Model for Terrain-induced Slug Flow for High Viscous Liquids

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1 Model for Terrain-induced Slug Flow for High iscous Liquids Andrea Shmueli 1, Diana C. González 2, Abraham A. Parra 2* and Miguel A. Asuaje 2 1 Department of Energy and Process Engineering, Norwegian University of Science and Technology, Trondheim, Norway 2 Department of Energy Conversion and Transport, Simón Bolívar University, Caracas, enezuela Received: 24 June 2014; Accepted: 1 September 2014 Abstract This paper presents the development and results of a simplified multiphase model for terrain-induced slugging analysis considering the viscosity effect of the liquid phase in the conservation equations. The phenomenon has been successfully modelled with numerical models (Fabre et al. 1990, Sarica et al 1991) that neglect the liquid viscosity effect. The objective of the present study is to clarify its importance in the numerical prediction of the characteristics of severe slugging. The results of the model where compared with experimental data reported in the literature showing a maximum percentage difference of 6% which represents a 7% improvement with respect to the prediction of existing models. The liquid viscosity has an effect on the duration of the severe slugging cycle. The cycle will reduce while the viscosity increases. Additionally, the effect of the riser inclination angle is analyzed showing a mitigation of the severe slugging phenomenon while decreasing the angle 431 NOMENCLATURE g gravitational constant (ms -1 ) Greek letters P pressure (Pa) a oid Fraction HL Hold up μ Dynamic viscosity (cp) f Darcy Friction Factor θ Inclination angle of the sloping pipe t Time (s) with respect to the horizontal elocity (m/s) β Inclination angle of the riser L Length of the negative slope pipe (m) Φ Pipe Diameter (m) i Counter space ε Pipe roughness (m) Δt Time Step (s) τ iscous stress Zr Riser Height (m) ρ Density (kg/m 3 ) m Mass (kg) x Liquid local height in the negative Subscripts slope pipe (m) L Liquid z Measurement of fluid in the riser (m) G Gas dzn Size of the differential volume (m) sl Liquid surface T Temperature (C) sg Gas surface Zj Position of the first gas bubble (m) g Gas R Universal gas constant o Conditions of entry Nm/kmol K r Riser Po Atmospheric pressure (Pa) p Negative slope pipe P sep Separator pressure (Pa) m Mixture C o Distribution Coefficient pe Penetration U o Drift elocity (m/s) slpen Fluid penetration rate in the riser (m/s) Superscripts slo Superficial velocity of liquid at the t Time inlet of the down pipe (m/s) Δt Time step * Corresponding author: aparra.bajausb@gmail.com

2 432 Model for Terrain-induced Slug Flow for High iscous Liquids sgo Superficial gas velocity at the h Horizontal inlet (m/s) v ertical Ma Molecular weight of the gas A Pipe cross sectional area (m 2 ) Re Reynolds number Zi Location of the differential control volume in the riser (m) 1. INTRODUCTION The transport of two phase gas liquid flows is one of the main challenges in oil production systems. A correct description of the phases transport behavior is required when designing multiphase flow lines and equipment. In offshore production systems, oil and gas are frequently transported from the wells to the surface facilities using pipeline-riser systems. Depending on the pipeline layout and the flow conditions a phenomenon called severe slugging can occur. Severe slugging is a cyclic type of flow that can cause structural damage to the pipes and equipment, flooding on the surface facilities, among others. The existence of this type of flow is conditioned by the following conditions: Upstream downwards pipe with stratified flow followed by an upwards pipe (riser). Low gas and liquid velocities and enough gas compressibility. On onshore transport facilities a similar phenomenon can occur due terrain layout and it is called terrain induced slugging. The severe slugging cycle was described by Schmidt et al. [1]. On their experiments they described 4 stages on the cycle: Slug generation, slug production, bubble penetration and blowout (See Figure 1). Some researchers have proposed transient mechanistic models to describe the severe slugging physics. Schmidt [2]. made a distinction between two types of slug flow: the terrain induced slug and hydrodynamic slug. He developed a mathematical model to simulate transient severe slug flow. Taitel [3] studied the conditions under which the quasi steady state severe slug can occur. Fabre et al. [4], based on the work by Schmidt et al [1], modelled the severe slug using the method of characteristics, local continuity and momentum instantaneous equations. Sarica & Shoham [5] developed a simplified model for transient severe slug. Hernandez et al. [6] proposed corrections to the mechanistic model proposed by Sarica & Shoham [5] on the calculation of the liquid density through the method of characteristics. On this work, the drift flux model of Fabre et al [4] is extended to take into account the liquid viscosity and riser angle inclination. The improvements and corrections proposed by Sarica & Shoham [5] and by Hernandez et al. [6] were taken into account. A sensitivity analysis of the liquid viscosity and riser angle inclination is performed aiming to recognize the difference on the severe and terrain slugging behaviour when operating with viscous flows. Sep Sep θ β θ β Stage 1: Slug generation Stage 2. Slug production Sep Sep θ β θ β Stage 3. Bubble penetration Stage 4. Gas expansion Figure 1: Severe Slug Cycle Proposals by Schmidt et al. [1]. Journal of Computational Multiphase Flows

3 Andrea Shmueli, Diana C. González, Abraham A. Parra and Miguel A. Asuaje MODEL A general con figuration of the pipeline-riser geometry is used to deduct the equations for each stage. The analysis was performed using a general case configuration. The geometric considerations and the coordinate reference system used are shown in Figure Stage 1 and 2: Slug generation and production During this stage the flow of gas into the riser is blocked by the liquid accumulation in the elbow and in the upwards pipeline. This accumulation results in increased pressure at the riser base. The end of this stage will be when the liquid level reaches the top of the riser. At this point the pressure at the base is the maximum that can be obtained from the cycle. The second stage will start once the liquid in the riser starts to move towards the separator or to the following pipe. This stage will last until the gas on the downwards pipe reaches the riser base. 2.2 Assumptions: -There is no mass transfer between the phases in the control volume -There is always stratified flow in the downpipe feeding the riser -The void fraction in the stratified pipe remains constant during the whole stage -The process is isothermal -No penetration of gas into the riser (Severe slugging type I) Following Fabre et al (1990) [4] the gas and liquid continuity equations written for the control volume 1 (C1) are presented below (See Figure 3) 2.3 Liquid mass conservation equation: = α slpen slo p dx dt (1) Negative slope pipe Positive slope pipe (riser) L Zr q b X Z Figure 2: Geometric considerations and coordinate reference system. sgo C1 slo P sep α p P p X(t) θ slpen β z(t) Figure 3: Proposed Control olume for Developing Mass Conservation Equations. Stage 1 and 2. olume 6 Number

4 434 Model for Terrain-induced Slug Flow for High iscous Liquids During the slug generation stage (Stage 1) the liquid penetration velocity to the riser slpen can be calculated as the variation in time of the riser liquid height. Therefore, Equation 1 can be particularized for this stage as follows: dx dz α p = slo dt dt (2) 2.4 Gas mass conservation equation: d dt ( ) ρ. A. α L x = ρ. A. g p go sgo (3) Assuming ideal gas behaviour is possible to rewrite Equation 3 as: ( ) d P L x = dt P. p o sgo α p (4) The gas and liquid continuity equations can be discretized using forward finite differences in time to produce equations 5 and 6 for stages 1 and 2 respectively. t+δ t t +Δ z = z +. Δ α t. x x slo t t t p ( ) (5) t+δ t t+δt o sgo t t p ( ) = Δ t+ Pp ( L x ) α p P L x P. (6) 2.5 Combined momentum equation The momentum equation for stage 1 and 2 is deducted using a control volume fixed to the liquid accumulated at the bottom of the pipes and this was considered for each time step (Figure 4). Inertial effects can be considered negligible in the momentum equation as compared with the gravity. Friction losses are considered in order to assess the effect of liquid viscosity. Equation 8 shows the balance of forces in the direction of the pipe at the base of the riser. P p.a t + ρ L.g.x(t).sin (θ)a t = P sep.a t + ρ L.g.z(t). sin (β).a t + τ o.f.π.(x(t) + (z(t)) (7) The viscous stress in terms of Darcy s friction factor is shown in Equation 8. ρl. f.. τ o = 8 (8) The friction factor was calculated using the correlation of Colebrook for turbulent flow and the Blasius equation for laminar flow. Particularizing Equation 7 for stages 1 and 2 it is obtained Equations 9 and 10 respectively. Journal of Computational Multiphase Flows

5 Andrea Shmueli, Diana C. González, Abraham A. Parra and Miguel A. Asuaje 435 X(t) C2 θ β z(t) ρl g x(t) sin(θ) A ρl g z(t) sin(β) A Psep A Z Pp A τ0 π φ z(t) τ0 π φ x(t) Figure 4. Control olume and force diagram for combined momentum equation of Stage 1. P + ρ. g. x t. sin θ = P + p L sep ρ.. g zt. sin L () ( ) f. ρl.. β + xt + zt 2. φ () ( ) ( () ()) (9) () ( ) P + ρ. g. x t. sin θ = P + p L sep ρ.. g zt. sin L f. ρl.. β + xt Zr 2. φ () ( ) ( () ) (10) The term represents the rate of penetration of liquid in the riser. Particularizing the Equations 9 and 10 to the time t + Δt t+δt ρ P = P + ρ g z sin( β) x sin( θ) + f.... L.. x + p sep L z 2. ( t+δ t) ( t+δ t) ( t+δ t) ( t+δ t) φ ( ) ( ) ( ) +Δ ( t+δ t) ( t+δ t) f. ρ.. ( t+δ t t t ) P = P + ρ. g z. sin( β) x. L p sep L sin( θ) + x Zr 2. φ (11) (12) 3. STAGE 3 AND 4: BUBBLE PENETRATION AND GAS EXPANSION The bubble penetration stage begins at the moment when the first gas bubble enters into the riser. It is characterized by the gas expansion in the upward pipe. During this stage the pressure in the pipeline decreases rapidly. It ends at the moment in which the first gas bubble reaches the top of the riser. From this point the stage 4 starts. olume 6 Number

6 436 Model for Terrain-induced Slug Flow for High iscous Liquids Zj Zr Zi Figure 5: Spatial coordinates for the location of the differential control volume in stages 3 and 4. In stage 4, the gas flow produced is much larger than the gas flow entering the pipe due to gas expansion, and this causes the line pressure to decrease rapidly. When the gas velocity is insufficient to maintain the liquid in the riser, it begins to fall. This phenomenon is known as liquid fallback This stage marks the end of the cycle of severe slug. To model stages 3 and 4 continuity equation for each phase and combined momentum equation were built. Unlike stages 1 and 2, during the penetration and the expansion of gas in the riser, there are important variations of the properties in time and space. In consequences these equations were formulated for a riser differential control volume as show in Figure 5. The coordinates Zr, Zj and Zi represent the height of the riser, the position of the first node with gases in the riser and the position of any existing node in the domain. 3.1 Mass conservation equations: Liquid and gas mass conservation in a differential control volume can be written as in Equations 13 and 14 respectively. ( ) ( ) 1 α r sl = t z ( ) ρg. α ρg g = r. s t z (13) (14) 3.2 Combined momentum equation: To obtain the combined momentum equation for stages 3 and 4, Newton s second law was applied to a generic differential element of fluid in the pipe. The forces diagram is shown in Figure 6. As a phenomenon dominated by gravity, acceleration term is neglected in the momentum equation. The viscous stress is calculated using the Equation 8 using the same methodology as in stages 1 and 2. Equation 15 shows the result of the sum of forces on the axis Z after simplification. zj 4. τ o P Psep dzn ρl. g. sin ( β) ( 1 αr) dz ρl. g. sin ( β ) φ ( Zr Z j) = 0 zi (15) Journal of Computational Multiphase Flows

7 Andrea Shmueli, Diana C. González, Abraham A. Parra and Miguel A. Asuaje 437 A τ0 π φ dzn z Psep A zj ρl g sinβ A (1-αr)dz zi ρl g sinβ A (Zr-Zj) P A dzn (Spatial variation) Figure 6: Force diagram used for the development of the combined momentum equation, stages 3 and Closure relationship In order to close the system of equations a closure relationship is required. Zuber & Findlay [7] drift flow model equation was used. (Equation 16) G = = + + sg C0 ( sl s g ) Uo α r (16) C 0 represents the effect corresponding to the non-uniform flow distribution and void fraction. Zuber and Findlay [7] suggested that C 0 can be estimated with the constant value of 1.2 if the distribution of the void fraction is not taken into account. U 0 by Dumitrescu [8] was used as estimation of the ascent velocity of the Taylor bubble. U0 = 0, 35. g. φ (17) His correlation was developed using the potential flow theory which does not consider the effect of viscosity. Nevertheless Gokcal et al. [9] in a theoretical-experimental study showed that the high viscosity of the liquid phase significantly affects on the prediction of the drift velocity. They proposed a correlation that takes into account the viscosity of the liquid phase as the inclination angle of the riser. The correlation is valid for fluid viscosities less than 537cP. ν 0,7 h 1,5 0 ( 0) ( ( )) ( 0 ) ( ( β) ) U = U sin β + U cos (18) Where: ( U 0 ) h... drift velocity equation for horizontal pipes ( U 0 ) v drift velocity equation for vertical piping 4. METHOD OF CHARACTERISTICS Liquid and gas mass conservation Equations 13 and 14, respectively, constitute a system of two partial differential equations. Using the method of characteristics we can transform them in an ordinary differential equation. This equation has the particularity to be valid only over a line called characteristic curve. A new differential equation is obtained (Equation 19). This equation can be easily integrated [10]. olume 6 Number

8 438 Model for Terrain-induced Slug Flow for High iscous Liquids D( α ) C D ( ) r α ρ r g + (1 αr 0 ) = 0 Dt ρ Dt g (19) Equation 19 is valid only on the characteristic line, defined by: dz dt = sg α r (20) Applying the method of characteristics, the answer of Equation 19 will be the value k, which will be constant over the characteristic line ρgr αgr k = C α 1 0 gr (21) For the case study, the characteristic curve determines the position of the front of gas for a given time. The real gas velocity does not remain constant over time or space. This means that the gas particles in a time t had a particle location and real velocity different to a time t+δt. This implies that the slope of the characteristic curve of the system will not remain constant. The surface velocities of liquid and gas for each position and time can be calculated using Equations 22 and 23. t+δt sg z+δz t+δt r z+δz α t+δt t s z z ρ C 1 t t 0 αr z z Δ g g +Δ +Δ t αr z Δt ρg z = t t t t z ρ z 1+ C0 αr z z Δ +Δ +Δ g 1 +Δ t t Δt +Δ ρ g z+δz t+δ t z+δz +Δt +Δz (22) 1 t+δt = U C 0 αr z+δz t+δ t sl z+δz t+δt sl z+δz t+δ t 0 sl z+δz (23) Z dt = const Z 3,3 Z 2,3 Z 3,3 Sep t 3 dz 3,3 k1 Z 2,3 Sep t 2 Z 2,2 Z 2,2 k 2 Z 1,t = 0 t 1 t 2 t 3 t Z i,t = Z node, time Figure 7: Illustration of the characteristic method. Journal of Computational Multiphase Flows

9 Andrea Shmueli, Diana C. González, Abraham A. Parra and Miguel A. Asuaje NUMERICAL SOLUTION For the numerical resolution the void fraction a p must be calculated. An iterative procedure based on the model proposed by Taitel & Duckler [11], for stratified flow balance was used. The model is valid for slightly inclined pipes (-10 < θ < 10 ), steady state fully developed flow and use the dimensionless parameter of Lockhart and Martinelli X2 and the dimensionless parameter-tilt angle Y as part of its calculation procedure. The resolution method for estimating the void fraction can be consulted in [12]. At all stages the pressure in the separator is assumed to be constant. For each cycle stage, a Δt mesh dependency study was required so the results did not depend on the variation in time. A detailed explanation of the numerical solution procedure can be found in [10]. 5.1 Numerical solution for Stage 1 The Equations 5, 6 and 11 constitute a system of three equations with three unknowns (P t+δt, x t+δt, z t+δt ). The fluid penetration rate in the riser can be calculated as the variation in time of the liquid level in the riser. Two initial conditions are required to solve this system: The initial level of liquid in the riser and in the downwards pipe represents the amount of fluid that returns after completion of the stage 4. Estimating the remaining liquid in the pipe is difficult, so for the first cycle, it was proposed that the riser and downward pipe had the same liquid level and these were equivalent to 5% of the height of the riser. For the second cycle, the liquid returns could be estimated more accurately with the volume of liquid remaining in the riser at the end of the previous cycle. The separator pressure, the gas and liquid superficial velocities at the inlet of the downwards pipe remain invariant over time. When z t+δt = Zr, it is considered that stage 1 has finished and stage 2 begins. The system of equations was solved with the Newton Raphson method. The greatest difficulty presented during the resolution of this step was to estimate the viscous force in the momentum equation, because the initial friction factor value is not known. To solve this problem an iterative algorithm which allowed the calculation of the Darcy friction factor was made, and simultaneously solved the stage. This algorithm is shown in detail in [10]. 5.2 Numerical solution for Stage 2 The equations system form by 1, 6 and 12 with three unknowns (P t+δt, x t+δt, ) was solved similarly to stage 1, with the Newton Raphson method. The Darcy friction factor was updated for each time step so as in step 1, an iterative procedure was required in order to obtain the exact value of the viscous force The initial conditions required for step 2 are the final conditions obtained in step 1 for the pressure, liquid level in the negative slope line, fluid penetration rate in the riser. When x t+δt 0 stage 2 is completed and stage 3 starts. 5.3 Numerical solution for Stages 3 and 4 The equations used for stage 3 and 4 are ideal gas state equation to calculate the gas density and Equations 15, Boundary conditions The boundary conditions applied on the domain for stages 3 and 4 were: -Rise base position: z(1,t) = 0. -Surface velocities of liquid and gas at the riser inlet: Obtained from Equations 1 and 3 particularized to this stage. slpen = d 1 = P g α ( P ( L x)) dt P slo sgpen o s o p p p (24) (25) olume 6 Number

10 440 Model for Terrain-induced Slug Flow for High iscous Liquids As the gas has reached the top of the riser at stage 4, the same pressure P(i, t) is set as the separator pressure and the level remained equal to the riser length Initial conditions The initial conditions required for stage 3 are the final conditions obtained in stage 2. From the final terms in stage 2, it is necessary to calculate: - Initial pressure at the base of the riser, using Equation 27. -Density of gas at the base of the riser (Ideal gas Equation). -Superficial liquid and gas velocity (boundary conditions) Equations 24 and 25. -oid fraction. (Equation 16). -Calculate the constant k. (Equation 21). -For stage 4 the initial conditions are the conditions at the end of stage Stop conditions When the gas has reached the height of the riser Zi = Zr, stage 3 ends and stage 4 begins. The end of stage 4 occurs when the liquid velocity at the top of the riser is negative. This marks the return of the liquid to the base of the riser and the beginning of a new cycle Calculation procedure Having defined the initial conditions, the numerical resolution for each time can be summarized in the steps shown below: For each solution node: -Establish coordinates z(i,t) (Equation 20) -Assume a viscous term value Ev(i,t) -Calculate in descending from the top node to the base node: -Calculate the pressure P(i,t) (Equation 15) -Calculate the gas density ρ g (i,t) with ideal gas Equation -Calculate the void fraction a r (i,t) (Equation 21) -If i=1, calculate the constant k with the void fraction values obtained with Equation 21. For nodes that need it, apply the correction in the calculation of densities. (Hernández, 2008) Calculate the superficial velocities of gas and liquid from the node at the base of the riser to the top node. (Equations 24 and 25) Determine the iscous term Ev(i, t) in each node using the previously calculated variables. The viscous term is shown in Equation 26, obtained from Equation 8 in the combined momentum equation. = ρ L f L L Ev dzn (26) 2 Φ The lengths of the elements dzn shown in Equation 15 corresponds to the size of the differential volume that owns the node solution. The solution information of each node represents the average value of the variables associated with the differential volume. Figure 8 shown an example of the discretization used on the riser. After calculating the new value of the viscous term for each node, this is compared with the initially assumed value. The iterative cycle for calculating the nodal unknowns corresponding to a particular time step culminate once the difference between the values assumed and calculated for the viscous term is less than 1e ALIDATION In order to validate the present model a comparison is made with experimental data existing in the literature: Fabre et al. [4] and Tengesdal et al. [13]. A mesh dependency study was carried out to guarantee that the results are not dependent on grid size. The optimal mesh was selected when the percentage difference between results of two successive grids was less than 3% [10]. 6.1 Fabre s experimental data Fabre et al [4] conducted experiments in a test bench of 2 ID. The pipe was 23 meters long with a slope of 0.57 Riser followed by 13.5m high. The working fluids used in the liquid and gaseous phase were water and air respectively. Journal of Computational Multiphase Flows

11 Andrea Shmueli, Diana C. González, Abraham A. Parra and Miguel A. Asuaje 441 Differential control volume associated with each node 1 dzn(i-1) dzn(i) dzn(i+1) i 1 i i+1 z(i)-z(i-1) 2 z(i+1)-z(i) 2 Figure 8: Discretization of the riser in Differential Control olumes Associated to Resolution Nodes Fabre et al. (1990) exp. Model Pressure at riser base (Pa) Time (s) Figure 9: Comparison between the model and experimental data of Fabre. The pressure at the base of the riser was used to compare the model with the experimental data. The results are presented in Figure 9. The proposed model underestimates the severe slug cycle by 8.5 s. However, it accurately predicted stages 1 and 2, obtaining a maximum percentage difference of 2% between the model and the experimental results. The predicted duration of stages 3 and 4 are lower than the experimental one by a percentage difference of 6%. 6.2 Tengesdal s experimental data Tengesdal et al. [13] conducted several experiments with a 3 ID test pipe. The test loop had 19.8m inclined pipe and can be tilted from 0 to -5. Following, a 14.9m riser height. The working fluids for liquid and gas were Crystex AF-M (refined mineral oil) and air respectively. Two cases, reported on Table 1, were studied. olume 6 Number

12 442 Model for Terrain-induced Slug Flow for High iscous Liquids Table 1. Operating conditions, Tengesdal et al. (2003) Case a Case b Superficial Liquid elocity[m/s] 0,32 0,50 Superficial gas velocity [m/s] 0,35 1,00 Inclination [ ] -1,00-5,00 Figure 10 (a) and (b) shows the comparison of the pressure in the riser base of the model calculated and reported by Tengensdal et al [13]. A good agreement is observed, with a maximum percentage derivation of 6% for stages 1 and 2, and 3% for stages 3 and RESULTS AND DISCUSSION 7.1 Effect of the liquid viscosity In order to evaluate the effect of the viscosity on the severe slug cycle, different liquids were used with viscosities greater and densities lower than water. Table 3 shows a list of the fluids used and their main properties. The pipeline-riser configuration is shown in Table 2. Presión en la base del riser (Pa) Exp_Case a Model Time (s) Case a Pressure at the riser base (Pa) Exp_Case b Model Time (s) Case b Figure 10: Comparison between the model and experimental data of Tengensdal et al. [13]. Table 2. The pipeline-riser configuration for simulated cases Parameter Unit alue Superficial Liquid elocity [m/s] 0,2 Superficial Gas elocity [m/s] 0,48 Pipeline length [m] 19,8 Pipeline inclination [ ] 3 Pipeline diameter [in] 3 Riser length [m] 14.9 Riser inclination [ ] 90 Riser diameter [in] 3 Table 3. Physical properties of the liquids used as reference for the Liquid Phase on the C Liquid Density ρ [Kg/m3] μ [cp] Water 1000,0 0,998 kerosene 809,00 1,6 Oil Oil Oil Oil Journal of Computational Multiphase Flows

13 Andrea Shmueli, Diana C. González, Abraham A. Parra and Miguel A. Asuaje 443 Figure 11 shows the pressure in the bottom of the riser for U sl = 0.2 m/s and U sg = 0.48 m/s for the fluids in Table 3. In general all the tested oils shown a shorter period in comparison with the water case. This reduction is clearly observed on the first two stages of the severe slugging cycle. The density of the fluid affects directly the maximum cycle pressure. However when increasing the viscosity, the maximum cycle pressure increases slightly. The biggest differences can be seen on the first stage of the severe slugging cycle. Big differences can be noticed when comparing Oil 1 (μ = 1.6 cp) with Oil 4 (μ = 311 cp). These differences do not seem to change in a linear way. When solving the cycle equation, the liquid viscosity is included in the momentum equations viscous term and in the calculation of the liquid hold up in the downwards pipe with stratified flow. A comparison of the current model with the original proposed by Fabre et al [4] was done in order to verify the importance of the viscous term compared to the gravity term on the momentum equation. Figure 12 (a) and (b) shows the effect of the inclusion of the viscous term in the resolution of the equations. From Figure 12 was determined that the inclusion of the viscous term for liquids with low viscosities will not have a significant effect. However, the minimum pressure (end of the cycle) will be underestimated if the viscous term is not included on the momentum equations (13% for the simulated case). Neglecting this term for liquids with viscosities greater than 311cp can cause greater deviation in the duration of stages 3 and 4. Pressure at the riser base (KPa) Oil_4 311 cp Oil_3 200 cp Oil_2 92 cp Oil cp Water Kerosene Time (s) Figure 11: Pressure at riser bottom vs time for different fluids. (a) Usl = 0,2m/s y Usg = 0,48m/s. Pressure at the riser base (KPa) With viscous term Without viscous term Pressure at the riser bae (KPa) With viscous term Without viscous term Time (s) Time (s) Water (m = cp, r = 1000 Kg/m 3 ) Oil_4 (m = 311 cp, r = 878 Kg/m 3 ) Figure 12: Comparison of the Pressure at Riser Bottom vs Time with and without the viscous term. (a) water (b) oil. olume 6 Number

14 444 Model for Terrain-induced Slug Flow for High iscous Liquids Table 4. Physical properties of the extra liquids used as reference for the liquid Phase on holdup C Liquid Density ρ [Kg/m 3 ] μ [cp] Oil Oil Oil The liquid hold up in the downwards pipe with stratified flow is estimated using the model from Taitel and Duckler [11]. Liquid hold up values for all the tested fluids in Table 3 and for the fluids shown in Table 4 are calculated and show an increment when increasing the liquid viscosity. This increment causes an augmentation of the compressibility of the gas during the stage 1 promoting a decreasing in the duration of the stage. On Figure 13 the calculated hold up values are plotted against the superficial liquid Reynolds number. It is possible to distinguish two regions on the plot. The liquid hold up increases rapidly for Re <100. The viscosity associated with this Reynolds number is close to 150 cp. The change of the line slope in Figure 13 can be used to determine when the viscous forces will start having a significant effect on the behavior of the severe slug cycle for a determined pipeline-riser configuration. 7.2 Effect of the riser inclination angle The inclination angle of the riser was varied to study the effect of this parameter on the slug flow behavior. Figure 14 shows the pressure at the base of the riser at different angles with water, the superficial liquid velocity was U sl = 0,127 m/s and the superficial gas velocity was U sg = 0,2 m/s. The behavior of the cycle was similar for all the studied liquids. The severe slug cycle time decreases when the inclination angle of the riser decreases. The smallest angle reported for each case corresponds to the minimum possible before being a case of continuous gas penetration into the pipe. 7.3 Drift velocity correlation: The effect on the severe slugging cycle of two drift velocity correlations was compared. The prediction from the drift velocity correlation by Gokcal et al. [9] is that the velocity will decrease while increasing the liquid viscosity. Dumistrescu [8] correlation does not consider the viscosity effect. Significant differences (up to 50%) were found when comparing the drift velocity values from both correlations with different riser inclinations. This is attributed that Dumistrescu [8] correlation was developed for vertical flows Liquid hold up (-) Re sl (-) Figure 13: Liquid hold up vs. Re sl. Journal of Computational Multiphase Flows

15 Andrea Shmueli, Diana C. González, Abraham A. Parra and Miguel A. Asuaje 445 Pressure at the riser base (KPa) For the geometrical conditions and studied liquid viscosities, no differences were found on the severe slugging cycle behaviour when using either Dumistrescu [8] or Gokcal et al. [9] in the present model closure relationship. 8. CONCLUSIONS This paper introduces and extends a model for the study of the phenomenon of the severe or terrain slugging. This model adapts the work of Hernandez [6] viscous effects of the liquid phase. -The model was validated with experimental results reported by Fabre et al [4] and Tengesdal et al. [13] and obtained a good comparison between the two. The maximum percentage difference was 6%. -A study of mesh dependency was made for each cycle stages to ensure that the results obtained by the method are independent of the numerical discretization used. -The liquid viscosity will have an important effect on the severe slugging cycle. The significance of this effect will be conditioned by the liquid hold up value on the downwards pipe with stratified flow. -For the studied conditions and fluids the biggest impact due to the liquid viscosity will be occur for superficial Reynolds numbers smaller than Increasing the density of the liquid phase at constant viscosity implies that the maximum pressure and cycle duration also increase. -The viscous term in the combined momentum equations is negligible for low liquid viscosities. -Neglecting the viscous term in the equations for liquids with viscosities greater than 311 cp may cause significant differences on the characteristics of severe slugging cycle. -In order for the model to more reliably predict the physics of the problem, Gokcal et al [9] correlation for the drift velocity was used. This correlation considers the effect of viscous liquid phase. -Dumitrescu [8] correlation do not correctly predicts the drift rate for angles different that 90. -The correlation of the drift rate used has no influence on severe slug behavior for the studied geometric, operating conditions and liquid viscosities. -For liquids with viscosities from 1 cp to 311 cp, as the riser angle inclination decreases, so do the severe slug cycle duration and maximum cycle pressure. REFERENCES [1] Schmidt, Z., Doty, D. R., & Dutta-Roy, K. Severe slugging in offshore pipeline-riser pipe system. SPE Journal (12334), (1985) [2] Schmidt, Z. B. Experimental Study of Severe Slugging in a Two Phase Flow Pipeline Riser Pipe System. SPE (8306- PA), (1980) [3] Taitel, Y. Stability of Severe Slugging. International Journal of Multiphase Flow, 12 (2), (1986) [4] Fabre, J., Peresson, L. L., Corteville, J., & Odello. Severe slugging in pipeline/riser system. SPE (16846), (1990) olume 6 Number Time (s) Figure 14: Pressure at Riser Bottom for different inclination angles β.

16 446 Model for Terrain-induced Slug Flow for High iscous Liquids [5] Sarica, C., & Shoham, O. A simplified transient model for pipeline-riser systems. Chemical Engineering Science, 46 (9), (1991) [6] Hernández, G., Asuaje, M., Kenyery, F., & Tremante, A. Two Phase flow Transient Simulation of Severe Slugging in Pipeline-Riser Systems. Computational Methods in Multiphase Flow I. (2007) [7] Zuber, N., & Findlay, J. A. Average volumetric concentration in two-phase flow systems. J. Heat Transfer, C87, (1965). [8] Dumitrescu, D. T. Strömung an einer Luftblase im senkrechten Rohr. Mat Mech, 23, (1943) [9] Gokcal, B., Al-Sarkhi, A. S., & Sarica, C. Effects of High iscosity on Drift elocity for Upward Inclined Pipes. SPE (115342). (2008) [10] Shmueli, A. Simulación del Flujo Tapón Severo e Inducido por el Terreno Para Líquidos iscosos. MSc. thesis Caracas, enezuela: Universidad Simón Bolívar. (2009) [11] Taitel, Y., & Duckler, A. E. A Model for Predicting Flor Regime Transitions in Horizontal and Near Horizontal Gas Liquid Flow. AIChE Journal, 22, Pp (1976) [12] Shoham, O. (2005). Mechanistic Modeling of Gas-Liquid Two-Phase Flow in Pipes. Texas, United States: Society of Petroleum Engineers [13] Tengesdal, J. Ø., Thompson, L., & Sarica, C. A Design Approach for a Self-Lifting Method To Eliminate Severe Slugging in Offshore Production Systems. SPE ( 84227). (2003) Journal of Computational Multiphase Flows