 'liquid' slug and its associated 'gas' bubble, being continuously introduced into the system at a fixed location considerably remoted from the system outlet. The flow parameters are predicted as the slugs traverse the system. There are of course interaction effects between successive slug units and outlet effects (disturbances) can be transmitted upstream. Liquid hold-up in the 'liquid' slug and 'gas' bubble is accounted for as gas compressibility. It is assumed that there is no mass transfer between phases.

To validate the approach used, the model was tested against a homogeneous flow condition with a steady air - water flow system. Results are also presented for trains of slug units negotiating the pipeline riser pipe system.

Key Words: oil-gas, slug flow, pipeline-riser, slugging, gas blowdown

Nomenclature A Area D pipeline diameter g gravity acceleration L length P pressure

Subscripts g gas plug sep separator

1. Introduction

R gas constant V velocity Vo volume Z gas compressibility factor

liquid slug

The presence of slug flow in a transportation pipeline can be particular troublesome. Slug flow gives rise to an abrupt fluctuations in pipeline pressure as well as gas and liquid flow rates. Such flow pattern can cause vibration of tubes and etc. The prediction of the onset of slug flow is of considerable industrial importance in that it can provide design engineers with the means to avoid this flow regime.

Theoretical models to predict the onset of slugging have been given by Kordyban and Ranov [1), WaIIisand Dobson [2) and etc. Various slug flow models have also been proposed by Dukler and Hubbard [3). Vermeulen and Ryan [4). This steady state models which consider, to a greater or lesser extent, the details of the flow of individual slugs. However Brill et al [5) have presented correlations based on anaIysisof data obtained from the Prudoe Bay pipeline in Alaska.

142

In relatively short horizontal lines it has been established that for steady conditions liquid slug length of the order of20 -25 pipe diameters can be expected (normal slug flow). However, in long undulating pipeline - vertical riser system, large slugs have been shown to occur. When slug lengths are greater than the height of the riser this is often termed as severe slugging. A typical offshore production system is show in Fig I. As most flowline pass over an undulating surface, the presence of slugs in such 'hilly terrain' flowline - riser systems give rise to a phenomenon know as 'severe' or 'terrain' induced slugging. When the liquid and gas flow rates are relatively low, liquid tends to accumulate at the lower sections of the pipeline to form long liquid slugs. Gas upstream is trapped and compressed until its pressure is high enough to blow the liquid slug out. For flowline - riser configurations, the length of the liquid slug generated at the base of the riser can vary from one to several riser pipe lengths. Such severe slugging is highly undesirable as it is associated with abrupt fluctuations in pipe line pressure and gas and liquid flow rates. Production instead of being smooth, takes the form of alternate liquid slug and gas bubble flow into the process plant. Slugging can also present problems for riser pipe work design due to the induced dynamic loading. The existing steady state models for two phase flow do not adequately describe the dynamic process of slugging.

This paper outlines a relatively simple dynamic model for slug flow which should be of assistance in the design of gas - oil production systems. Details of modelling are presented in section3, it is important to mention here that no attempt has been made to determine what influences slug generation nor to determine slug characteristics for a particular system and flow condition.

2. Previous work

Schmidt et al [6,7) studied slug flow using a simulated pipeline - riser system and found that severe slug flow depends on the geometry of the pipeline - riser system. It was stated that severe slugging was obtained if the pipelin~ was downward inclined immediately upstream of the riser and in stratified flow. A dynamic model was developed to describe the slugging cycle. The proposed model was tested against experimental data and good agreement was obtained.

Bendiksen et al [8) developed a dynamic, one-dimensional two phase flow pipeline model (OLGA) . The authors reported that slip effects due to the different local mass distribution in the two phase are incorporated into the simulations to account for different flow regimes. Comparison of predictions with Schmidt's experimental result were very good except at extremely low liquid flow rates.

Simple simulation models have been developed as reported by Mackay [9) for slugs flow in a pipeline riser system. The dynamics of a single bubble/liquid slug unit approaching and negotiating a riser was predicted. The dynamic model is basically a spring-mass system with damping where the compressible gas in analogous to the spring, the liquid slug to the mass, and damping to the wall friction. Base of riser pressure and slug velocity variations were predicted. The results show that as the slug enters the riser the following gas bubble is being compressed to produce the required hydrostatic head to lift the slug up the riser. As the slug tail enters the riser the hydrostatic head rapidly decreases and the pressure generated in the gas bubble is available to accelerate the liquid slug up the riser.

Matsui et al [10] studied pressure wave propagation in liquid slug - gas bubble systems in horizontal and vertical pipes. The liquid - gas system was modelled as a series of spring - mass systems accounting for liquid interia and gas compressibility. Liquid hold up effects were not account for. Qualitative agreement was obtained between prediction and experiment.

Criteria for severe slugging have been suggested by Taitel [11]. The author proposed a criterion for the conditions which give rise to severe slugging. He proposed that severe slugging would exist in a pipeline riser system when the liquid in the riser is unstable and gas penetrates into the riser. This resulted in an unstable blowout and a cyclic process. When the liquid column is stable a steady state situation exists.

Fuchs [12] presented a criteria giving the maximum system pressure above which terrain slugging does not occur. He noted that the numerical value of his proposed slugging parameter is system dependent and must be obtained experimentally.

Linga [13] conducted terrain slugging tests in a closed loop system with a horizontal length followed by a downward sloping length immediately upstream of a riser. A simulated bump could be introduced into the horizontal section. It was observed that blockage at riser base was initiated by long transient slug units propagating along the pipeline. This is at variance with Schmidt's statement that upstream flow must be stratified. Linga observed in his conclusion that as terrain slugging is a strongly system dependent phenomenon the experimental results obtained cannot be extrapolated to an arbitrary pipeline - riser system. He stressed the requirement of a dynamic two phase flow simulator.

3. Modelling

3.1 Description of the transient slUG flow model

For 'steady state' normal slug flow, liquid slug over-runs the liquid fIlm associated with the downstream gas bubble and entrains gas into the slug along with the liquid of the fIlm. Liquid and gas are shed from the tail of the liquid slug. For normal slug flow the momentum difference between the fluid entering and leaving the slug is of significance. For long slugs, induced by terrain or riser effects, these momentum effects may be small relative to momentum of the slug itself. In normal slug flow it may be assumed that the momentum of the slug is constant with time, whereas in cases where the slug as a whole may be accelerating or decelerating, allowance must be made for the change of momentum with time. Because the interest of the present study was in transient slug flow with relatively long slug units it was thought reasonable to treat the liquid slug and associated gas bubble as two units of fixed mass.

The flow is modelled by considering trains of slug units, each consisting of 'liquid' slug and its associated 'gas' bubble, being continuously introduced into a system ata fixed location considerably removed from the pipe outlet. The ' liquid' slug and' gas' bubble each have an average hold-up and since the liquid is assumed incompressible variations in these hold-ups with pressure arise because of the compressibility of the gas. It is assumed that there is no mass transfer between phases.

Within each 'liquid' and 'gas' unit (fixed masses) the flow is assumed homogeneous and isothermal and the variation in gas properties is described by the gas equation of state. The governing equations for the units are formulated in a Lagrangian frame of reference.

Because of the compressibility of the gas in the 'liquid' slugs and 'gas' bubbles the leading and trailing faces of these will have different velocities and thus will alter in length as they traverse the system. The momentum equation for each unit account ofinteria, pressure, gravity and wall friction forces. To evaluate the friction term in the momentum equation the average of the leading and trailing face velocities is used along with average homogeneous flow properties values.

The system of equations for all the units within a pipeline system form a set of coupled, non-linear, ordinary differential equation which must be solved simultaneously, taking account of the inlet and outlet boundary condition. Parameters such a slug velocity, bubble velocity, pressure, slug/bubble length and their position in the system can be predicted as the slug units pass through the system. Pressure, velocity, hold up and mass rate variation with time at a fixed location can be obtained.

3.2 Equation for slUG units within the system

The 'liquid' slug momentum equation is ( see Fig 2 for nomenclature)

( dV'1 I dt}J = M, [(PdJ-PUJ+I)A -'tLltDL,-L~p""Ag] (I)

where M, is the mass of the 'liquid' slug,

M, =ALJpLH", +Pg(l-H",J]

where H", is the liquid hold up in the slug as it introduced into the system.

143

The average slug velocity and pressure are defined as simple arithmetic means of upstream and downstream values.

VgfJ+VifJ (2) V'J=--2-

Pd,+Pu)+1 PLJ = 2

Applying the state equation (~Ee)~: ~::~ [~~;~iJSlUg gives,

where Vo is the volume of gas and from geometry

( d~;L ), = (V'Jj- V g/j)A

The two-phase homogeneous density is

p"" = (I-HdPg +HLSPL

where _ X,Vg HLS= I

X,V g + (1- x,)vI

is the homogeneous hold up in the slug, x, being the dryness fraction.

A similar approach is used for the ' gas' bubble.

The momentum (~;:r=i~g [(PUJ -Pd)A -'tgltDLg -L",PgmAg]

where Mg is the mass of the 'gas' bubble,

Mg = A Lg[PLHLg + Pg(l-HLg )]

where HLg is the hold up in the bubble as it introduced into the system

Pgm = (1-HLg)pg + HLgPL

and

The average velocity and pressure are V gfJ + V,IJ-I

V,,= 2

PUJ+PdJ P"=-2-

The state equation for the gas( :;gt)he ~ ~;g~U(b:~o~i)ves,

dl J Va" dl J

where ( dVOg ) _ dt J -(Vg/j-V,jj_I)A

3.3 Equation for slug units leaving the .system

For 'gas' bubbles leaving the system (referred to as gas blow-down phase)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

where mg = pgmAVgup is the leaving gas mass flow rate and Vgup = 2V" - V,IJ-I

The suffix sep refers to the pipe exit.

The above equation is derived for a control volume formed by the moving upstream face of the gas bubble and the fixed location of the pipe outlet. Thus mass and momentum within the control volume vary with time. Wall friction is accounted forooting that the 'wet' wall reduces as the bubble progressively leaves the system. For the purpose of this study it has been assumed that the static pressure at the exit section is constant, i.e. the separator pressure. If fluid inflow to and outflow form the phase separation unit can give rise to pressure fluctuation within the separator then the governing equations for the separation system can be solved simultaneously with the set of equation for the slug units in the pipeline.

The state equation for the gas in the' gas' bubble gives:

( dPg ) _ I. [ dMg dVO,,] (if J- Vo" ZRTxgTt-P"1iI (12)

and geometry gives

( dVOg) 1 -- =-V A+-(I-x)m dl J 'IJ -I PL g g

(13)

A similar set of equations apply when the 'liquid' slug is leaving the system.

3.4 Boundary and initial conditions

3.4.1 Initial condition

The average hold up for the slug unit about to be introduced into the system can be defined in terms of the 'liquid' slug and 'gas' bubble hold up and their respective lengths.

HLSLs+H.L. HL ="~.--=--~

L. Hence the bubble length to liquid slug length ratio can be expressed as

L. HLS-HL L;= HL -HLg

The average holdup (HL ) can be evaluated in following section.

(14)

(15)

Brill et al [5] have presented a simple empirical liquid slug length correlation:

In(Ls) = -2.66:3+ 5.441(lnd)o.5 + 0.059(ln Um )

where d (pipe diameter) is in inches and Ls is in feet and Um (mixture velocity) in

fl/s.

Brill found that slug lengths were variable and that they followed a log - normal distribution. Thus it is possible to ~imple choose a slug length to study its effect on a particular system. The effects of different lengths of slug could then be investigated.

Brill et al also presented the following correlation for hold up in the gas bubble.

HL• = 1.0-0.01 exp[a +b In(U,.)+c In(U,/]

where a = 4,47108-0. 1369IU"

b = --0.05831 + 0.0807U"

c = 0.02124-0.01 I 69U"

where U,. and U" are in fl/sec.

Gregory et al (14) presented an empirical correlation relating the liquid slug hold up to the mixture velocity Um in (fl/sec) .

I + (0.0352U S 39

The above hold up correlations all inherently take account of slip between the phases. If it is assumed that these values give reasonable estimates of the respective hold up just prior to introduction of a slug unit tlien the bubble length corresponding to a particular liquid slug length and the masses M, and M. can be obtained for use in the analysis.

3.4.2 Pressure Boundary Condition

The downstream pressure is taken as being constant. It is assumed that this pressure is independent of whether gas or liquid units are leaving the system.

The upstream boundary conditions are modelled by assuming that slug units, consisting of a 'liquid' slug and its associated 'gas' bubble, are pushed into the pipeline past a fixed datum location by a piston moving with a constant velocity. The tail pressure of the' gas' bubble adjacent to the 'piston' face can be obtained using equations (3) and (8) by considering the total number of slug train in the pipeline as:

p. "' P", •• =rL(2P8I -2PL)]+P"p

3.4.3 Piston velocity

(16)

The total average mass flow rate upstream of the chosen datum location must be the same as the total average mass flow rate d~wnstream.

m total upstream = mass . in slug/bubble unit time taken to introduce the unit

U + U LS[PLHLS+P.(1-HLS)]+L.[PLHL.+P.(1-HL.)] PL ,I p. '. (L +L)IV'

(17) S g puton

Using the definition of the average hold up over the entire slug unit this reduced to

Vp,,,., PL~L~' :'~;~~L) (18)

However, the individual phases average mass flow must also satisfy the continuity relationship. Thus,

PLA U" = PLHLA Vp",.. (19)

p.A U,. = p.(1- HL)A Vp",.. (20)

144

combining equations (19) and (20) give,

U" HL =--- (21)

U,,+U,.

The average holdup for the slug unit was shown to be equal to the no slip (homogeneous) holdup in equation (21). Substitute equation (21) into (18) shows that

~-=~+~=~ ~~ The piston velocity was shown to be equal to the two phase mixture velocity from the continuity balance. Using the homogeneous hold up HL along with HLS and

H L. from the correlation, the initial liquid slug to gas bubble length ratio can be

obtained from equation (15). To note that this constant 'piston' velocity can be vary from unit to unit if the piston face pressure varies.

3.5 Method of Solution

Initially the complete system downstream of the datum is considered to be at a pressure equal to the outlet pressure. The frrst slug unit is introduced and the equations solved for each time interval until the 'piston' reaches the datum where upon a second slug unit is introduced and the set of coupled equations for the two slug units in the system are solved until the 'piston' again reaches the datum. This procedure continues until a sufficient number of slug units have been introduced to produced ' steady' results, i.e the effects of the initial conditions have disappeared. A fourth order Runge-KUlla method of solving the differential equations were used.

4. Results and discussion

4.1 Comparison of result for homogelleous flow

To validate the approach used in the transient slug flow model, the model were tested under the homogeneous flow condition with a steady homogeneous air-water flow system. Homogeneous air-water flow over a 25 mm pipe diameter with 31 m horizontal pipeline. The liquid mass flow is 2.42 kg/s and gas mass flow of 2,4508E-4 kg/so The inlet pressure is 518.6 KN/M"2. The pressure at the end of the pipeline predicted by the steady flow model was 277.58 KN/M1I2.

In the transient slug flow model, an initial identical 'liquid' slug and 'gas' bubble length of 2.4 m to making up a slug unit. i.e homogeneous flow. The separator pressure was constant at 277.58 KN/M"2.

Figs 3 to 5 show calculated results for homogeneous air - water flow. Figs 3 (a) and 4 (a) show the mixture velocity and pressure variation with position calculated using a steady homogeneous flow model. Figs 3 (b) and 4 (b) show these results transposed to show the variation with time as a fluid package moves through the pipe. Figs 3 (d) and 4 (d) show the corresponding result obtained using the transient slug flow model. Comparison of result in Figs 3 (b) and 3 (d) show that the trend and magnitude are similar but the transient analysis give a fluctuating velocity variation. These variation were thought to arise because of the constant piston velocity used in the analysis. To investigate this, a variation in piston velocity during slug unit introduction was used which followed the variation in the homogeneous mixture velocity over the distance moved by the piston. Figs 3 (c) and 4 (c) show the result obtained which, considering the approximation made in the analysis, is in good agreement with the steady flow result.

Fig 5 shows the variation of liquid, gas mass flow rate at pipeline outlet and pressure variation at a fixed position 14 m away from pipe outlet. The result was calculated by the transient method with variable 'piston' velocity. The mass flow rate predicted at the pipeline outlet is in good agreement with the inlet gas. liquid mass flow rate.

It is noted that the speed of the piston controlling the mass flow rate. A variable piston velocity give a better result. Since the mixture velocity gradually increase in the downstream direction because of gas expansion. However, from the result obtained it seen not unreasonable to assume a constant 'piston' velocity during slug injection, as the pressure at piston face vary, the constant velocity can vary from unit to unit especially when the slug length are not the same.

4.2 A pipeline riser system

The system considered was a typical pipeline riser system with ISO m vertical riser followed by a 60 m horizontal pipe length leading to the separator. Pipeline diameter was 0.2 m and the separator pressure was assumed constant at 13.8 bar. For a given gas and liquid flow rate, the two phase steady flow model was used to determine the pressure, temperature, mixture velocity and the liquid properties at the base of the riser. As shown in equation (22) that the piston velocity is equal to the two phase mixture velocity, hence the mixture velocity at the base of riser has used as the constant 'piston' velocity. The initial liquid slug to gas bubble length ratio can be obtained form equations (14) and (15).

Regular slug now in pipeline-riser

Figs 6 to 11 are for oil-gas flow in a pipeline-riser system with an initial liquid slug length of 91.4 m.

Figs 6 and 7 show the average slug velocity, liquid slug hold up, liquid slug length and gas bubble average velocity, pressure, gas length and hold up ~ariation.s with time as a slug unit passes through the system. The result are for a stngle umt forming part of a train of units completely filling the pipe.

Fig 6 (a) shows how the velocity of the liquid slug varies as it passes along the pipe. The velocity variations are caused by the acceleration of preceding liquid slugs as they exit from the pipe, and as slugs negotiate the bends at the base and the top of the riser. The amplitude of the velocity variation decrease with distance from the outlet due to the damping effect of wall friction.

Fig 7 shows corresponding results for the gas bubble following the liquid slug. Just before the last of the liquid slug leaves the system the average gas bubble pressure is higher than the outlet pressure and hence when the liquid ftnally leaves the pipe the gas blows- down to a lower average pressure and the velocity decrease fairly rapidly. The subsequent increase in velocity is caused by the acceleration of the following liquid slug.

Figs 6 (b) and 7 ( c) show the variation of liquid length and gas bubble length as it passes a series of expansion and compression as it travel through the pipeline-riser system. The gas bubble length change more rapid than the liquid slug length.

Fig 8 shows the average slug velocity, mixture velocity, slug front and slug tail velocity variation with time. The model assumed that liquid film distributed homogeneously into the gas bubble with an average homogeneous holdup, variation of the slug front and slug tail velocity are caused by the compressibility of the gas. The result shows that both these velocity are quite close to the average slug velocity. The comparison between the mixture velocity and the average slug velocity is good. To note that the mixture velocity varies as a resultof gas expansion due to friction loss. The average slug velocity calculated consider both the effect of slug acceleration, deceleration and gas expansion.

Fig 9 (a) and (b) show the variation of liquid and gas mass flow rates at the pipe outlet. The mass flow rate shows alternate liquid production periods at the riser outlet. The instantaneous flow rate of each phase is many times greater than the average flow rate. The oil-gas separator and control systems must be designed to cope with such disturbances. This information would be useful for the design of phase separation facilities.

Fig 10 shows mass flow rate and velocity variations at base and top of riser. This information would be useful for the pipe work structure design.

Fig II shows the pressure variations with time at a fixed location. The initial variation shows results form the fact that the pipe is initially assumed to contain only gas. When the pipe becomes full of slug units the pressure variation is regular with time.

5. Conclusions

Based on a number of simplifying assumptions a relatively simple dynamic model on slug flow in a pipeline-riser system is developed based on transient Lagrangian formulation. This approach is shown to be useful for predicting the influence of the successive slug unit as the preceding slug unit leaving the system or negotiates the pipeline - riser configuration.

The dynamic model were tested under homogeneous flow condition with a steady state homogeneous flow model, good agreement result is obtained.

The slug velocity calculated by the transient model included the interaction effect of the successive slug train, slug unit was found to undergo a series of acceleration and deceleration as it traverse through the pipeline - riser system. The average slug velocity predicted by the model was in agreement with the mixture velocity. A high liquid velocities are achieved as liquid leaves the system.

The results obtained from the model are in agreement with what would qualitatively be expected. The proposed model could easily be integrated with a dynamic model of the receiving facilities.

The model can be used to assess the effect of slug flow on the dynamic loading of systems and an aid to the design of phase-separation facilities. In order to validate the model comparison with field or laboratory data is required.

Acknowledgement

This work was undertaken for Britoil pic and the authors appreciate their assistance and financial support.

145

References:

[I] Kordyban, E.S. and Ranov, T. : "Mechanism of slug formation in horizontal two phase flow". J. Basic Eng., 92, 857 - 864, Dec 1970.

[2] Wallis, G.B. and Dobson : "The onset of slugging in horizontal stratified air - water flow". Int. J. Multiphase Flow, I, 173 - 193, 1973

[3] Dukler, A.E. and Hubbard, M.G. " A model for gas - liquid slug flow on horizontal and near horizontal tubes". Ind. Eng. Chern. Fund., Vol 14, No. 4, pp 337 -47, 1975

[4] Vermeulen, L.R. and Ryan, J.T. :"Two phase slug flow in horizontal and inclined tubes". Can. J. Chem, Eng., Vo. 49, pp 195 - 20,1971

[5] Brill, J.P., Schmidt, Z., Coberly, W.A., Herring, J.D. and Moore, D.W.: "Analysis of two - phase tests in large diameter flow lines in Prudhoe Bay field". Society Petroleum Engineering, 363- 78, June 1981.

[6] Schmidt, Z., Brill, J.P. and Beggs, H.D.: "Experimental study of severe slugging in a two -phase flow pipeline - riser pipe system". Society Petroleum Engineering Journal, 407 - 14,Oct 1980

[7] Schmidt, Z. Doty, D.R. and Dutta, R. :" Severe slugging in offshore pipeline riser pipe system". Paper SPE 12334, Sept 1983.

[8] Bendiksen, K., MaInes, D. and Nuland, S. : "Severe slugging in two phase flow system". Report prepared for Norwegian State Oil Co., Kjellen. April 1982.

[9] Mackay, D.C. : "Dynamic' model predicts slugging flow effect in offshore production facility". Oil & Gas Journal, Sept 14,1987.

[10] Matsui, G. and Sugihara, M. " Propagation charactenstics of pressure wave through gas -liquid plug train system". Bulletin ofJSME, Vol 22, No. 173, pp 1562 - 1569, 1979.

[11] Taitel, Y. : "Stability of severe slugging". International Journal Multiphase Flow, Vol. 12, No.2, 1986

[12] Fuchs, Per. : " The pressure limit for terrain slugging". 3rd International Conference on Multiphase Flow. The Hague, Netherlands, May 18 -20, 1987.

[13] Linga, H. : "Terrain slugging phenomena". Some experimental results obtained at SINTEF two phase flow laboratory, 3rd International Conference on Multiphase Flow, The Hague, Netherlands. May 18 -20. 1987.

[14] Gregory, G .A., Nicholson, M.K. and Aziz, K. : "Correlation ofliquid volume fraction in the slug for horizontal gas -liquid slug flow". Int. Journal of Multiphase Flow, 4, No I, pp 33-39,1978.

.... ~

Pll I-I

0L11\:8.'l 3eparelOr

Fig. I Flow line, riser and oil - gas separator configuration

P: Pressure V: Velocity

HL : Liquid holdup

j : Number of slug unit L: Length

Ptlj-I

PU 1+ 2

Pu 1+1 Ptl 1+1

1+ Lr 'j_ Ls ---0{

Fig. 2 Nomenclature for slug train

I--:==:::J P,e p

I l3V

1 L3

Fig. 3

(u) '1.98

00'1.95

" I: ~'I.93

>-I- '1.90 u 0'1.68 -.J w >'1.85

0 7 1'1 21 28 35

01 ST ANCE (M)

'1.98 (b)

00'1.95

" I: ~'I.93

>-I- '1.90 u 0'1.88 -.J w >'1.85

0.00 1.'10 2.80 '1.20 5.60 7.00

TIME (S)

'1.98 (c)

(1)'1.95

" ~'I.93 >-I- '1.90

u 0'1.88 -.J w > .... 85

11.00 12 .... 0 13.80 15.20 16.60 18.00

TIME (S)

.... 98 (d)

00 .... 95

" I: ~"'.93

>-I- .... 90

g .... 88 -.J

~"'.85 11.00 12 .... 0 13.80 15.20 16.60 18.00

TII1E (S)

Air - water homogeneous flow a), b) Eulerian method for steady flow

c) Lagrangian transient method with variable "piston" velocity d) Lagrangian transient method with constant "piston" velocity

(J) , <!)

~

3< 0 -I LL

(J) (J)

< ~

(J)

< (!)

Fig. 5

2.50 (J) , <!)

~ 1.88 3< 0

it 1.25 (J) (J)

~ 0.625

a ~

::J 0 C?J

-I 0 5 10 15 20

TIME (5)

0.000250

0.000188

0.000125

0.0000625

0

0 5 10 15 20

TIME (5)

100

N +- 365 ~ , z ~ 330

w ~

~ 295 (J) w ~

a.. 260

0 5 10 15 20

TIME (5)

Transient method with variable "piston" velocity a), b) mass flow rate at pipe outlet Vs. time

c) pressure at pipeline location Vs. time

(a)

25

(b)

25

<c)

25

(u) 15 i I 15

r.. (f) 12

<n 12 " :c

" 9 1::

9 >->-

I-- 6 I- 6 u ..... a 3 u a 3

.J

.J W

W > 0 > 0 1-100 1560 1720 1880 2010 2200

HOO 15-10 1680 1820 1960 2100 TIME (S) TIME (S)

(b) 2200

'" <c-:r:: 1950 " z y

1700

w

'" 1150 => (f) (f) w 1200

'" "- -100 1560 1720 1880 2010 2200

TIME (S)

~ I 500

(c)

.... 120 ~ QC 375

90 :r:: 1::

60 :r: 250

::r: I--I-- '" (!J z 125 z 30 w W .J .J 0

0 1"'100 1560 1720 1880 2010 2200

1-100 15-10 1680 1820 1960 2100 TIME (S)

TIME (S)

0.1-10 (d)

0.6-10

~ 0.125

0.605 "-

"- => 0.110 5 0.570

0 .J

.J a a ::r: 0.0950 ::r: 0.535

0.0800 0.500

HOO HOO 1560 1720 1880 20"'\0 2200

15-10 1680 1820 1960 2100

TIME (S) TIME (S)

Fig. 6 History of one slug unit of a multi-slug train moving through Fig. 7 Histo~y o~ one.slug unit of a multi-slug train moving through

the pipeline-riser system the p~pel~ne-r~ser system

a) average slut;: ve.bc ity b) slug length c) holdup in ljquid slug a) average gas vel oei ty 0) average gas pressure c) gas length d) holdup in g.as plug

.... ~

Fig. 8

;;; , J:

>-t: u 0 -' w >

15

12

3

AVERAGE SLUG VELOC ITY SLUG TAIL VELOCITY SLUG FRONT VELOC I TY MIXTURE VELOCITY

a I I I I I , ( I iii i i 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950 2000 2050

TIME (S)

Velcoity history of one slug unit moving through the pipeline -riser system

Fig. 9

G2:::: 200

~ 150 (9

'" ~ 0

ti 100 (f) (f) <: r: 0

50 :::J C>

-'

01 ,hJ'\ j~LJ~I\l\j\J\J~hf 0 500 1000 1500 2000 2500

TIME (S)

8 ~

6 (f) , r: ~

~ 0 -' 4 LL

(f) 1\ 1\ j\ 1\ il j\ /\ /\ n 1\ j\ /1 (f)

<: r: (f) 2 <: (9

v o I I I VI VI VI VI VI VI VI VI VI VI iii i i

o 500 1000 1500 2000 2500

TIME (S)

Mass flow rate variation at pipe outlet Vs. time

.... lS

(/) , :c

>­I­.... U o --I W >

(J) , :c

>­I-....

12

9

6

3

o I I -

o 500

Ii

10

7

3

1000 1500 2000 2500

TIME <5>

U o --I W > O+I------+------.------~----~r_----~

100

(JJ

'- 75 :c

:. 50 o --I IL 2S (J)

o SOO

(J)

< :c o I ",

200

(J)

'- 150 :c

o 500

1000 1500 2000 2500

TIME <5>

<c>

1000 1500 2000 2500 TIME (5)

<d)

:. 100 o ...J IL 50 (J) (J)

< :c o 1 WillllillJ o sao 1000 1500 2000 2SOO

TIME <5>

Fig. 10 Nass flow rate and velocity variation with time a) velocity at base of riser b) velocity at top of riser c) total mass flow rate at base of riser d) total mass flow rate at top of riser

N 2200 +-:c '- 1950 z :>l

1700 w ~ ~ 1't50 (/) (JJ w ~ Q..

1200 0 500 1000 1500

TIME (5)

2000 N +-:c 1800 , z :>l

1600 w ~

liOO ~ (J) (J) w 1200 ~ Q.. 0 500 1000 1500

TIME (5)

2000 N +-:c , 1800 z :>l

1600 w ~ liOO ~ (J) (J) w 1200 ~ Q.. 0 500 1000 1500

TIME (5)

1600 N +-:c 1537 , z :>l

li75 w ~

li12 ~ (J) (JJ w 1350 ~ Q..

0 500 1000 1500 TIt:E (5)

Fig. II Pressure at pipeline location Vs. time aJ 1066 m upstream from base of riser bJ 457 m upstream from base of riser cJ base of riSer pressure d) top of riser pressure

2000 2500

<b>

2000 2500

<c>

2000 2500

<d)

2000 2500