2D CFD modelling of the drift flux in two-phase Air ...

1

2D CFD modelling of the drift flux in two-phase Air-Newtonian slug-flow pattern flow along horizontal pipelines

Mariana Alarcón López

Department of Chemical Engineering, Universidad de los Andes, Bogotá, Colombia

General objective To study the behavior of the drift flux considering two-phase slug-flow pattern flow along horizontal pipelines for a high viscosity fluid (greater than 0.3 Pa*s), analyzing the effects of the physical properties, such as density, dynamic viscosity and surface tension, as well as the pipe length. Specific objectives

• To find an accurate setting in CFD to model the system comparing the results with experimental data.

2

2D CFD modelling of the drift flux velocity in two-phase Air-Newtonian slug-flow pattern flow along horizontal pipelines

Mariana Alarcón López

Department of Chemical Engineering, Universidad de los Andes, Bogotá, Colombia

ABSTRACT The present study analyzes the drift velocity of a synthetic oil in horizontal two-phase slug flow pipelines, by evaluating the effect of some physical properties, such as density and dynamic viscosity, and pipeline characteristics, such as the length of the pipe, due to its applications in various industries as in the O&G industry processes. This was achieved by using Computational Fluid Dynamics (CFD) tool approaches. The STAR-CCM+ software was utilized to simulate a half circular pipeline with a symmetry plane in a 2D mesh model, analyzing three different turbulence models. This model was fixed with a mesh independence test to obtain an accurate number of cells for the grid. The CFD results were compared with the experimental data gathered by the Tulsa University Fluid Flow Project (2018) research group. The drift velocity results achieved with a reasonable accuracy level in the pipeline, with error values under 15%. A dimensionless analysis for the experimental and CFD Reynolds numbers was done, concluding that the drift velocity within the pipe is dominated by viscous forces that overcome the inertial forces.

KEYWORDS: Drift flux, CFD, VOF, slug flow, two-phase flow, flow patterns.

NOMENCLATURE

API

American Petroleum Institute RANS Reynolds-Averaged Navier-Stokes

CFD

Computation Fluid Dynamics r Radius of bubble cap [m]

CFL

Convective Courant number Re Reynolds number [-]

Co Distribution parameter or flow coefficient [-]

S Slip ratio [-]

D

Pipe diameter [m] T Temperature [°C]

EMP

Eulerian Multiphase Model U" Freestream velocity [m/s]

EOR

Enhanced Oil Recovery VFG Volume Fraction of Gas [-]

Eo

Eötvös number [-] VOF Volume of Fluid

Fr

Froude number [-] v$ Drift Velocity [m/s]

g

Acceleration due to gravity [m/s2] v% Surface mixture velocity [m/s]

GDP

Gross Domestic Product v Velocity [m/s]

ID

Pipe Inner Diameter [m] v& Surface velocity [m/s]

IMD

Interface Momentum Dissipation v' Translational velocity [m/s]

K

Kinetic energy [J] x Two phase quality [-]

L

Pipe Length [m] ∆t Time step [s]

3

MMP

Eulerian Multiphase Mixture Model ∆x Length interval [m]

Nvis

Viscosity number [-]

Greek Letters α

Void Fraction [-] ρ Density [kg/m3]

ε

Dissipation rate [J/kg*s] ρ Averaged density [kg/m3]

σ

Surface tension [N/m] τ Shear stress averaged in time [Pa]

Γ Mass transfer at the interface [kg/m3*s]

τ' Shear turbulent stress [Pa]

µ

Dynamic Viscosity [Pa*s] ω k − ω turbulence model [-]

θ

Inclination angle [°]

Sub-indices

G

Gas L Liquid

h

Horizontal m Mixture

i

i-phase v Vertical

k Phase (dispersed phase either continuous phase) [-]

1. INTRODUCTION Nowadays, the O&G industry has had a significant intake in the national Gross Domestic Product (GDP). However, this sector is being the focal point of some environmental problems. Colombia has available light oil reserves that one day must run out (PwC, 2014). Therefore, there is a need of looking for technologies of enhanced oil recovery methods (EOR) for heavy and extra heavy oils, which represent approximately a 70% of the available reserves in the world (Moreiras et al. 2014, Guo et al. 2016). Is necessary to discover a way to recover this type of oils, to improve the recovery percentage, to deploy techniques for non-conventional reservoirs and methods for offshore drilling. Despite these technologies are very expensive, they are needed to enhance the development of the sector. The heavy and extra heavy oils are identified because of their high viscosity that inhibits its easily flow during its extraction. Consequently, it makes more difficult its recovery at normal reservoir conditions. In addition to the viscosity, the specific gravity is also an important property to classify oils. The results of this division is the differentiation of the different types of flows. There could be light, heavy or extra heavy oils. It is measured based on the American Petroleum Institute (API), called API degrees (Guo et al. 2016). Most of the encountered crudes are non-Newtonian fluids, the ones that have an apparent viscosity that changes depending on the shear rate and the shear stress applied to the fluid (Rodrigues, 2015). While the study of non-Newtonian fluids is very important, this study is going to focus on Newtonian fluids with the aim of modeling the drift flux in CFD with a 2D mesh model. All the Computation Fluid Dynamics (CFD) studies are commonly made in 3D mesh, but these simulations require an extensive use of computational resources.

4

Thus, the main objective is to design two-phase flow models that could give an accurate prediction of the drift velocity depending on the flow, whether it is a low viscosity fluid or a high viscosity fluid (greater than 0.3 Pa*s). This is really important because this parameter affects directly the behavior of both phases and the phases interactions within the pipe. Also, is important to recognize all the constrains and limitations that each model has and that they work in a specific range of data, that can be thus analyzed by using dimensionless numbers analysis. These numbers take into account parameters as density, viscosity, velocity and gravity, among others. Its importance is that during the drilling process, there are going to appear many fluids; however, the main phases are going to be the liquid phase (Newtonian fluid) and the gaseous phase. Some of the important constrains are the angle inclination and the pipe orientation and diameter. 2. LITERATURE REVIEW 2.1 Translational velocity As this study is going to be focused on two-phase slug-flow pattern flow, it is important to understand that this consists of a slug unit, that is the liquid slug, and the gas bubble, as shown in Fig 1:

Figure 1. Schematic translational velocity in two-phase slug-flow pattern flow (Baba et al. 2018).

The translational velocity is defined as the velocity of the whole slug unit, Nicklin et al. (1962) proposed a model, defined as the mixture velocity (v%) multiplied by a distribution coefficient (Co), plus the drift velocity. The first term (C7v%) represents the maximum mixture velocity of the slug unit (Moreiras et al. 2014). The expression is shown in Eq. (1).

v' = C7v% + v$ (1)

Where the mixture velocity is defined as the sum of gas and liquid surface velocities, as shown in Eq. (2).

v% = v&: + v&; (2)

This relation is meaningful for analyzing the velocity of the whole unit within the pipe, specifically the bubble behavior. However, for a better quantitative analysis of the flow, the concept of void fraction is more accurate, and it will be explained in section 2.5. As it can be seen in the equation, the value of the drift velocity affects directly the translational velocity, so it is an important parameter to study. 2.2 Drift flux The drift flux or drift velocity is the velocity at which the bubble travels along the pipe, penetrating the stagnant liquid. This displacement disturbs the liquid that at the beginning is static within the pipe. The drift flux is a model applied as a dispersed multiphase flow, which means it assumes the whole as a single fluid rather than a separated flow (Taitel & Barnea, 2015). This assumption implies the reduction of the six original differential equations to four, eliminating one momentum and one energy equation. The drift flux model assumes that the relative motion between the phases is in the calculation of the kinematic constitutive equation, that is the mixture momentum equation. This model and its equations are only valid and accurate when the motions of both phases are strongly coupled. This last assumption is in accordance when treating the mixture as a whole rather than

5

two separated phases, the interaction between phases allows the model thus, to reduce the number of equations and also the computational time (Ishii & Hibiki, 2011). According to Moreiras (2014), this model is specified by four dimensionless numbers, the Reynolds (Re), Eötvös (Eo), Froude (Fr) and Viscosity (Nvis) numbers; which are defined in Eq. (3), Eq. (4), Eq. (5) and Eq. (6). These parameters are useful for the drift velocity calculation. It is important to denote that these numbers are useful when the density and viscosity of the gas are small compared to those of the liquid (Taitel &Barnea, 2015), and that two of them can be used at the same time due to the relation between them.

Fr = v$ρ;>.@ gD ρ; − ρ:C>.@

(3)

E7 = gDE ρ; − ρ: σCF (4)

NHIJ = µ gDK ρ; − ρ: ρ; C>.@ (5)

Re = v$ρ;Dµ;CF (6)

The drift flux model is defined through a set of equations that include continuity for one phase, that is usually the gaseous phase when modelling two-phase flow systems; and three conservation equations which are continuity, momentum and energy, for the mixture (Ishii & Hibiki, 2011). This model is commonly used for miscible mixtures, where the phase interactions can be tracked and studied in detail. In other words, the phases share a well-defined interface free surface region. As this model undertakes constant temperature, the energy equation is eliminated from the set of equations to solve, so the final three field equations are defined by Ishii & Hibiki (2011) and Pico et al. (2017) as follows:

Ø Continuity equation for the mixture:

∂ρ%∂t

+ ∇. ρ%v% = 0 (7)

Ø Continuity equation for the secondary (dispersed) phase in terms of its void fraction:

∂αEρE∂t

+ ∇. αEρEv% = ΓE − ∇. αEρEv% (8)

Ø Mixture momentum equation:

∂ρ%v%∂t

+ ∇. ρ%v%v% = −∇ρ% + ∇. τ + τ' − αQρQvQ%vQ%E

QRF (9)

Analyzing these equations closely, when the void fraction of the dispersed phase (gaseous phase) has a value of 0 or 1, the mixture term vanishes and now both phases are modeled individually, whereas for the other cases the mixture is modeled as a pseudo-fluid, where both phases are co-existing in the tube (Pico et al., 2017). The drift flux model is a simplified model, very useful for solving engineering problems due to its facility to understand a certain system, and solving it easily obtaining accurate results. Treating the mixture as a single fluid, with a dispersed phase and a continuous phase, reduces the computational time and allows a better comprehension of the phenomena within the pipe, the phases behaviors and the phases interactions along the whole geometry. 2.3 CFD two-phase flow models

6

In this and other similar software, there are many multiphase flow models that can be applied to two-phase flow systems. Two-phase flow systems are characterized by the existence of one or several contacts between phases and discontinuities in fluid properties at the interface. The interactions between the two phases causes the interface values to change continuously, which may result in enhancement or reduction in the efficiency of transfer of mass or energy in the two-phase flows. Consequently, the solution of the Reynolds-Averaged Navier-Stokes (RANS) equations is affected due to these discontinuities. The drift-flux model is extremely important for predicting two-phase flows because the drift velocity and distribution parameters are directly related to the physical structure of the flow (Ghajar et al, 2014). There are two general approaches for solving these flows and interface interactions: the dispersed flows and the stratified flows. So, a phase is considered dispersed if it occupies disconnected regions of space, otherwise it is continuous. STAR CCM+ provides distinct models to meet the requirements of these two categories of flow. Some of them are Eulerian Multiphase Models (EMP) as the Eulerian Multiphase Mixture Model (MMP), the Multiphase Segregated Flow Model and the Volume of Fluid (VOF) model. The others are the Lagrangian models as the Lagrangian Multiphase Model (Siemens, 2018). The chosen model uses the Eulerian-Eulerian approach because the distinct physical substances are represented by the Eulerian phases, which are modeled in a Eulerian framework. The distinction of a Eulerian phase includes the set of models that apply to the phase material and the relevant material properties (Siemens, 2018). In the first model, the Eulerian Multiphase Mixture Model, mass momentum, and energy are treated as mixture quantities rather than phase quantities. It solves transport equations for the mixture as a whole, and not for each phase separately. The MMP is projected to use it as a replacement for the more computationally expensive EMP. However, in some cases, large variations in the quantities between phases are not well-resolved. This model is not suitable for use when the mixture properties are not good approximations of the real properties of the phases. It is recommended to use it in cases where the grid is too course to resolve the interface between the phases. Consequently, the MMP model cannot be expected to solve a sharp interface, even on a fine grid. Apply this method where the weighted physical properties could be significant to represent a mixture of phases (Siemens, 2018). The other model, the Multiphase Segregated Model, it is based on a Eulerian-Eulerian formulation where each current phase has its own set of conservation equations; thus phases are considered to be mixed on length scales smaller than the length scales to resolve, and co-exist everywhere in the flow domain. This concept of co-existing phases is called “interpenetrating continua”. The concept assumes inherently that the important item is the time averaged of the flow, rather than the instantaneous behavior. Here, each phase has its own velocity and physical properties. The “segregated” term refers to the fact that the solution algorithm uses a SIMPLE-approach, which has separate pressure and velocity solvers; in brief, the phases are not in equilibrium. The conservation equations for each existing phase variable requires closure by the definition of phase interactions at each phase interface. This definition consists of suitable models for the interfacial area, and for the rates of interphase transfer of mass, momentum and energy. Is very difficult to know the accurate value for the parameters (Siemens, 2018). And the used in this study, the Volume of Fluid model, first it can be considered a special case of MMP, with a specialized convection scheme and no diffusion. This model is suited for simulating flows of several immiscible fluids on numerical grids capable of resolving the interface between the phases of the mixture. So, these models fit well in air-oil flow systems. As said in section 2.2, there is no need for extra modeling for the interphase interaction, thus the model assumes that the phases involved share pressure, velocity and temperature profiles. Due to the high numerical efficiency of the model, it is appropriate for simulations of flows where each phase constitutes a large structure, with a relatively very small total contact area between them. The spatial distribution of each of them at a given time, is defined in terms of the volume fraction. In order to obtain a sharp interface between the phases is recommended to use the 2nd term for the convergence, and modify the sharpening factor, setting it in 1, to obtain the most sharped interface. In other words, to ensure sharp resolution of the free surface region, the CFL number is limited to 1 (Siemens, 2018). 3. THE STATE OF THE ART

7

The analysis of previous drift flux models is very important to achieve significant results. Thus, the previous CFD and experimental studies are going to be stated in this section. 3.1 Previous experimental studies For the case of experimental studies, it includes several different studies, within which there are studies for the viscosity, the diameter and length of the pipe, the fluids properties, especially the liquid properties for two-phase flows, as well as for the inclination angle effects. To begin, for the determination of the drift velocity, Dumitrescu (1943) and Davies and Taylor (1950) performed experiments to find the drift velocity using an inviscid potential flow analysis for vertical flow. Inviscid potential flow theory inherently neglects the effect of surface tension and viscosity. They assume that the coordinate system is based on the bubble motion, thus, it is in steady state. Both researchers used the Pi of Buckingham theorem, and derived Froude number as the dimensionless group to relate the upward bubble velocity, which has a constant value. Davies and Taylor estimated the constant value as 0.328. Dumitrescu theoretically determined this value as 0.351 which agreed well with the air/water experimental data of Nicklin et al. (1962). Then, Zukoski (1966) experimentally investigated the effects of liquid viscosity, surface tension, pipe inclination on the motion of single elongated bubbles in stagnant liquid column with varying pipe diameter. He found a drift velocity correlation which adjusted to all the parameters evaluated primarily for vertical tubes. Finally, Joseph (2003) found that for having a good prediction of the drift velocity, is necessary to consider the liquid viscosity, the surface tension, and the characteristics as the shape of the bubble-nose. He estimated that for spherical bubble-nose shapes, the surface tension term vanishes, and the equation just depends on the oil viscosity and the radius of the spherical-cap bubble (Jeyachandra, 2011). Further studies are required in the case of not having a spherical cap bubble. Additionally, for evaluating the effects of viscosity liquids on drift velocity, Colmenares et al. (2001) studied pressure drop models for horizontal slug flow for viscous oil, using preexisting data of Taitel & Barnea (2015). They did a modification of this correlation obtaining accurate results. Rosa et al. (2004) experimentally investigated the influence of liquid viscosity on two-phase horizontal slug flow. They concluded that the bubble front velocity and slug frequency increased with an increase in liquid viscosity (TUFFP, 2010). Gokcal (2005) investigated two-phase flow characteristics of oil with viscosity in a 0.181 – 0.585 Pa*s range, obtaining a definition for the drift velocity in this whole range. Also, they had into account the liquid and gas flow rates for the evaluating its effects in flow pattern, pressure gradient and liquid holdup. As the liquid viscosity increased, the importance of viscosity decreased. Finally, Jeyachandra (2012), studied experimentally the effect of high-viscosity oils in several pipe diameters, with the aim of expanding the investigation made by Gokcal (2008). He found a relation that could be applied accurately to various viscous liquids and different pipe inclinations. Conversely, an inviscid flow theory was proposed by Alves et al. (1993), which studied the surface tension effects in inclined pipelines, using the correlation obtained by Benjamin for horizontal flow, extending it for vertical cases. This assuming the shape of the bubble nose region. In the study of specifically horizontal slug flow, Dukler & Hubbard (1975) claimed that there is no drift velocity for horizontal flow since gravity cannot act in the horizontal direction. They used the liquid properties and the Froude number for the calculation of the drift velocity and pressure drop. However, Nicholson et al. (1981), Weber (1981), and Bendiksen (1984) showed that drift velocity exists for the horizontal case and the value of drift velocity can exceed the vertical drift velocity value. There were performed experiments to determine the effect of tube inclination on bubble motion with Reynolds and Froude numbers of the liquid phase, and pipe diameter, as the most important parameters. In addition, Benjamin (1968) stated that drift velocity in horizontal slug flow is the same as the velocity of the penetration of gas when liquid is drained out of a horizontal pipe. The results are in well agreement with the ones obtained by Zukoski (1966). Is important to clarify that he considers energy transfer phenomenon and heat losses affect directly the results.

8

Also, for the specific case of inclined pipelines, Zukoski (1966), Bendiksen (1984), Weber et al. (1986), Hasan & Kabir (1986), and Carew et al. (1995) experimentally studied drift velocity of Newtonian and non-Newtonian fluids and found that increases for an increasing inclination angle and decreases to its lowest value for vertical flow. The maximum value for the inclination angle is between 30° and 50° from horizontal. In his study, he showed how viscosity reduces the inclination dependence of the Froude number of the bubble. Shosho & Ryan (2001) performed experiments and investigated the effects of the pipe size, also fluids including Newtonian and non-Newtonian on drift velocity for vertical and inclined pipelines. They studied a huge range for the viscosity, from 0.003 to 0.883 Pa*s and concluded that as the angle of inclination from the horizontal increases, Froude number increases, reaches a maximum, and then decreases for the fluids tested. The maximum Fr value occurs at larger angles of inclination when the fluid is non-Newtonian and is similar for vertical tubes. 3.2 Previous CFD studies Although there are not as many CFD studies as there are for experimentation, to determine the drift velocity, Andreussi & Bonizzi (2009) studied high-viscosity oils in a commercial CFD software (FLUENT) and concluded that due to the complexity of the flow, because of these viscosities, the drift velocity values decrease significantly along the pipeline in the time. They utilized a VOF approach for the simulations for resolver the Navier-Stokes governing equations. The same happened with Ramin & Henkes (2012), they did a dimensionless analysis for a wide range of the Eo and Re numbers obtaining good results that were in accordance with the analytical and gathered experimental data. They analyzed the effects of the viscosity of the fluids and the surface tension of each fluid. Lu (2015) analyzed the gas-liquid two-phase slug flow of large horizontal pipelines experimentally and in CFD software with the purpose of studying the slug initiation, growth and collapse along the tube. He utilized six CFD software (STAR-CD, CFX, FLUENT, LedaFlow, TRIOMPH and TransAT), to conclude about the constrains and limitations of each one and see which the best was to predict accurately the drift velocity within the pipeline. For the grid of the simulations, he used a polyhedral mesh due to its capability for reducing the error, and the lowest number of cells possible. These last results were in agreement with the ones obtained by Sanderse (2015). He found accurate values for the pipeline flushing and slug flow using with an adjustment of the drift velocity correlation of Benjamin (1968); however, the drift velocity was nor accurately predicted. Similarly, Yamoah (2015) focused his study in vertical gas-liquid two-phase flow in pipelines to study the drag, lift, wall lubrication and turbulent dispersion forces. He predicted very well the interface interactions, the results yielded satisfactory agreement with the experimental data. Likewise, Kroes & Henkes (2014) studied the flow behavior of pipelines full of liquid using the FLUENT software and observed two different flow regimes, for the first one the drift velocity was constant along the pipe, whereas for the second one this value decreased over time. Found that not just the drift velocity values changed, also the bubble shaped was altered. He developed a correlation for the drift velocity in horizontal pipelines that depends on the liquid viscosity. 3.3 Existing drift velocity correlations In the chart below, there are the most relevant drift velocity correlations and the application range of each one.

Table 1. Previous correlations for the drift velocity. Reference Drift velocity Inclination angle

Bendiksen (1984)

v$ = v$Scosθ + v$Hsinθ

0° < θ < 90°

Bendiksen (1984)

v$ = 0.54 gD 0°

Benjamin (1968)

v$ = 0.542 gD 0°

Davies & Taylor (1950) v$ = 0.328 gD 90°

9

Dumitrescu (1943)

v$ = 0.35 gD 90°

Hasan & Kabir (1986)

v$ = v$H sin θ 1 + cos θ F.E 0° < θ < 90°

Jeyachandra (2012)

v$ = 0.53eCFK.cdefgh.ijklmh.n gD 0°

Joseph (2003)

v$ = −43µ;ρ;+

49gr +

169µE

ρ;E

90°

Kroes & Henkes (2003)

v$ = 0.51 gD 0°

Moreiras (2014)

v$ = 0.54 −NHIJ

0.01443 + 1.886NHIJgD

0°

Nicklin et al. (1962)

v$ = 0.351 gD 90°

Weber (1981)

v$ = 0.544 − 1.76E7>.@q gD 0°

Zukoski (1966)

v$ = 0.542 gD 0°

4. MATERIALS AND METHODS According to the objective of this study, the materials and methods are going to be explained in this section. For the first stage, it is important to clarify that the experimental procedure was done by the Tulsa University Fluid Flow Project research group. 4.1 Experimental tested fluids As stated before, the experimental procedure was performed by the Tulsa University Fluid Flow Project research group (TUFFP, 2017). There, the drift velocity of a synthetic oil was measured at different temperatures to analyze its effects in the flow behavior within the pipe. The changes in temperature have the effect of changing the viscosity of the oil. The properties of the synthetic oil are review below in Table 2.

Table 2. Properties of synthetic oil at 101325 Pa. Properties Temperature [°C] Density [kg/m3] Viscosity [Pa.s]

1 27 845.9 0.48 2 22 849.2 0.625 3 22 849.9 0.678 4 22 850.1 0.686

Also, they measured a surface tension of 0.03 N/m for the synthetic oil, used for all the cases of study. The surface tension is in the calculation of some dimensionless numbers as the Eötvös number; consequently, it is in the drift velocity calculation. It is a very important parameter that models the interface interactions, also determines the tension forces between the existing phases. Thus, it helps to determine how miscible the phases are. 4.2 CFD modelling To design the CFD model, the commercial software STAR CMM+ v13.04.011 was used to perform the simulations to evaluate each case of study. This software is based on the finite volumes method. It works with a mesh model that has a certain number of cells in which the equations will be iterated and solved in the whole

10

geometry. Thus, with a given quantity of cells, the system will have an accurate solution for the phase interactions and the phases behavior. 4.2.1 Spatial discretization First, the geometry of a 12.43-m long and 50.8-mm ID pipeline was built in Autodesk Inventor Professional 2018®. It was taken into account that for the mesh transformation, from a 3D model to a 2D model, the inlet and outlet faces had to be in the YZ plane, and the pipe had to be the half so that the symmetry condition could be applied. In Fig. 2 is shown the 3D isometric view of the pipeline geometry.

Figure 2. 3D pipeline geometry.

The system was modelled setting an automated mesh, using the Surface Remesher, the Polyhedral Mesher and the Prism Layer Mesher. First, the surface remesher was selected because it helps to rebuild the damaged cells in the surface mesh as it retriangulates it. The polyhedral mesh was selected over the other types of mesh, because this polyhedral meshing model utilizes an arbitrary polyhedral cell shape in order to build the core mesh. The CFD software creates at first a tetrahedral mesh, and then by a dualization process, the polyhedrons are created. This is useful because when the equations are being solved in each cell, they will have more information received from the cells next to each one (typically each cell has 6 faces in 2D). Likewise, this polyhedral mesher can obtain additional edges on the boundary, which can be added in order to avoid warped boundary faces and to enhance uniformity among cells. Lastly, the prism layer mesher is used to created orthogonal cells next to the boundaries and wall surfaces. (Siemens, 2018). This mesher reduces a particular numerical discretization error near the wall boundary known as numerical diffusion. This method produces disturbances on the velocity gradients in the prism layers to improve the resolution (Pico et al., 2017). In the controls section, the base size was set based on the mesh independence test. The surface curvature was left with the default value, this setting fits the cells mesh according with the angles and curvature of the geometry, if they are circles or squares. The other main value was the number of prism layers, this represents the layers that are created between the walls and the first polyhedral cells of the mesh in the outer real walls of the pipe. The summary of the values of the mesh are presented in Table 3. The mesh showed a constant change of the cells size between its close cells in the whole geometry (Siemens, 2018). This ensures that the obtained data can predicted very well the drift velocity results.

Table 3. Values for the automated mesh. Mesh specifications Set value

Base size [cm] 0.3 Minimum surface size [%] 24

Surface curvature [pts] 36 Number of prism layers [-] 8

Prism layer total thickness [%] 24 A surface control was created with the aim of eliminating the prism layers in the inlet, outlet and symmetry boundaries. The reason for eliminating them, is that they are not real walls of the system. In Fig. 3, is shown the mesh obtain for all the simulations done.

11

Figure 3. Mesh. (a) Inlet face - 3D, and (b) Symmetry plane - 2D.

Fig. 4 shows a near-view of a mean cell of the mesh, where for the symmetry plane is more defined than for the inlet face.

Figure 4. Cell quality. (a) Inlet face, and (b) Symmetry plane.

4.2.2 Physical model selection For the physical setting of the pipeline, the EMP and VOF approaches were used due to their capability for calculating the behavior of both phases and the tracking along the tube of the phases interactions (Siemens, 2018). As stated in section 2.3, these models allow to make an accurate calculation of the drift velocity and also a well-described development of the gaseous phase penetrating the liquid phase. Additionally, the VOF model helps to solve the equations, especially for the interface, using mixture properties, assuming the whole as a single fluid rather than two separated phases. This allows to reduce the computational time and aids to improve the results, due to its ease for tracking this interface. The following models were selected according to the complexity of the flow and its development within the tube first, as the results depend on the time, the implicit unsteady model was selected. For modeling the material, is selected the Eulerian Multiphase Model as stated in previous sections. As it is a single fluid of co-existing phases, the system cannot be modeled with the multicomponent liquid model. Then, as stated in section 2.3, the VOF approach is the most accurate because can capture the system behavior along the pipeline. It also reduces the computational, solves 3 equations for the mixture and its physical properties, and one for the gaseous phase. As the implicit unsteady model was chosen, the flow available is the segregated flow. This model solves the flow equations, two components for the velocity and one for the pressure (Pico et al., 2017). The implication of using the constant density equation of state (incompressible flow) is in agreement with the segregated flow equation. As it is a very complex flow and the liquid is a very viscous fluid, the turbulent model is selected to have more accurate results for the drift velocity. The k − ε model lies within the calculation and solution of the RANS equations. This model was preferred over the k − ω because solves transport equation for the turbulent kinetic energy 𝑘 and the turbulent dissipation rate 𝜀. Also, it has certain models to solve flows with low Re numbers. Experimentally, the Re numbers were between 2 and 6.

12

In a further section, the turbulence model selection is going to be explained in detail with its implications. Finally, the two optional models were selected with the aim of improving the physical solution of the flow. The first one was the gravity, it changes significantly the velocity because this parameter is affected primarily by the inclination angle. The gravity is specified by its components in both axes (x, y). The other additional model was the cell quality remediation. This model is very important to fix and recalculate the solution in cells where it is significantly different from the solution of other cells. It helps to get the solution in a poor-quality mesh; once these cells and their neighbors have been marked, the computed gradients in these cells are modified in such a way as to improve the robustness of the solution (Siemens, 2018). To define the Eulerian phases, both gaseous and liquid, the constant density equation of state was selected because the fluid is assumed incompressible. Furthermore, for the phase interaction was used the VOF-VOF model, the slip velocity is assumed negligible, and the optional models used were the surface tension force due to the effect of the surface tension between the phases, the multiphase material because there are two different materials in the system, and finally the interface momentum dissipation since adds extra momentum dissipation in the proximity of the free surface to vanish parasitic or false currents. These wrong currents are caused by discretization errors that appears due to the discontinuities of the solution across the free-surface interface (Siemens, 2018). Finally, the segregated flow model was fixed with the MUSCL 3rd order convection and the settings of the VOF model were fixed to improve the solution, taking into account the high viscosity of the fluid. These parameters support the convergence in the CFD software (Pico et al., 2017). In Table 4 are shown the specific values used for the synthetic oil simulations.

Table 4. VOF specifications for the synthetic oil. Parameter Set value

Sharpening factor 1 Angle factor 0.05

Lower Courant number 0.5 Upper Courant number 1

IMD ON 4.2.2.1 Turbulence models As the experimental results are available, it is possible to know the real regime flow within the pipe for each case. Thus, one important part was to evaluate some models for the CFD turbulence modeling, for low Re numbers. For this reason, the k − ε turbulence model was selected. Then, the available models were analyzed a priori to choose the 3 best ones and then analyze them in CFD. The first selected model was the Abe-Kondoh-Nagano (AKN) k − ε Low-Re because is known to work well for a wide range of complex flows. Also, is a good choice where the Reynolds number are low, but the flow is relatively complex (Siemens, 2018). For this model, the wall treatment needed to be specified as All y+ or Low y+ to evaluate the wall definition. The Low y+ Wall Treatment was pretty accurate because resolves the viscous sublayer and needs few information to predict the flow across the wall boundary. The computational time is significantly affected with this method, so it used in specific cases for Low-Re numbers. Whereas, the All y+ wall treatment is a hybrid method which takes into account both Low y+ and High y+ Wall Treatment methods, for coarse meshes (Siemens, 2018). The second selected model was the Realizable k − ε Two Layer, mainly because contains and additional transport equation to solve the turbulent dissipation rate ε. Also, a critical coefficient of the model is expressed as a function of mean flow and turbulence properties, rather than assumed to be constant as in other models. As is said in its name, it combines the main model with the Two-Layer approach that is an alternative to the Low-Re number approach, which allows the model to be applied in the viscous-affected layer, including the viscous sub-layer which depends only in the fluid density, and the buffer layer which is the transition between the viscous sub-layer and the log-law layer (dominated by viscous and turbulent effects) (Siemens, 2018). The Two-Layer All y+ Wall Treatment method is selected by default due to the implications of the model.

13

The third and last selected model was the Standard k − ε Low-Re. This model, as its name expresses, combines the Standard k − ε model with the Low-Re approach. The standard model involves the two common transport equations, for the kinetic energy k and the dissipation rate ε. It also takes into account effects such as buoyancy and compressibility. This model is better than the simple one because has more damping functions that enable it to be applied in the viscous-affected regions near walls (Siemens, 2018). Finally, as stated before, the wall treatment approached used was the Low y+ Wall Treatment. 4.3 Initial and boundary conditions With the geometry parts identified, the regions were created by creating a region for each part and a boundary for each part surface. The pressure and velocity are known so they were defined in the system, the outlet boundary was set as a pressure outlet with a reference pressure of 101325 Pa (the pressure of Tulsa, Oklahoma), and the velocity as 0 m/s (Siemens, 2018). Likewise, the symmetry region boundary was defined as a symmetry plane; thus, it simulates only half of the pipeline, it assumes that the behavior in both halves is the same, it is symmetric to this plane. The initial performance was a pipeline full of liquid (VFG = 0, VFL = 1), and the gas was entering completely across the outlet region, therefore, this boundary was fixed with a VFG of 1 (VFG = 0). The initial view is shown in Fig. 5.

Figure 5. Initial conditions for the pipeline.

4.4 Time-step setting A time-step setting was necessary to have the time that last each iteration under control in all the simulations. For setting this parameter, the experimental results for the drift velocity results were taken into account. The calculation is defined in Eq. (10). The CFL Courant number was defined as 1 to ensure that the numerical calculations would develop in all the cells of the mesh. It is recommended to simulations that used the Eulerian-Eulerian multiphase approach.

C =U" ∗ ∆t∆x

(10)

4.5 Error analysis A statistical analysis is necessary to see in which grade the results obtained in the 2D mesh simulations are similar to the ones obtained in previous experimental studies. For this reason, it is used the absolute error, to see how much the results are deviated from the experimental ones, as shown in the Eq. (11)

e =sim − exp

exp (11)

5. RESULTS AND DISCUSSION 5.1 Experimental drift velocity The results for the drift velocity measured experimentally in a PVC 50.8 mm ID pipeline are shown in Fig. 6.

14

Figure 6. Experimental drift velocity.

The results obtained by the TUFFP research group are in agreement with the conclusions of Moreiras (2014) and Jeyachandra (2012) as the drift velocity decreases with increasing viscosity. The behavior of the synthetic oil within the pipe has the same tendency for all the viscosity conditions. Also, the velocity decreases significantly in the first 4.089 meters of the tube. When the viscous oil is reaching the outlet face, the drift velocity tends to stabilize, and its changes are less than at the beginning. Although they tested at two different temperatures for the oil, they acquired more than one measure due to the changing value of the density, and the most important, the dynamic viscosity. As stated in section 3.1, the viscosity affects directly the drift velocity. The change of the drift velocity is proportional to the change on viscosity values.

Figure 7. Reynolds numbers found experimentally.

As stated by Zukoski (1966), the viscous effects in drift velocity are negligible for Re numbers greater than 200. If not, is very important to analyze the viscous effects within the tube. So, that is why in Fig. 7 are shown the Reynolds numbers found during the experimentation. For all the experiments in all the distances measured, these values are between 1 and 7, so the viscosity of all of them has to be studied. The Reynolds number is proportional to the drift velocity, so the tendency is similar for all the experiments. The Reynolds number decreases for an increasing viscosity. For similar viscosities, the Reynolds numbers are also similar. 5.2 CFD drift velocity In this section are going to be shown and explained the results obtained for the CFD drift velocity. 5.2.1 Mesh independence test A mesh independence test was performed with the objective of analyzing the effect of the numbers of cells on the drift velocity results. This allows to see how the error and the computational time vary in regard to the total number of cells. The conditions of the simulations for this test are shown in Table 5.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

051015D

rift v

eloc

ity (m

/s)

Distance from the outlet (m)

0.48 Pa.s0.625 Pa.s0.678 Pa.s0.686 Pa.s

0

1

2

3

4

5

6

7

051015

Re

Distance from the outlet (m)

0.48 Pa.s0.625 Pa.s0.678 Pa.s0.686 Pa.s

15

Table 5. Fluid properties utilized for the mesh independence test. Fluid Synthetic oil

ρ [kg/m3] 845.9 µ [Pa.s] 0.48 σ [N/m] 0.03 D [m] 0.0508

Distance at which the velocity was measured [m] 1.537

Turbulence model AKN k − ε Low-Re For this analysis there were selected five values for the base size, and with the results according to the number of cells, the best size was chosen to use that value in all the simulations. The summary of the parameters of each test are shown in Table 6, as follows:

Table 6. Mesh specifications.

Name Base size [cm] Target surface size [cm]

Minimum surface size [cm]

Prism layer total thickness [cm]

I 0.30 0.30 0.072 0.072 II 0.25 0.25 0.06 0.06 III 0.20 0.20 0.048 0.048 IV 0.15 0.15 0.036 0.036 V 0.10 0.10 0.024 0.024

Fig. 8, shows the results obtained, where for the III, IV and V simulations the computational time and the drift velocity error do not change significantly, although the number of cells increases twice its magnitude. In the case of the I and II simulations, the error and the computational decreases. So, between these two simulations, the best one is the accurate one because has the lowest error and the lowest computational time. In this simulation the number of cells to decrease was 23410 cells in the 2D mesh, which is the number of cells obtained using a base size of 0.3 cm. For all the simulations of this test, the drift velocity was measured at a distance of 1.537 m, the first plane measured experimentally, to obtain the error between those values using less computational time and that the results could be comparable.

Figure 8. Mesh independence test results.

5.2.2 Drift velocity for synthetic oil In this section are going to be shown the results obtained in the CFD simulations for the drift velocity. As was stated before, the model is a 2D mesh, given that these simulations require a lot of computational time due to the selected models and the pipe length. The longer the tube, more computational time the model will requiere. Taking this into account, the drift velocity in the last two measured planes was calculated with the error obtained by extrapolating the behavior of it. So, with the 3 first points graphed, the adjusted trend was calculated with an exponential trend line approximation with the aim of estimating the expected error and with these values, finding the drift velocity for the last 2 points.

I IIIII IV V

0

10

20

30

40

50

60

0

20

40

60

80

100

120

140

160

180

20000 40000 60000 80000 100000 120000 140000

Com

p. T

ime (

h)

Erro

r (%

)

Number of cells

Drift velocity error

Computational time

16

The results of the adjustments obtained are shown in Fig. 9. The exponential trend line adjusts very well to the results; the behavior could be tracked along the pipeline. This was proved obtaining the coefficient of the determination. All the values for these parameters are greater than 0.95, this is in agreement with the error values obtained. The equations for the error of each model utilized are shown in Annex 1 and Annex 2 with their respect value of the coefficient of determination.

Figure 9. Drift velocity error vs. distance. (a) Condition 1, and (b) Condition 2.

The left side of the Fig. 9(a), are the results gathered for the first condition evaluated for the synthetic oil and Fig. 9(b), are the results for the second condition evaluated experimentally for the synthetic oil. The error decreases as the bubble is moving through the tube. At the beginning the velocity is greater in one order of magnitude and then it comes slower after crossing the third measured plane (5.562 m). After having the equation for the error of each model, the drift velocity was calculated clearing this term of the error equation, shown in Eq. (11). These values were plotted in regard to the distance from the outlet face to each measured plane. As shown in Fig. 10, The drift velocity decreases rapidly during the first section, then it continues the trend, but the change is less. As for the turbulence models used, for the first measured situation the most accurate model is the Standard k − ε Low-Re. Although the error is relatively high, this model has the lowest error between the three analyzed models. This applies for the beginning of the pipeline, then as the velocity decreases more than expected, the error is very low and is within the 15%. As the velocity stabilizes at the end, all the three models are accurate for the drift velocity solution. As specified in section 4.2.2.1, this model is different and stands out because takes into account the solution of the affected viscous sub-layer in the near-wall regions and also the damping functions for low Re numbers. In the second situation, the most accurate model was the AKN k − ε Low-Re, obtaining the finest results at the beginning of the pipeline. Like for the first case, in the second case, the drift velocity stabilized at the end of the tube after crossing the third measured plane. The error in the last measured planes is within the 15% of error. This model takes into account the Low-Re approach and tends to work for a wide range of complex flows.

Figure 10. Drift velocity vs. distance. (a) Condition 1, and (b) Condition 2.

As stated before, the viscous effects are very important for the synthetic oil, due to its high viscosity. In Table 7, are shown the maximum CFD and experimental Re numbers obtained, to confirm that all of them are values

020406080

100120140160180200

051015

Erro

r (%

)

Distance from the outlet (m)(a)

Drift velocity error AKNDrift velocity error RealizableDrift velocity error StandardRealizable adjustedStandard adjusted

(a)

0

20

40

60

80

100

120

051015

Erro

r (%

)

Distance from the outlet (m)

Drift velocity error AKNDrift velocity error RealizableDrift velocity error StandardAKN adjustedStandard adjustedRealizable adjusted

(b)

00.020.040.060.08

0.10.120.140.160.18

0.2

051015

Drif

t vel

ocity

(m/s

)

Distance from the outlet (m)

ExperimentalCFD AKNCFD RealizableCFD Standard

(a)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

051015

Drif

t vel

ocity

(m/s

)

Distance from the outlet (m)

ExperimentalCFD AKNCFD RealizableCFD Standard

(b)

17

lower than 20. These values mean that the viscous forces are much greater than the inertial forces related with the drift velocity of the synthetic oil. So, the viscosity will dominate the penetration of the gas bubble, modifying the drift velocity. This is in agreement with the variation of the drift velocity as the gas bubble travels along the pipe, given the strong viscous forces associated with the fluid studied.

Table 7. Maximum CFD and experimental Reynolds numbers. Fluid Dynamic viscosity [Pa.s] CFD Re Experimental Re

Synthetic oil 0.48 16.1558 5.7296 Synthetic oil 0.625 9.4595 2.8990 Synthetic oil 0.678 - 2.6109 Synthetic oil 0.686 - 2.4551

To sum up the results presented before, Fig. 11 shows the CFD drift velocity for the two conditions evaluated. The drift velocity at the start of the tube is relatively high compared with the experimental gathered data; however, the tendency is in accordance with these data. For lower viscosities, as already mentioned, the drift velocity increases proportionally. To conclude, based on the dimensionless analysis of the Reynolds number, the inertial forces cannot overcome the viscous forces of the synthetic oil; thus, the oil viscosity will dominate the drift velocity as the gas bubble is penetrating the tube. These presented drift velocities are the ones obtained with the best turbulence model for each condition.

Figure 11. CFD drift velocity results.

6. CONCLUSIONS To conclude, the drift velocity behavior along horizontal pipelines stabilizes from approximately half of the pipe, in the second section obtaining accurate results with an error under 15%. Although at the start of the pipeline the error overcomes the 100%, the tendency along the pipe stays coherent. This means that for experimental and CFD results, the drift velocity values kept the behavior in time. The dimensionless analysis of the Reynolds numbers allows to conclude that the regime along the pipe is laminar; however, it is a very complex flow because of the viscosity of the fluid and the pipe length. Furthermore, due to the high viscosity of the synthetic oil, the drift velocity is slow (less than 1 m/s) and decreases with and increasing viscosity. For all the gathered data, the drift velocity decreases significantly in the first 5.562 meters and then continues decreasing but with a lower changing rate. The most accurate turbulence model for the synthetic oil in general for the conditions evaluated, is the Standard k − ε Low-Re model which takes into account the Low-Reynolds approach for these specific systems, with the Low y+ Wall Treatment to solve the viscous sub-layer. Thus, is a good model because the drift velocity values are significantly different from the other two models studied, and similarly the error is significantly less compared with the experimental data. For future studies, it is recommendable to study deeply the damping relaxation factors of the Standard k − ε Low-Re model to obtain better results for the drift velocity. Also, to evaluate in CFD more viscosities to conclude about the functional range of the selected model. 7. ACKNOLEDGMENTS

0

2

4

6

8

10

12

14

16

051015

Re

Distance from the outlet (m)

0.48 Pa.s

0.625 Pa.s

18

Chemical Engineering professor at Universidad de los Andes, Mr. Nicolás Ríos Ratkovich; Petroleum Engineering professors at The University of Tulsa, Mr. Eduardo Pereyra and Mr. Tae-woo Kim, and Master’s degree research assistants at Universidad de los Andes, Mr. Juan Valdés Ujueta and Miss Paula Pico Viviescas are acknowledged for their theoretical and practical support on this study. 8. REFERENCES [1] PwC, "Colombia Oil & Gas Industry 2014: An Overview," 2014. [2] K. Guo, H. Li and Z. Yu, "In-situ heavy and extra heavy oil recovery: A review," Fuel, pp. 886-902,

20 August 2016. [3] P. E. Rodrigues and M. L. Batzle, "Shear modulus of heavy oils: Confinement effect in rheometer

measurements," Fuel, pp. 520-525, 26 March 2015. [4] R. A. Eghorieta, T. A. Afolabi and E. Panacharoensawad, "Drift flux modeling of transient high-

viscosity-liquid and gas two-phase flow in horizontal pipes," Journal of Petroleum Science and Engineering, pp. 605-617, 20 July 2018.

[5] A. J. Ghajar and S. M. Bhagwat, "Flow patterns, void fraction and pressure drop in gas-liquid two phase flow at different pipe orientations," in Frontiers and Progress in Multiphase Flow I, L. Cheng, Ed., Oklahoma, Springer, 2014.

[6] L. Marquez and J. Ochoa, "Comparison of different void fractions correlations for different flow patterns and pipe inclinations," Bogotá, 2017.

[7] Y. D. Baba, A. M. Aliyu, A. E. Archibong and M. Abdulkadir, "Slug length for high viscosity oil-gas flow in horizontal pipes: Experiments and predictions," Journal of Petroleum Science and Engineering, pp. 397-411, 8 February 2018.

[8] Siemens, "The Steve Portal," 25 October 2017. [Online]. Available: https://thesteveportal.plm.automation.siemens.com/articles/en_US/FAQ/How-do-I-decide-which-multiphase-model-has-to-be-used-in-my-simulation. [Accessed 15 September 2018].

[9] J. P. Valdés Ujueta, P. D. Pico Viviescas, E. Pereyra and N. Ríos Ratkovich, "Experimental and CFD modelling of the drift flux in two-phase air-(non)Newtonian slug-flow pattern flow along horizontal and inclined pipelines," Bogotá, 2017.

[10] A. Livinus, P. Verdin, L. Lao, J. Nossen, M. Langsholt and H.-G. Sleipnæs, "Simplified generalised drift velocity correlation for elongated bubbles in liquid in pipes," Journal of Petroleum Science and Engineering, pp. 106-118, 14 October 2017.

[11] M. Ishii, "One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase flow regimes," in Thermo-fluid dynamics of two-phase flow, Second ed., Illinois, Springer, 2011, pp. 10-50.

[12] M. Ishii and T. Hibiki, Thermo-fluid dynamics of two-phase flow, Second ed., Illinois: Springer, 2011.

[13] Y. Taitel and D. Barnea, Encyclopedia of two-phase heat transfer and flow I: Modeling of gas liquid flow in pipes, J. R. Thome, Ed., Lausanne: World Scientific, 2016, pp. 185-194.

[14] Tulsa University Fluid Flow Project, "Drift velocity experiments," Tulsa, 2017. [15] T. B. Benjamin, "Gravity currents and related phenomena," La Jolla, 1968. [16] K. H. Bendiksen, "An experimental investigation of the motion of long bubbles in inclined tubes,"

Elsevier, Oslo, 1984. [17] M. Weber, "Drift in intermittent two-phase flow in horizontal pipes," Quebec, 1981. [18] M. E. Weber, A. Alarie and M. E. Ryan, "Velocities of extended bubbles in inclined tubes,"

Pergamon Journals, Quebec, 1986. [19] R. M. Davies and S. G. Taylor, "The mechanics of large bubbles rising through extended liquids and

through liquids in tubes," Jstor, London, 1950.

19

[20] D. T. Dumitrescu, "Strömung an einer Luftblase im senkrechten Rohr," Z. angew. Math. Mech, Bukarest, 1943.

[21] A. J. Ghajar and S. M. Bhagwat, Frontiers and Porgress in Multiphase Flow I, L. Cheng, Ed., Aarhus: Springer, 2014, pp. 157-175.

[22] A. R. Hasan and C. S. Kabir, Predicting Multiphase Flow Behavior in a Deviated Well, North Dakota: SPE Production Engineering, 1986, pp. 474-482.

[23] B. C. Jeyachandra, "Effect of pipe inclination on flow characteristics of high viscosity oil-gas two-phase flow," Tulsa, 2011.

[24] B. C. Jeyachandra, B. Gokcal, A. Al-Sharki, C. Sarica and A. Sharma, "Drift-velocity closure relationships for slug two-phase high-viscosity oil flow in pipes," SPE Journal, Tulsa, 2012.

[25] D. J. Nicklin, J. O. Wilkes and J. F. Davidson, "Two-phase flow in vertical tubes," Trans. Instn Chem. Engrs., Cambridge, 1962.

[26] E. E. Zukoski, "Influence of viscosity, surface tension, and inclination angle on motion of long bubbles in closed tubes," J. Fluid Mech, Pasadena, 1966.

[27] D. D. Joseph, "Rise velocity of a spherical cap bubble," J. Fluid Mech., Minneapolis, 2003. [28] J. Colmenares, P. Ortega, J. Padrino and J. L. Trallero, "Slug flow model for the prediction of

pressure drop for high viscosity oils in a horizontal pipe," SPE Journals, 2011. [29] The University of Tulsa, "Seventy Fourth Semi-Annual Advisory Board Meeting Brochure and

Presentation Slide Copy," Tulsa, 2010. [30] B. Gokcal, "Effects of high oil viscosity on two-phase oil-gas flow behavior in horizontal pipes,"

Tulsa, 2005. [31] B. Gokcal, "An experimental and theoretical investigation of slug flow for high oil viscosity in

horizontal pipes," Tulsa, 2008. [32] B. Gokcal, Q. Wang, H. -Q. Zhang and C. Sarica, "Effects of high oil viscosity on oil/gas flow

behavior in horizontal pipes," SPE Journals, Tulsa, 2008. [33] I. N. Alves, O. Shoham and Y. Taitel, "Drift velocity of elongated bubbles in inclined pipes,"

Chemical Engineering Science, Tulsa, 1993. [34] A. E. Dukler and M. G. Hubbard, "Slug frequency in horizontal gas-liquid slug flow," Ind. Eng.

Chem. Process Des. Develop., 1975. [35] M. K. Nicholson, K. Aziz and G. A. Gregory, "Intermittent two phase flow in horizontal pipes:

predictive models," The Canadian Journal of Chemical Engineering, vol. 56, pp. 653-663, December 1978.

[36] P. S. Carew, N. H. Thomas and A. B. Johnson, "A physically based correlation for the effects of power law rheology and inclination on slug bubble rise velocity," Int. J. Multiphase Flow, vol. 21, no. 6, pp. 1091-1106, 1995.

[37] C. E. Shosho and M. E. Ryan, "An experimental study of the motion of long bubbles in inclined tubes," Chemical Engineering Science, pp. 2191-2204, 4 June 2001.

[38] R. F. Kroes and R. Henkes, "CFD for the motion of elongated gas bubbles in viscous liquid," BHR Group, Delft, 2003.

[39] Siemens, "The Steve Portal," 2018. [Online]. Available: https://documentation.thesteveportal.plm.automation.siemens.com/starccmplus_latest_en/index.html#page/STARCCMP/GUID-40F39626-68FC-4928-9695-77394F6B83D8=en=.html. [Accessed 16 October 2018].

[40] Siemens, "The Steve Portal," 2018. [Online]. Available: https://documentation.thesteveportal.plm.automation.siemens.com/starccmplus_latest_en/index.html#page/STARCCMP/GUID-440BD096-05C8-438C-B102-36EE986C8C06=en=.html. [Accessed 16 October 2018].

[41] Siemens, "The Steve Portal," 2018. [Online]. Available: https://documentation.thesteveportal.plm.automation.siemens.com/starccmplus_latest_en/index.html#

20

page/STARCCMP/GUID-4AAA5D04-D290-467B-9E13-D57039DD85F8=en=.html. [Accessed 16 October 2018].

[42] Siemens, "The Steve Portal," 2018. [Online]. Available: https://documentation.thesteveportal.plm.automation.siemens.com/starccmplus_latest_en/index.html#page/STARCCMP/GUID-812361C5-EEF9-4028-8A9C-EF5F730917B6=en=.html. [Accessed 16 October 2018].

[43] Siemens, "The Steve Portal," 2018. [Online]. Available: https://documentation.thesteveportal.plm.automation.siemens.com/starccmplus_latest_en/index.html#page/STARCCMP/GUID-82CD7ECC-1BB5-4DA1-B024-273814BB8E3D=en=.html. [Accessed 16 October 2018].

[44] Siemens, "The Steve Portal," 2018. [Online]. Available: https://documentation.thesteveportal.plm.automation.siemens.com/starccmplus_latest_en/index.html#page/STARCCMP/GUID-478651F1-9901-41B6-8193-3052E3CCE9DB=en=.html. [Accessed 16 October 2018].

[45] Siemens, "The Steve Portal," 2018. [Online]. Available: https://documentation.thesteveportal.plm.automation.siemens.com/starccmplus_latest_en/index.html#page/STARCCMP/GUID-AD444CAC-8F38-4825-8F38-EE6FA6EFDEF7=en=.html. [Accessed 16 October 2018].

[46] J. Moreiras, E. Pereyra, C. Sarica and C. F. Torres, "Unified drift velocity clousure relationship for large bubbles rising in stagnant viscous fluids in pipes.," Journal of Petroleum Science and Engineering, pp. 359-366, 16 September 2014.

[47] H. Pineda, T. Kim, E. Pereyra and N. Ratkovich, "CFD modeling or air and highly viscous liquid two-phase slug flow in horizontal pipes," Chemical Engineering Research and Design, pp. 638-653, 26 June 2018.

[48] P. Andreussi, M. Bonizzi and A. Vignali, "Motion of elongated gas bubble over a horizontal liquid layer," BHR Group Multiphase Production Technology, Pisa, 2009.

[49] M. Ramdin and R. Henkes, "Computational fluid dynamics modeling of Benjamin and Taylor bubbles in two-phase flow in pipes," Journal of Fluids Engineering, vol. 134, pp. 2-8, April 2012.

[50] M. Lu, "Experimental and computational study of two-phase slug flow," London, 2015. [51] B. Sanderse, M. Haspels and R. A. W. M. Henkes, "SImulation of elongated bubbles in a channel

using the two-fluid model," Journal of Dispersion Science and Technology, vol. 36, pp. 407-1418, 21 May 2015.

[52] S. Yamoah, R. Martínez-Cuenca, G. Monrós, S. Chiva and R. Macián-Juan, "Numerical investigation of models for drag, lift, wall lubrication and turbulent dispersion forces for the simulation of gas-liquid two-phase flow," Chemical Engineering Research and Design, no. 98, pp. 17-35, 13 April 2015.

21

ANNEXES Annex 1. Drift velocity adjusted error for condition 1.

Model R2 Equation # Realizable k − ε Two-Layer 0.99291 Error = 271.26eC>.K>qw (12)

Standard k − ε Low-Re 0.99884 Error = 203.88eC>.Exyw (13) Annex 2. Drift velocity adjusted error for condition 2.

Model R2 Equation # Realizable k − ε Two-Layer 0.9999 Error = 200.25eC>.KzEw (14)

Standard k − ε Low-Re 0.9988 Error = 211.91eC>.xFw (15) AKN k − ε 0.9967 Error = 181.13eC>.KyFw (16)