Accepted Manuscript Mechanistic modeling of critical sand deposition velocity in gas-liquid stratified flow Ramin Dabirian, Ram S. Mohan, Ovadia Shoham PII: S0920-4105(16)31228-1 DOI: 10.1016/j.petrol.2017.06.006 Reference: PETROL 4018 To appear in: Journal of Petroleum Science and Engineering Received Date: 5 December 2016 Revised Date: 13 April 2017 Accepted Date: 2 June 2017 Please cite this article as: Dabirian, R., Mohan, R.S., Shoham, O., Mechanistic modeling of critical sand deposition velocity in gas-liquid stratified flow, Journal of Petroleum Science and Engineering (2017), doi: 10.1016/j.petrol.2017.06.006. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT RI PT Mechanistic Modeling of Critical Sand Deposition Velocity in Gas-Liquid Stratified Flow SC Ramin Dabirian Tulsa University Separation Technology Projects McDougall School of Petroleum Engineering The University of Tulsa Tulsa, OK, USA ramin-dabirian@utulsa.edu M AN U Ram S. Mohan Tulsa University Separation Technology Projects Mechanical Engineering Department The University of Tulsa Tulsa, OK, USA ram-mohan@utulsa.edu Abstract TE D Ovadia Shoham Tulsa University Separation Technology Projects McDougall School of Petroleum Engineering The University of Tulsa Tulsa, OK, USA ovadia-shoham@utulsa.edu EP Among the different regimes in multiphase pipelines, stratified flow is most prone to sand AC C deposition, owing to the low liquid velocity and the lack of mixing. The main goal of this study is to predict the critical sand deposition velocity under stratified flow. The critical sand deposition velocity is defined as the minimum velocity that keeps all particles moving at all time, above which there is no stationary bed at all. Two models with similar approaches are developed based on the forces acting on a particle, such as the drag, turbulent, apparent gravity, van der Waals and lift forces for prediction of the transitions between moving/stationary dunes and moving/stationary bed. Dune pilling up ACCEPTED MANUSCRIPT and collapsing mechanism considering a torque balance on a rolling particle located at the top of the dune is adopted for stationary dunes. Torque balance applied to a rolling particle located on the lowest stratum of the moving bed layer is used to predict the minimum liquid film velocity Comparison between the predictions of the two RI PT for particle to grow a stationary bed. mechanistic models and the experimental data show good agreement, with critical sand deposition velocity absolute relative errors at 16.4% for stationary dunes and 12.6% for SC stationary bed. M AN U Introduction Solid particle hydraulic transport is one of the methods that has been applied by the industry for many years for handling sand generated during petroleum production. This mode of operation is environmentally friendly and also has low operation and maintenance costs. Research on sand transport in multiphase flow has recently gained attention in the Petroleum TE D Industry, as it has some important applications. These include cutting transport in wellbore, abrasive slurry jet drilling (Padsalgikar, 2015), and sand transport in multiphase flow pipeline EP (Dabirian et al. 2016). Several researchers attempted to extend the modeling of liquid-solid flow to gas-liquid-solid flow. However, in multiphase flow, owing to the complexity of the physical AC C phenomena, various flow patterns occur, such as stratified, slug, annular and dispersed bubble. As the each of the flow patterns exhibits a different sand transport mechanism, extension of models developed for liquid-solid flow to gas-liquid-solid flow is questionable. Liquid-Solid Flow: Durand and Condolios (1952) developed the first correlation for the minimum liquid velocity needed to suspend particles in horizontal pipeline. A series of equations to predict the critical velocity for slurry flow was developed by Thomas (1962), considering the ratio the particle diameter to viscous sub-layer thickness. Later, a model based on the forces ACCEPTED MANUSCRIPT acting on a single particle was developed by Wicks (1971), suggesting that the moment created by lift, drag, and buoyancy forces that cause a single particle to rotate should overcome moment created by gravity force. Oroskar and Turian (1980) developed a correlation for critical velocity RI PT to keep a particle suspended, based on isotropic turbulence theory, considering the component of the energy in the vertical direction versus gravity. A model for minimum velocity to keep particles completely suspended in the liquid phase was developed by Davies (1987). He stated SC that only eddies with approximate size of particles are effective in particle transport. M AN U A two-layer model for prediction of the “limit of stationary deposition velocity” (critical velocity) was developed by Wilson et al. (1992). The model is based on an equilibrium between driving and resisting forces, which act on a sand bed. It is widely applied for hydraulic transport of solid in pipeline, especially for the sliding bed regime. Similarly, Doron et al. (1987) developed a 2-layer model for the prediction of limit of stationary deposition velocity. Later, TE D Doron and Barnea (1993) extended the Doron et al. (1987) model to a 3-layer model, including both a stationary bed layer and a moving bed layer on top of it. The 3-layer model resolved the poor prediction of the critical velocity, obtained by the 2-layer model, for low slurry flow rate. EP Lahiri (2010) developed a model to predict the critical suspension velocity based on 800 The model was developed utilizing artificial neural AC C data points collected from literatures. network and support vector regression. More recently, Miedema (2012) published a comprehensive book on slurry transport fundamentals and historical overview. Also presented is an outstanding framework for slurry transport, which is applicable for dredging operation using water as the carrier liquid and sand or gravel as the solids. The framework includes a methodology to determine the slurry head loss, as well as the limit deposit velocity, for spatial volumetric concentration. ACCEPTED MANUSCRIPT Gas-Liquid-Solids Flow: Holte et al. (1987) modified the Wicks (1971) model developed for liquid-solid flow, extending it to stratified flow. A similar approach was presented by Salama (2000), for the critical sand deposition velocity to completely suspend particles in the liquid 0.04 ⋅d 0.17 P ⋅υ −0.09 L  ∆ρ  ⋅    ρL  0.55 ⋅ D0.47 .......................................................................................... (1) SC VM ,C V  =  SL   VM  RI PT phase, by applying the liquid superficial and mixture velocities, as follows where VM,C is the critical mixture velocity needed to suspend all particles. Note that the effect of M AN U particle concentration is not considered in this model. Another extension of single phase liquid model for the critical velocity to gas-liquid flow, utilizing the drift flux model, was presented by Danielson (2007). The author assumed constant (terminal) velocity between the solid particles and the liquid phase, ignoring the effect of particle TE D concentration. The developed correlation for the critical velocity, VC , for particle suspension is as follows −1 / 9 EP VC = 0.23υ L ⋅ d P1/ 9 ⋅ ( g ⋅ D ⋅ (s − 1)) 5 / 9 ............................................................................................ (2) where s is the density ratio of solid to liquid phases. A model for the prediction of the minimum velocity needed to completely suspend the AC C particles, based on force balances on a particle in vertical and horizontal directions, was published by Ponagandla (2008). The drag and lift force coefficients were solved iteratively, utilizing the Stokes’ equation and the turbulent velocity fluctuations. Hill (2011) developed a correlation to predict the minimum velocity required to keep all particles moving, extending the Oroskar and Turian (1980), which utilizes the liquid film velocity instead of the single phase full pipe velocity, namely ACCEPTED MANUSCRIPT = 1.85CV g d P ( ρ P / ρ L − 1) where N Re = 0.1536 Dρ L g d P ( ρ P / ρ L − 1) µL (1 − CV ) 0.3564  dP    D −0.378 N Re 0.09 ................................................. (3) RI PT VC ..................................................................................................... (4) where CV is the particle volume concentration. SC Ibarra et al. (2014 and 2016) developed an empirical correlation to predict the minimum mixture velocity for sand deposition in stratified flow, combining both Oroskar and Turian M AN U (1980) and Salama (2000) models. The correlation includes four empirical constants obtained by regression analysis of the acquired experimental data and a modification of Chisholm (1976) liquid holdup, yielding −0.378  VSL  −35.49  d P  0.09   = 1.3277⋅  ⋅ (1 − CV ) ⋅  ⋅ N Re, ........................................... (5) crit  g d P ( ρ P / ρ L − 1) D  VM ,crit  where NRe,crit is given in Equation (4). TE D VM ,C As can be seen from the literature review, previously published models were developed EP for “bed” in general, not distinguishing between Stationary Dune and Stationary Bed. Based on the current study data (refer to Figure 1) these two configurations have different particle AC C deposition mechanisms. The current study analyzes these two configurations separately, developing the deposition velocities for these two sand flow regimes. The critical sand deposition velocity is defined as the minimum velocity that keeps all particles moving at all time, above which either the dune or bed is sliding, whereby below it either one is stationary. The previously published studies for gas-liquid-solid are either empirical correlations, extensions of liquid-solid models or semi-mechanistic models. On the other hand, the current study includes two mechanistic models, based on the physical phenomena, for Stationary Dunes ACCEPTED MANUSCRIPT and Stationary Bed separately. In general, mechanistic models can be scaled up with more confidence, as compared to either empirical or semi-mechanistic models. The developed models include the important flow variables, some of which were neglected by previous studies, such as RI PT liquid viscosity, particle size and concentration, stratified flow characteristics (liquid holdup and M AN U SC phase velocities). TE D Figure 1. Stratified Flow Sand Regimes for Stationary Particles Modeling EP In this study, two models are developed for prediction of the critical sand deposition velocity, including the transitions between Moving Dunes - Stationary Dunes and Moving Bed - AC C Stationary Bed. Both proposed models are based on a torque balance derived from the forces acting on a particle. For the transitions between Moving Dunes - Stationary Dunes the analyzed particle is located on top of the dune, while for the transition between Moving Bed - Stationary Bed the particle is located at the bottom of the moving bed layer. Dabirian (2016) may be referred for more details. Figure 2 presents the flow chart of the approach applied for the prediction of the critical sand deposition velocities. The proposed models are based on five steps: (1) For a given ACCEPTED MANUSCRIPT superficial liquid velocity, the superficial gas velocity is reduced step by step. Correspondingly, the liquid holdup ( H L ) is predicted by using the Taitel and Dukler model (1976) with stratified wavy interfacial shear stress with an interfacial friction factor of f I = 0 .0142 ; (2) The liquid RI PT film velocity, which is the critical sand deposition velocity, is calculated by VL =VSL / HL ; (3) The forces acting on the particle, including the drag, turbulent, apparent weight, van der Waals and lift forces are determined; and (4) Equating the net torque balance to zero, the resulting liquid AC C EP TE D M AN U SC velocity is identified as the critical sand deposition velocity. Figure 2. Computational Algorithm for Critical Sand Deposition Velocity ACCEPTED MANUSCRIPT Liquid Holdup Taitel and Dukler (1976) model is the most applicable model used for characterizing stratified wavy flow in industry. The model is developed from a combination of gas and liquid RI PT momentums, and the combined momentum is an implicit equation for the liquid level in the pipe. The model is applicable for steady-state, fully-developed, and Newtonian flow, and the wettability effect is ignored. Gas-liquid interface is assumed to be flat, and the liquid holdup is τ WG M AN U momentum of gas and liquid phases calculated as SC calculated based on the liquid thickness. The liquid height is calculated based on the combined  1 SG S 1   − (ρ L − ρ G )g sin (α ) = 0 .......... .......... .......... .......... ..... (6) − τ WL L + τ I S I  + AG AL  AL AG  where S G , S L and S I are the perimeter lengths of the gas, liquid and interface, respectively. Also, AG and AL represent the cross-sectional area occupied by the gas and liquid phases, TE D respectively. ρ G and ρ L symbolize the gas and liquid densities, respectively. τ WG , τ WL and τ I are gas, liquid and interfacial shear stresses, respectively, and they are determined as 1 f G ρ GVG2 ................................................................................................................... (7) 2 1 τ WL = f L ρ LVL2 ...................................................................................................................... (8) 2 1 τ I = f I ρ G (VG − VI )2 .......................................................................................................... (9) 2 AC C EP τ WG = where V L is the liquid velocity, VG is the gas velocity, and V I is the interface velocity, which is ignored since VI << VG . f G , f L and f I represent the friction factors for gas, liquid and interface, respectively. The friction factors for gas and liquid phases are calculated based on Blasius equations. Taitel and Dukler (1976) suggested that the interfacial shear stress for stratified smooth is the same as the gas wall shear stress. For stratified wavy flow with small ACCEPTED MANUSCRIPT amplitude waves, a constant value for f I = 0 . 0142 originally suggested by Cohen and Hanratty (1968) and later used by Shoham and Taitel (1984) is applied for all cases. RI PT Forces Acting on Particle The forces acting on a single particle can be divided into two groups. The first group includes the forces, such as drag, lift and turbulent, which promote the movement of the particles SC along the pipe. The second group consists of the forces that resist the movement of the particles. These forces are considered for modeling of the critical sand deposition velocities for both • M AN U Stationary Dunes and Stationary Bed. The equations of these forces are presented next. Drag Force ( FD ) This force is exerted on a particle by the fluid in the same direction of the flow and is given as TE D 1 2 FD = ρ L CD APP VLL ......................................................................................................... (10) 2 where ρ L is the liquid density, VLL is the local liquid velocity, APP is the area of the particle EP that is projected on a plane normal to the flow direction and CD is the drag coefficient. The drag coefficient is determined by White (1991), namely, 24 6 + + 0.4 ................................................................................................ ReP 1 + Re2P AC C CD = (11) The particle Reynolds number, ReP , is defined as ρ LVLL d P .................................................................................................................. (12) µL where dP is the particle diameter and µL is the liquid dynamic viscosity. It is assumed that the ReP = movement of the particle is affected by the velocity profile upstream of it. The local velocity, ACCEPTED MANUSCRIPT VLL , is calculated based on the “law of the wall” developed by Leif (1972), which is valid through the viscous sublayer and the turbulent boundary layers. It is defined as + RI PT 1 1 1  +  + + + + + y + = VLL + a ekVLL −1 − kVLL − (kVLL ) 2 − (kVLL )3 − (kVLL )4  .................................. (13) 2 6 24   + where a = 0.1108 and k is 0.4. The dimensionless distance, y , and V LL , the dimensionless local flow velocity, are, respectively ............................................................................................................................... (14) υL VLL = SC + y Vτ VLL Vτ ............................................................................................................................. (15) M AN U y+ = υ L is kinematic viscosity and the friction velocity, Vτ , is calculated by Vτ = VLL f 8 ......................................................................................................................... (16) The moody friction factor, f , is calculated based on Hall (1957) as follows TE D 1   106  3   4 ε f = 0.00551 + 2 ×10 +   .................................................................................... (17) D Re     For the calculation of the drag force from Eq. 10, the determination of the local velocity, AC C EP VLL , follows Ramadan et al. (2001), as shown schematically in Figure 3. Figure 3. Velocity Field ahead of Particle ACCEPTED MANUSCRIPT As shown in the figure, the distance y is taken from the center of the analyzed particle, as the particle is partially situated between the particles beside it. In Eq. 10, APP is particle projected area to the flow direction of the upper part of the particle, which is determined by RI PT  πd 2   β sin(2β )  APP =  P  0.5 + + ...................................................................................... (18) 180 2π   4  where β is the repose angle of the particles. Finally, it is proposed that the roughness (ε) used Turbulent Force ( FT ) M AN U • SC in the calculation of the friction factor in Eq. 17 is approximated by the radius of the particle. Portion of the turbulence generated by the fluid suspends the particles against gravity. According to Oroskar and Turian (1980), the turbulence is negligible when either laminar flow exists in the pipe or the particle size is less than the viscous sublayer ( δ ) . Otherwise, the TE D turbulent force is calculated based on the fraction of the turbulent energy that is available to suspend a particle in the liquid phase. According to this model, it is assumed that part of the turbulent energy is dissipated as heat, which will not contribute to suspending the particles. Thus, EP the turbulence portion that causes particle suspension is reduced by a factor, w. The turbulent force and the factor, w, are given by and AC C if Re < 2100 or d P < δ 0,  FT =  1 2 otherwise  4 ρ LVτ AP w, ...................................................................................  4γ 2  ∞  4γ 2   2  2  + ∫ exp − dγ  ................................................................ γ exp −  π π  π  γ  π   The flow Reynolds number is defined as w= (19) (20) ACCEPTED MANUSCRIPT Re = ρ LVL d L µL .................................................................................................................. (21) where d L is the liquid phase hydraulic diameter and γ is given by VS (1 − CV ) 2 υL .................................................................................................................. (22) RI PT γ= C V is the sand concentration and VS is settling velocity, which is calculated as gd P2 ( ρ P − ρ L ) .............................................................................................................. (23) 18µ L SC VS = • Apparent Weight Force (FG ) M AN U where ρP is the particle density. The apparent weight force is the difference between the weight and buoyancy forces, which is given as follows • π d P3 6 g(ρP − ρL ) .......................................................................................................................... (24) TE D FG = Van der Waals Force (FVDW ) EP This is attraction force between particles is significant for small particle sizes. According AC C to Cabrejos (1991) the van der Waals is defined by  A  d  FVDW =  H   P2  ............................................................................................................... (25)  12   S  where AH and S are, respectively, the Hamaker. Constant and the separation length between the particle and the wall, which are given by Rabinovich and Kalman (2009), as follows AH = 6.50×10−20 ................................................................................................................ (26) S = 6.20 ×10−9 .................................................................................................................... (27) ACCEPTED MANUSCRIPT • Lift Force (FL ) The fraction of pressure and shear stress acting on the surface of the particle constitutes RI PT the lift force. According to Saffman (1965), small particles experience lift force when the particle is moving through a relatively high viscous fluid in a low shear flow, as given by µ V d P2  dVLL    FL = 1.615 L LL υ L 0.5  dy  ........................................................................................... (28) dVLL is the local velocity gradient, which is calculated based on Ramadan et al. (2001), as dy SC where 0.5 dVLL Vτ2 dVLL = dy υ L dy + where dV LL dy + + M AN U follows ................................................................................................................ (29) + is given by Leif (1972) as + dVLL = dy+ 1 TE D ............................................... (30) 1 3 + 2 1 4 +3   kVLL+ + 2 1 + a ke − k − k VLL − k VLL − k VLL  2 6   where a = 0.1108 and k is 0.4. Wang et al. (1997) modified Saffman’s equation based on the experimental results collected by Hall (1988), namely, 0.5 EP µLVLLd P2  dVLL    F (r + ) .......................................................................................... (31) FL = 1.615 0.5 υL  dy  + AC C + where F (r ) , as given below, is valid for 1.8 < r + < 100 F (r ) = 20.9 (r + ) 2.31 101.962 log r + + 1.412 .................................................................................................... (32) and r + is given as   dP  υL + r = 2  dV LL  dy        −0.5 .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... ........ (33) ACCEPTED MANUSCRIPT Prediction of Critical Sand Deposition Velocity for Stationary Dunes Several mechanisms have been proposed for sand transport in pipelines, including RI PT suspension, sliding (dragging), lifting and rolling. According to Ramadan et al. (2001), the main mechanisms of transporting particles in flowing fluid are either lifting or rolling. Stevenson et al. (2002) developed correlations for incipient particle motion velocity for sliding and rolling. SC Ibrahim et al. (2003) considered incipient particle motion velocity for the mechanisms of suspension, sliding and rolling. Three models for dragging, lifting and rolling were developed by M AN U Byron et al. (2016), based on force balances for initiating particle motion. According to this study, two mechanisms, namely, saltation (bouncing) and rolling (climbing) play important roles AC C EP TE D in sand transport, as shown schematically in Figure 4. Figure 4. Schematic of Rolling and Saltation Mechanisms Force balances along the flow direction and normal to the flow direction are considered corresponding to the sliding and lifting forces, which are given, respectively, by FSliding = FD − FG sin(θ + α ) − F f ............................................................................................ (34) FLifting = FL + FT − FG cos(θ + α ) − FVWD .......... .......... .......... .......... .......... .......... .......... .... (35 ) where (θ ) is the front angle of the dune and (α ) is the pipe inclination angle. Substituting the ACCEPTED MANUSCRIPT different forces, as defined previously in Eq. (34) and Eq. (35) results in π d3 1  A  d p  2 FSliding = CD ρL APPVLL − FG sin(θ + α) − f K { P g(ρP − ρL ) cos(θ + α) +  H   2  2 6  12   S  µL VLL d p2  dVLL  1   F(r + )} .............................................................. (36) − ρL (Vτ ) 2 AP w − 1.615 0.5 4 υL  dy  FLifting = 1.615 µ L VLL d p2  dVLL  υ L 0.5 RI PT 0.5 0.5 π d3 1   F (r + ) + ρ L (Vτ ) 2 AP w − P g ( ρ P − ρ L ) cos(θ + α ) 4 6  dy  SC  A  d p  −  H   2  ............................................................................................................... (37)  12   S  M AN U where f K is the kinematic friction coefficient accounting for the effect of wet lubrication friction between two particles. The saltation mechanism is caused by the resultant of the lifting and sliding forces. The saltation happens only when the resultant angle of FLifting and FSliding is greater than the summation of TE D the front angle of a dune (θ ) and pipe inclination angle (α ) , as given by   > (θ + α ) ............................................................ (38)  EP FD − FG sin(θ + α ) − F f  arctan   FL + FT − FG cos(θ + α ) − FVWD Based on the acquired experimental data, Eq. 38 is not valid for dunes, and as a result, saltation AC C does not occur in the transition between Moving Dunes - Stationary Dunes. According to the study of US Geological Survey (Colver, 1998), sand dunes creep on the pipe bottom based on pilling up and collapsing mechanisms, which are derived by particle rolling. Figure 5 shows a schematic of the pilling up and collapsing mechanism. As shown in the figure, the sand particles role and pile up on top of the dune until the front of a dune is so steep that it collapses under its own weight. The repeating cycle of sand moving up and collapsing ACCEPTED MANUSCRIPT causes the sand dunes to move forward in the direction of the flow. Collapsing of a dune occurs when the front angle of dunes (θ ) exceeds a specific value. Based on the study of Colver (1998), this angle for wet sand is θ = 45º. In this study, it is assumed that the repose angle of M AN U SC RI PT particle is β = 60°, assuming that the particles are closely packed with nearly equal diameters. Figure 5. Schematic of Pilling up and Collapsing Mechanism of Sand Dunes TE D The directions of the forces depend on the repose angle ( β ) , dune front angle ( θ ) and pipe inclination angle ( α ) . Figure 6 shows the directions of the forces acting on a particle located AC C EP on the dune. M AN U SC RI PT ACCEPTED MANUSCRIPT Figure 6. Forces Acting on Single Particle for the Case of Sand Dunes The net torque can represent the rolling tendency of a particle located on the top of a sand TE D dune. The torque balance around point O (see Figure 6) is given by Tnet = L D ( FD ) − LG ( FG ) + LT , L ,VWD ( FT + FL − FVWD ) .......... .......... .......... .......... .......... .... (39 ) EP For the dune case, the following equations represent the levers of the drag force (LD ) , turbulent force ( LT ) , lift force (LL ) , Van Der Waals force (LVWD) and apparent weight force (LG) , LD = AC C namely, dP sin (β ) ..................................................................................................................... (40) 2 LT = LL = LVWD = LT ,L,VDW = LG = dP cos(β ) ................................................................................ (41) 2 dP cos(α + θ − β ) ......................................................................................................... (42) 2 Substituting for the forces and their corresponding levers results in the final form of the net ACCEPTED MANUSCRIPT torque, as follows  πd 3  dP d 1 2 sin( β )  C D ρ L APPVLL  − P cos (α + θ − β )  P g ( ρ P − ρ L ) .................... (43) 2 2  2  6  1 µ L VLL d p2 dP 2  + cos( β ) ρ (V ) AP w + 1.615 4 L τ 2 υ L 0.5   dVLL   dy 0.5  A  F (r + ) −  H  12   d p   2  S      RI PT Tnet = To find the transition from Moving Dunes to Stationary Dunes, two conditions must be satisfied: SC 1) The net torque balance is zero, which means that no particle rolling happens. In such conditions, the rolling up torque is equal to the torque that tends to rotate the particle M AN U downwards. 2) The sand dune is stable, which means the angle of front dune ( θ ) needs to be 0 ≤ 45°, otherwise, the dune collapses under its weight. Thus, incorporating the two conditions presented above, Eq. 43 is solved implicitly for VL TE D , which represents the critical sand deposition velocity. The effect of concentration on the critical velocity is considered only in the turbulent force. Since the experiments were conducted at low concentrations less than 0.00404 m3/m3, the effect of concentration on the turbulent force is not EP significant and can be neglected. Therefore, referring to Figure 4, for this study, the pilling up and collapsing mechanism is not a function of the particle concentration and the size of sand AC C dune; however, it is strongly depends on the steepness of the dune front. Prediction of Critical Sand Deposition Velocity for Stationary Bed The modeling approach for the prediction of the critical sand deposition velocity for Stationary Bed is similar to the one developed for the Stationary Dunes case, which was derived base on the particle rolling mechanism. However, for the case of Stationary Dunes the analyzed particle is located on top of the dune. On the other hand, for the Stationary Bed case, the ACCEPTED MANUSCRIPT analyzed particle is located at the bottom of the lowest stratum of the moving bed layer, and on a stationary mono-layer of particles, as shown schematically in Figure 7. Thus, for the Stationary Bed case, the weight of the other particles in the moving bed that are located on the top of the RI PT analyzed particle is added in the analysis. This gives the value of the maximum liquid velocity, which causes the first sand particles to stop moving and deposit as a stationary mono-layer of TE D M AN U SC particles at the bottom of the pipe. EP Figure 7. Forces Acting on Analyzed Particle for the Case of Sand Bed The weight force on the particle placed at the lowest stratum of the moving bed layer AC C includes the weight of the particle itself and other particles in the moving bed layer on top of it. Therefore, the average number of solid particles that exists in the moving bed should be considered. Following Doron and Barnea (1993), y MB − d P y + 1 = CMB MB + (1 − CMB ) ...................................................................... (44) dP dP and the total submerged weight is found as N = CMB ACCEPTED MANUSCRIPT   π d P3 y FG = N FG = C MB MB + (1 − C MB ) g ( ρ P − ρ L ) ........................................................ ( 45) dP   6 where C MB is the maximum packing of particles (CMB = 0.74) , and y MB is the height of the RI PT moving bed layer. It is assumed that the moving bed height ( yMB) is proportional to the particle concentration (CV ) . The acquired experimental results show a correlation between the yMB ) , and the particle concentration (CV ) , namely, dP SC dimensionless height of a moving bed, ( M AN U y MB 0.5 = 337.8 (CV − 0.001) ............................................................................................... (46) dP The force levers of the lift, drag, turbulent and van der Waals are calculated as before (Eqs. 40 and 41), while the force lever for the apparent weigh is LG = dP cos(α − β ) ...................................................................................................................... (47) 2 Also, the torque balance for the Stationary Bed case is similar to the torque balance TE D calculated for Stationary Dunes (Eq. 43). The following equation is the final form of the torque balance for the Stationary Bed case:  π d3  dP y 1 2 d sin(β )  CD ρ L APPVLL  − P cos(α − β ) P g (ρ P − ρ L ) CMB MB + (1 − CMB ) 2 dp 2  2   6   EP Tnet = AC C 0.5 1 µ L VLL d p2  dVLL  dP  AH   d p   2 +    + cos(β ) ρ L (Vτ ) AP w + 1.615 F ( r ) −    2  ....................... (48) 0.5  dy  4 2 12 υ    S     L  Similarly, the net torque balance is set to zero and the equation is solved for V L , which presents the critical sand deposition velocity for Stationary Bed. Comparison Study Two computer codes are developed utilizing MATLAB, based on the models presented in the previous section. The input for these models/codes include the superficial gas velocity, ACCEPTED MANUSCRIPT superficial liquid velocity, pipe diameter, pipe inclination angle, liquid viscosity, and particle size and concentration. The output provides the critical sand deposition velocities for both the Stationary Dunes and Stationary Bed cases, and their corresponding liquid hold up are RI PT calculated. The developed models are compared with the experimental data collected by Dabirian et al. (2016), as well as the correlation by Ibarra et al. (2014 and 2016). Both studies were SC conducted in a 4 in. diameter horizontal pipe, utilizing air–liquid stratified flow. The Ibarra et al. M AN U (2014 and 2016) empirical correlation is selected for comparison with the developed mechanistic models predictions, because it utilizes the same definition for the critical sand deposition velocity as in this study. The current study data include the air-water-solid and air–PAC/water–solid under stratified flow. Poly Anionic Cellulose (PAC) is added to the water to increase its viscosity to 5 TE D cP. The experiments were conducted with three particle sizes of 45–90 µm , 125–250 µm and 425–600 µm . The liquid and gas superficial velocities are, respectively, VSL = 0.05 and 0.1 m/s EP and VSG = 3 m/s to 15.5 m/s. The experiments are conducted at low particle concentrations in the ranges of .0000404 to 0.00404 m3/m3. Comparisons between the experimental data and the AC C predictions of the current study model and Ibarra et al. (2014 and 2016) empirical correlation of the critical sand deposition velocities are presented in Figures 8 through 13 for air–water-solid flow, and for air–PAC/water-solid in Figures 14 to 19. • Air–Water–Solid Flow ACCEPTED MANUSCRIPT For particle size of 45–90 µm, only stationary dunes are observed for both Vsl = 0.05 and 0.1 m/s over the entire sand concentration, as shown in Figures 8 and 9, respectively. The developed stationary dunes model predicts a fairly constant critical sand deposition velocity for RI PT these data. As explained before, the pilling up and collapsing mechanism exhibits a weak function of sand concentration for the transition between Moving Dunes and Stationary Dunes. However, the experimental results show that the critical liquid film velocity increases slightly SC with concentration. In general, reasonably good agreement is exhibited between model predictions and the experimental data. On the other hand, Ibarra et al. (2014, 2016) correlation M AN U predicts a weak linear trend for the critical velocity over the entire sand concentration range. It AC C EP TE D over-predicts the critical velocity for Vsl = 0.1 m/s, while it is closer to the data for Vsl = 0.05 m/s. Figure 8. Comparison between Experimental Data and Predicted Models for Air–Water–Solid Flow with 45–90 µm at VSL = 0.05 m/s M AN U SC RI PT ACCEPTED MANUSCRIPT Figure 9. Comparison between Experimental Data and Predicted Models for Air–Water–Solid Flow with 45–90 µm at VSL = 0.1 m/s TE D Figures 10 and 11 presents the comparison for particle size of 125–250 µm, for Vsl = 0.05 and 0.1 m/s, respectively. Stationary dunes are observed only at concentrations less than 0.001 m3/m3, while for higher concentrations stationary bed occurs. As can be seen in Figure 10, the EP model for stationary dunes cannot predicts the trend of experimental data for Vsl = 0.05, showing a flat trend. On the other hand, the model for stationary bed predicts well the experimental trend AC C for the two superficial liquid velocities. Ibarra et al. (2014, 2016) correlation, again, shows a monotonic trend over entire concentration range; it over-predicts for stationary dunes, and underpredicts for stationary bed data. M AN U SC RI PT ACCEPTED MANUSCRIPT AC C EP TE D Figure 10. Comparison between Experimental Data and Predicted Models for Air–Water–Solid Flow with 125–250 µm at VSL = 0.05 m/s Figure 11. Comparison between Experimental Data and Predicted Models for Air–Water–Solid Flow with 125–250 µm at VSL = 0.1 m/s ACCEPTED MANUSCRIPT For particle size of 425–600 µm, the experimental data shows that only stationary bed is observed for all sand concentrations. As can be seen in Figures 12 and 13, the model underpredicts the critical velocity for concentrations greater than 0.0015 m3/m3. The under-prediction RI PT of the model is owing to under prediction of the drag and lift forces for these concentrations. The predictions of the model is based on a closure relationship developed for the bed height. Future measurement of the bed height will improve the predictions of the model. Again, Ibarra et al. AC C EP TE D M AN U SC shows a linear trend, exhibiting a poor agreement with experimental model. Figure 12. Comparison between Experimental Data and Predicted Models for Air–Water–Solid Flow with 425–600 µm at VSL = 0.05 m/s M AN U SC RI PT ACCEPTED MANUSCRIPT Figure 13. Comparison between Experimental Data and Predicted Models for Air–Water–Solid Flow with 425–600 µm at VSL = 0.1 m/s Air–PAC/Water-Solid Flow TE D • The effect of liquid viscosity on the critical sand deposition velocity is presented in EP Figures 14 to 19. Note that increase in viscosity results in two competing phenomena: on the one hand, it results in an increase in the drag force, which promotes the movement of a particle; AC C however, on the other hand, it causes a reduction of the turbulent force, which promotes particle deposition. The predictions of the new model generally show that for both Stationary Dunes and Stationary Bed, higher liquid viscosity results in a better agreement between the model predictions and experimental data. For the 45–90 microns particle size, as shown in Figures 14, and 15 for Vsl = 0.05 and 0.1 m/s, respectively, only stationary bed occurs. This is owing to the effect of the viscous sublayer, ACCEPTED MANUSCRIPT whereby the small particles are trapped in the layer and are not exposed to the turbulence created by the outer layer and gas-liquid interface. The model suggested by Oroskar and Turian (1980) predicts appropriately that the turbulent force is negligible for small particle. For this case, very RI PT good agreement is observed between the acquired experimental data and the model predictions. As can be seen, Ibarra et al (2014, 2016) under-predicts the critical velocity for all particle EP TE D M AN U SC concentrations. AC C Figure 14. Comparison between Experimental Data and Predicted Models for Air– PAC/Water–Solid Flow with 45–90 µm at VSL = 0.05 m/s M AN U SC RI PT ACCEPTED MANUSCRIPT Figure 15. Comparison between Experimental Data and Predicted Models for Air–PAC/Water–Solid Flow with 45–90 µm at VSL = 0.1 m/s TE D The predictions of the developed models for Stationary Dunes and Stationary Bed for the 125–250 microns particle size are presented in Figures 16 and 17 for Vsl = 0.05 and 0.1 m/s, respectively. As mentioned before, a constant critical velocity is predicted for the Stationary EP Dunes case, while the data show a slight increasing trend for concentration less than 0.001 AC C m3/m3. For the Stationary Bed, the model predicts the trend of the experimental data correctly. Ibarra et al. (2014 and 2016) shows poor agreement with the stationary bed case. M AN U SC RI PT ACCEPTED MANUSCRIPT AC C EP TE D Figure 16. Comparison between Experimental Data and Predicted Models for Air–PAC/Water–Solid Flow with 125–250 µm at VSL = 0.05 m/s Figure 17. Comparison between Experimental Data and Predicted Models for Air–PAC/Water–Solid Flow with 125–250 µm at VSL = 0.1 m/s ACCEPTED MANUSCRIPT Figures 18 and 19 presents results for the 425–600 microns particle size for Vsl = 0.05 and 0.1 m/s, respectively. As for the previous cases, a fair good agreement is observed between the RI PT model predictions and the experimental data for both Stationary Dunes and Stationary Bed. The effect of viscosity on the model predictions for both Stationary Dunes and Stationary Bed is as follows: For Stationary Dunes, the model is not sensitive to viscosity, predicting SC almost the same values for water and PAC/water flows. On the other hand, for the Stationary Bed case, the trend of the model predictions is increasing critical sand deposition velocity with M AN U increasing liquid viscosity, which is in a good agreement with the experimental data collected. Ibarra et al. (2014) model under-predicts the critical velocity over entire sand concentrations for all three particle sizes and two superficial liquid velocities. This phenomena happens because the correlation is developed based on experimental data collected for air-water, and the effect of AC C EP TE D viscosity on the critical velocity was not investigated experimentally. ACCEPTED MANUSCRIPT M AN U SC RI PT Figure 18. Comparison between Experimental Data and Predicted Models for Air–PAC/Water–Solid Flow with 425–600 µm at VSL = 0.05 m/s TE D Figure 19. Comparison between Experimental Data and Predicted Models for Air–PAC/Water–Solid Flow with 425–600 µm at VSL = 0.1 m/s Conclusion Two models are developed for the prediction of the transition between Moving Dunes to EP Stationary Dunes and Moving Bed to Stationary Bed under gas-liquid stratified flow. The models are based on the forces acting on a solid particle, such as the drag, turbulent, apparent gravity, AC C van der waals and lift forces. The liquid holdup is calculated based on Taitel and Dukler (1976) model with f I = 0 .0142 . The model developed for Stationary Dunes is based on the pilling up and collapsing mechanism of a dune, considering a torque balance on a particle located on the top of a dune. The model for Stationary Bed is based on a similar torque balance applied to a particle located on the lowest stratum of the moving bed layer. The models predict the minimum liquid film ACCEPTED MANUSCRIPT velocity that causes the analyzed particle to stop moving, and deposit as Stationary Bed/Dunes. The models enable prediction of the critical sand deposition velocity and are in good agreement when compared to the data acquired at different experimental flow conditions, such as RI PT different phase velocities, particle sizes, and particle concentrations. A comparison between model predictions and the experimental data of the critical sand deposition velocity show SC absolute relative errors are 16.4% for Stationary Dunes and 12.6% for Stationary Bed. Nomenclature a = AG = Cross-sectional area Occupied by the Gas Phase, [ m ] AH = Van Dar Waals Force Coefficient, [J] AL = Cross-sectional area Occupied by the Liquid Phase, [ m ] AP = Pipe Cross Sectional Area, [ m ] APP = Upper Area of Particle Projected on a Plane Normal, [࢓૛ ] CD = Drag Coefficient, [-] CV = Sand Concentration, [v/v] dP = Particle Diameter, [µm] D = Inside Pipe Diameter, [m] fI = Interface Friction Factor, [-] = Kinematic Friction Coefficient, [-] = Drag Force, [kg.m/࢙૛ ] = Friction Force, [kg.m/࢙૛ ] FL = Lift Force, [kg.m/࢙૛ ] FT = Turbulent Force, [kg.m/࢙૛ ] FVDW = Van Dar Waals Force, [kg.m/࢙૛ ] FD Ff M AN U 2 2 EP TE D 2 AC C fk Leif Model Coefficient ACCEPTED MANUSCRIPT = Apparent Weight Force, [kg.m/࢙૛ ] g = Acceleration due to Gravity 9.81, [m/࢙૛ ] HL = Liquid Holdup, [-] k = Leif Model Coefficient LD = Drag Force Lever, [kg.m/࢙૛ ] LL = Lift Force Lever, [kg.m/࢙૛ ] LT = Turbulent Force Lever, [kg.m/࢙૛ ] LVDW = Van Dar Waals Force Lever, [kg.m/࢙૛ ] LG = Apparent Weight Force Lever, [kg.m/࢙૛ ] N = Weight of the Particle Itself and Other Particles in the Moving Bed Layer on = Solid-Liquid Density Ratio, [-] S = Van Dar Waals Force Coefficient, [m] SG = Gas Perimeter, [m] SI = Interface Length, [m] SL = Liquid Perimeter, [m] Tnet = Net Torque, [kg.m.l/࢙૛ ] VC = Critical Liquid Film Velocity, [m/s] VLL+ EP AC C VLL TE D s VL SC M AN U Top of It, [-] RI PT FG = Liquid Film Velocity, [m/s] = Local Liquid Velocity, [m/s] = Dimensionless Local Liquid Velocity, [-] VM = Mixture Velocity, [m/s] VM ,C = Critical Mixture Velocity, [m/s] VSG = Superficial Gas Velocity, m/s ACCEPTED MANUSCRIPT VSL = Superficial Liquid Velocity, m/s w = Factor assigned for Conversion of Turbulent Energy to Heat x = Fraction of Eddies with Instantaneous Velocities Equal or Greater than the RI PT Terminal Velocity of the Particle, [-] = Distance taken from the center of the analyzed particle, [m] y+ = Dimensionless Distance, [-] Greek Letters SC y = Pipe Inclination Angle, Radians β = Particle Repose Angle, Radians θ = Angle of front Dune, Radians γ = Factor for Energy Dissipation, [࢓ି૚ ] δ = Viscous Sublayer Thickness, [m] ε = Pipe Roughness, [-] µG = Gas Viscosity, [kg/m.s] µL = Liquid Viscosity, [kg/m.s] ρG = Gas Density, [kg/࢓૜ ] ρL = Liquid Density, [kg/࢓૜ ] ρP = Sand Particle Density, [kg/࢓૜ ] τI = Interfacial Shear Stress, [kg/m.࢙૛ ] = Wall Shear Stress, [kg/m.࢙૛ ] = Gas Phase Wall Shear Stress, [kg/m.࢙૛ ] = Liquid Phase Wall Shear Stress, [kg/m.࢙૛ ] = Liquid Kinematic Viscosity, [࢓૛ /s] τWG τWL υL TE D EP AC C τW M AN U α References ACCEPTED MANUSCRIPT 1. Cabrejos, F.J.: “Incipient Motion of Solid Particles in Pneumatic Conveying. MS Thesis, University of Pittsburg, Pittsburg, PA, USA, 1991. RI PT 2. Chisholm, D.: “A Theoretical Basis for the Lockhart-Martinelli Correlation for TwoPhase Flow”, Intl. J. Heat Mass Trans., No. 12, pp. 1767-1778, 1967. 3. Cohen, S.L., and Hanratty, T.J.: “Effects of Waves at a Gas-Liquid Interface on a SC Turbulent Air Flow”, J. Fluid Mech., No. 31, pp. 467, 1968. 4. Colver T J.: Pocket Reference. Littleton, Colorado: Sequoia Publishing, Inc., 1998. M AN U 5. Dabirian, R., Mohan, R., Shoham O., and Kouba. G.: “Critical sand deposition velocity for gas-liquid stratified flow in horizontal pipes”, Journal of Natural Gas Science and Engineering, 33, 2016. 6. Dabirian, R., Mohan, R., Shoham O., and Kouba, G.: "Sand flow regimes in air-water stratified flow in a horizontal pipeline”, peer review journal of SPE Oil and Gas TE D Facilities, SPE 174960 PA, 2016. 7. Dabirian, R., Mohan, R., Shoham, O., and Kouba, G.: “Sand transport in stratified flow in 2015 EP a horizontal pipeline”, SPE Annual Technical Conference and Exhibition, Houston, USA, AC C 8. Danielson, T.J.: “Sand Transport Modeling in Multiphase Pipelines”, Offshore Technology Conference, OTC paper 18691, 2007. 9. Davies, J.T.: “Calculation of Critical Velocities to Maintain Solids in Suspension in Horizontal Pipes”, Chemical Engineering Science, Vol. 42, No. 7, Pergamon Journals Ltd., pp. 1667-1670, 1987. 10. Doron, P., & Barnea, D.: “ A three layer model for solid liquid flow in horizontal pipes. International Journal of Multiphase Flow”, Vol. 19, No.6., 1029-1043, 1993. ACCEPTED MANUSCRIPT 11. Doron, P. and Barnea, D.: “Pressure Drop and Limit Deposit Velocity for Solid-Liquid Flow in Pipes”, Journal of Chemical Engineering Science”, Vol. 50, No. 10, pp. 15951604, 1994. RI PT 12. Hill, A.L.: “ Determine the Critical Flow Rates for Low Concentration Sand Transport in Two-Phase Pipe Flow by Experimentation and Modeling” M.Sc. Thesis, The University of Tulsa, Tulsa, OK, USA, 2011. SC 13. Holte, S., Angelsen, S., Kvernvold, O., and Raeder. J.H.: “Sand Bed Formation in Group Meeting, 1987. M AN U Horizontal and Near Horizontal Gas-Liquid-Sand”, The European Two-Phase Flow 14. Ibarra, R., Mohan, R., and Shoham, O.: “Critical Solid particles Deposition Velocity in Horizontal Stratified Flow”, SPE International Symposium and Exhibition on Formation Damage, 2014. TE D 15. Ibarra, R.J., Mohan, R. and Shoham, O.: "Investigation of Critical Sand-Deposition Velocity in Horizontal Gas/Liquid Stratified Flow," SPE Production & Operations Journal, 2016. EP 16. Ibrahim, AH., Dunn, PF., and Brach, RM.: “ Microparticle Detachment from Surfaces Exposed to Turbulent Air Flow: Controlled Experiments and Modeling”, Aerosol Sci, 34: AC C 765-782, 2003. 17. Lahiri, S. K.: “Study on slurry flow modelling in pipeline”. Durgapur, India: National Institute of Technology, Durgapur, India, 2009. 18. Leif, N.P.: “Boundry Layer Theory”, Tapir, Trondheim, Norway, pp. 83-130, 1972. 19. Miedema, S.A.: “ The Delft Head Loss & Limit Deposit Velocity Model”, www.dredgining engineering.com. Delft, The Netherlands, 2012-now. ACCEPTED MANUSCRIPT 20. Oroskar, A.R., and Turian, R.M., “The Critical Velocity in Pipelines Flow of Slurries”, AIChE Journal, Vol. 26, No. 4, pp. 550-558, 1980. 21. Padsalgikar, A., “Solid Particle Transport in Gas-Liquid Stratified Slurry Flow,” Ph.D. RI PT Dissertation, The University of Tulsa, 2015. 22. Ponagandla, Veerendhar: “ Critical Deposition Velocity Method for Dispersed Sand Transport in Horizontal Flow. The University of Tulsa, Tulsa, 2008. SC 23. Ramadan, A., Skalle, P., and Johansen, S. T., Svein, J., and Saasen, A.: “Mechanistic Model for Cutting Removal from Solid Bed in Inclined Channels”, Journal of Petroleum M AN U Science and Engineering 30, 129-141, 2001. 24. Saffman, P.G.: The Lift on Small Sphere in a Slow Shear Flow”, J. Fluid Mechanic. 22, 385-400, 1965. 25. Salama, M.M.: “Sand Production Management”, ASME Transactions – Journal of TE D Energy Resources Technology, Vol. 122, No. 1, pp. 29-33, 2000. 26. Shoham, O., and Taitel, Y.: “Stratified Turbulent-Turbulent Gas Liquid Flow in Horizontal and Inclined Pipes”, AIChE Journal, Vol. 30, pp. 377, 1984. EP 27. Soepyan, B., Cremaschi, S., Sarica, Cem., and Subramani.: “Solids Transport Models Comparion and Fine-Tuning for Horizontal, Low Concentration Flow in Single-Phase AC C Carrier Flow”, AIChE Journal, Volume 60, 76-122, 2014. 28. Stevenson, P., and Thorpe, R.B., “Velocity of Isolated Particles Along a Pipe in Stratified Gas-Liquid Flow”, AIChE Journal, Vol. 48, No. 5, pp. 963-969, 2002. 29. Stevenson, P., Thrope, R. B. and Davidson, J. F.: “Incipient Motion of a Small Particle in the Viscous Boundary-Layer at a Pipe Wall”, Chemical Engineering Science, 57, 4505-4520, 2002. ACCEPTED MANUSCRIPT 30. Taitel, Y., and Dukler, A.E.: “A Model for Predicting Flow Regime Transitions in Horizontal and Near-Horizontal Gas-Liquid Flow”, AIChE Journal, Vol. 22, No. 1, 47, 1976. RI PT 31. Thomas, DG.: “Transport Characteristics of Suspensions: Part VI. Minimum Transport Velocity for Large Particle Size. AIChE Journal, 8, 3, pp. 373-378, 1962. 32. Wang, Q., Squires, K.D., Chen, M., McLaughlin, J.B.: “On the Role of Lift Force in SC Turbulence Simulations of Particle Deposition”, Int. J. Multiphase Flow 23 (4), 749-763, 1997. M AN U 33. White, F.M.: “Viscous Fluid Flow”, McGraw-Hill, New York, pp. 203-493, 1991 34. Wicks, M.: “Transport of Solids at Low Concentrations in Horizontal Pipes”, Advances in Solid-Liquid Flow in Pipes & Its Applications, I. Zandi, ed., Pergamon Press, pp. 101124, 1971. TE D 35. Wilson, K. C.: “A Unified Physically based Analysis of Solid-Liquid Pipeline Flow”. AC C EP Hydrotransport 4 (pp. A1 1-16). Banff, Alberta, Canada: BHRA Fluid Engineering, 1976. ACCEPTED MANUSCRIPT HIGHLIGHTS • Two models developed to predict the minimum liquid film velocity that causes the • RI PT analyzed particle to stop moving, and deposit as Stationary Bed/Dunes. The developed models are based on rolling mechanisms considering a torque balance on a particle located on the top of a dune or located on the lowest stratum of the • SC moving bed layer. Comparison between model predictions and the experimental data of the critical sand AC C EP TE D M AN U deposition velocity.