Nsidibe Sunday
email: nsidibe.sunday@insa-cvl.fr

Khaled Chetehouna

Nicolas Gascoin

Sunday Ikiensikimama

Numerical Heat Transfer Study of Two-Phase Oil-Water Flow in different Flowline Inclinations using OpenFOAM

Keywords: Flow assurance, Flow pattern, Heat transfer, Flowline, Two-phase flow

Estimating the oil-water temperatures in flowlines is challenging especially in deep or ultra-deepwater offshore applications where issues of flow assurance and dramatic heat transfer are likely to occur due to the temperature difference between the fluids and the surroundings. Heat transfer analysis is very important for the prediction and prevention of deposits in oil and water flowlines, which could impede the flow and give rise to huge financial losses. Therefore, the development of a 3D Computational Fluid Dynamics predictive tool for heat transfer analysis is significant to serve the need in exploring the technologically demanding and hostile subsea deepwater fields. In this work, a non-isothermal two-phase flow model is first developed and then implemented in the InterFoam solver by introducing the energy equation using OpenFOAM ® code. The Low Reynolds Number (LRN) k-ε turbulence model is utilized to resolve the turbulence phenomena within the oil and water mixtures. The flow patterns and the local heat transfer coefficients (HTC) for two-phase oil-water flow at different flowlines inclinations are validated by the experimental literature results and the relative errors are also compared. Thereafter, HTC and flow patterns for oil-water flows at downward inclinations angles can be predicted by the models. The velocity distribution and temperature variation at the flowline cross-sections are simulated and analyzed in detail. Consequently, the numerical model can be generally applied to compute the global properties of the fluid and other operating parameters that are beneficial in the management of twophase oil-water transportation.

Introduction

The exploitation of hydrocarbon resources in hostile and subsea deepwater surroundings has been driven by the continual increase in energy demand and depletion of already existing conventional oil-gas reserves [START_REF] Oddie | Experimental study of two and three-phase flow in large diameter inclined pipes[END_REF][START_REF] Sunday | Numerical Study of the Behavior of Two-Phase Hydrocarbon Flow Phenomena in Horizontal Flowlines using OpenFOAM[END_REF]. Two-phase flow in pipelines is usually experienced in the nuclear, chemical, and petroleum industries, mostly in the power generation, combustion systems, and crude oil transportation through inclined and horizontal pipelines [START_REF] Oddie | Experimental study of two and three-phase flow in large diameter inclined pipes[END_REF][START_REF] Sunday | Numerical modeling and parametric sensitivity analysis of heat transfer and two-phase oil and water flow characteristics in horizontal and inclined flowlines using OpenFOAM[END_REF]. Transportation of this two-phase hydrocarbon through such environments (about 4-5 o C) which creates many flow assurance issues usually requires long flowlines [START_REF] Oddie | Experimental study of two and three-phase flow in large diameter inclined pipes[END_REF][START_REF] Sunday | Numerical Heat Transfer Analysis of Two-Phase Flow in Horizontal and Inclined Flowline using OpenFOAM[END_REF]. Flow assurance in the petroleum industry is one of the biggest problems bedeviling hydrocarbon production and this can cause serious challenges to subsea deep and ultra-deepwater field developments [START_REF] Sunday | An Overview of Flow Assurance Heat Management Systems in Subsea Flowlines[END_REF]. This type of surroundings is characterized by low seabed temperature and high hydrostatic pressures, which may result in wax deposition thereby affecting the flow of the produced two-phase hydrocarbon fluids (oil, water) along the flowlines up to the processing facilities [START_REF] Sunday | An Overview of Flow Assurance Heat Management Systems in Subsea Flowlines[END_REF]. Also, in Fig. 1 the flowlines have sections with different inclinations due to uneven seabed which affects the flow regimes [START_REF] Amundsen | An experimental study of oil-water flow in horizontal and inclined pipes[END_REF]. Therefore, a good understanding of the behavior of hydrocarbon fluids is important in oil production and field development. Subsea heat management technique is a crucial ingredient to the successful operations of flow assurance in deepwater petroleum fields [START_REF] Sunday | An Overview of Flow Assurance Heat Management Systems in Subsea Flowlines[END_REF]. It is required during design to maintain the temperature of the fluid inside the flowlines well above the surrounding temperature. Heat analysis encompasses both steady-state and transient studies during different stages of the field's life and must serve as a standard design tool for heating systems or thermal insulation selection to avoid the formation of hydrate and wax [START_REF] Sunday | An Overview of Flow Assurance Heat Management Systems in Subsea Flowlines[END_REF][START_REF] Sunday | Numerical study and sensitivity analysis of two-phase oilwater flow and heat transfer in different flowline orientations using OpenFOAM[END_REF]. Many technical and scientific research efforts have been established in the studies of heat transfer in pipelines. In the last five decades, most published works were reported on heat transfer coefficient correlations and experimental data for forced convection in two-phase gas-liquid in horizontal and vertical pipes. These developed correlations were only applied to some flow regimes within a specific range due to limited available experimental

SPE Brazil Flow Assurance Technology Congress

Rio de Janeiro, Brazil, November 15 -18 2022 data. Most of the published two-phase heat transfer studies focused on gas-liquid. Many technical and scientific research efforts have been established in the studies of heat transfer in pipelines. In the last five decades, most published works were reported on heat transfer coefficient correlations and experimental data for forced convection in two-phase gas-liquid in horizontal and vertical pipes. These developed correlations were only applied to some flow regimes within a specific range due to limited available experimental data. Most of the published two-phase heat transfer studies focused on gas-liquid. Two-phase heat transfer process is dependent on the hydrodynamic behavior of the flows. The flow pattern of oil-water flows is completely different from the gas-oil flows and the distinction is ascribed to the small buoyancy effect and large momentum transfer capacity in the oil-water flows [START_REF] Oddie | Experimental study of two and three-phase flow in large diameter inclined pipes[END_REF]. Also, the research on heat transfer for two-phase oil/water is limited when compared to gas/liquid heat transfer. Some experimental, empirical correlations and theoretical models have been reported for estimating the heat transfer of oil/water flow at different pipeline inclinations [START_REF] Boostani | Heat transfer to oilwater flow in horizontal and inclined pipes: Experimental investigation and ANN modeling[END_REF]. However, no reported CFD model in the open literature has been employed to compute the influence of different flowline inclinations on the HTC for two-phase oilwater flows. The objective of this paper is to develop a nonisothermal two-phase heat transfer model that is robust enough to span most of the fluid combinations, flow patterns, flowline orientations, and flow rates. The developed heat transfer model is validated with the experimental results for various flow patterns and HTC of oil-water twophase flow in inclined and horizontal pipes. To this end, HTC and flow patterns for two-phase oil-water flows at downward flowline inclinations are also examined.

Mathematical Model

Several studies on the hydrodynamics and thermal modeling of different flow behavior and flowline orientations are performed, centering on computing the HTC, velocity and temperature distribution, pressure drop, and flow patterns.

Governing Equations A 3D non-isothermal twophase Newtonian flow model has been developed in this section. The governing equations comprise mass conservation, momentum conservation, and energy conservation equation. From the preceding analysis, the mathematical models of nonisothermal two-phase Newtonian flow are expressed below: Mass conservation equation. At the flowline crosssection in the axial direction, the total mass of the oil-water phases is constant. The conservation equation is shown as follows.

+ ∇. (𝜌𝑈) = 0

(1) where ρ and U are the density of the velocity of the two-phase fluid. Momentum conservation equation. Assuming an incompressible two-phase fluid flow, the Navier-Stokes and the Volume of Fluid equations of a fully developed flow are expressed as:

+ 𝛻 . (𝜌𝑈𝑈) = 𝛻𝑃 + 𝛻 . 𝜇 (𝛻𝑈 + 𝛻𝑈 ) + (𝜌𝑔𝑠𝑖𝑛𝜃) + 𝐹 (2)
+ 𝛻. (𝛼𝑈) + 𝛻 . (1 -𝛼)𝛼𝑈 ) = 0 ( 3
)
where U is the velocity in the axial direction, 𝛻𝑃 is the pressure gradients in the stream's direction, g is the acceleration due to gravity, θ is the inclination angle, μeff is the effective dynamic viscosity which is the summation of the molecular and turbulent viscosity (μ + μt), Ur is the relative velocity between phases, α is the fluid volume fraction, and Fs is the body force in the axial direction.

Boundary Conditions

The simulation of twophase oil-water flow is considered axisymmetric. At the onset of the simulation (time = 0), an equal volume fraction of oil-water was infused at the inlet domain of the pipe with different superficial inlet velocities, the outlet permits the mixed flowing fluids to come out of the domain with a pressure of zero, while the no-slip condition for velocity was assumed at the pipe wall. A uniform wall heat flux value ranging from 15,100 -24,800 W/m 2 and at atmospheric pressure (room temperature of about 25 o C) were used. Since the flow problem was deemed axisymmetric, the computational domain was specified as half-pipe, and the symmetryplane boundary condition was used in the longitudinal pipe section (X-Z plane in Fig. 2). In summary, the appropriate boundary conditions are utilized according to Table 1.

Numerical Method

OpenFOAM v.5.0 was used to simulate different two-phase flow cases. The two-phase flow is calculated by using the developed solver based on the Volume of Fluid approach. Turbulence effects and their influence on the flow were performed for all simulations using RANS models. Discretization of the transport equation is executed using the Finite Volume Method. The Euler scheme is used for temporal and time discretization and spatial discretization is a standard second-order scheme.

Model Geometry and Mesh

The 3D flowline geometry was constructed with a diameter and length of 0.011m and 1.8 m respectively. The mesh consisting of structured hexahedral cells is bounded by three types of boundary faces (inlet, outlet, and wall), as seen with the 3D flowlines in Fig. 1. Both geometry and the structured mesh were created by the blockMesh utility in OpenFOAM. However, the HTC for horizontal flow is higher than the downward/upward flows at Reso values of about 1200-1800. This difference is traceable to the flow pattern observed at these Reso values and the difference in turbulence forces. In conclusion, the proposed model was able to compute the HTC results for two-phase oil-water flows and it is a function of the flow patterns.

Table 2. Flow pattern comparison of present work

with experimental data.

Results and Discussion

The developed numerical model is first implemented to generate validated numerical results of flow patterns and HTC for oil-water twophase flow at different flowline inclinations. Thereafter, the prediction of the hydrodynamic and thermal behavior of the two-phase flow is described As can be seen, the temperature of both oil and water phases increases because of the higher surrounding temperature through the constant wall heat flux and the convective heat transfer effects. As the oil superficial velocity increases at constant water superficial velocity during a transition from Po/w to Dw/o, the temperature of oil and water keeps increasing from the horizontal flowline wall to the fluid interface, and the temperature gradient in the radial direction of the water phase is smaller than that of the oil phase gradient. This is because oil has a lower specific heat transfer heat capacity (cp) than water and the phenomenon entails heat being transferred from the wall to the oil interface faster. water horizontal flowline increases from the wall to the fluid interface, and the temperature gradient in the radial direction of the water phase is smaller than that of the oil phase gradient. In upward flow, the temperature is higher than that of horizontal flow due to a higher liquid holdup by gravitational forces in the upward flow. In the end, the developed numerical model could be generally employed to compute fluid properties and other operating variables in the two-phase flowlines transportation system.

Figure 1 .

 1 Figure 1. Schematic representation of oil-water twophase subsea production

  Turbulence model. Because of the characteristics of turbulence in two-phase fluid flow, the turbulent fluctuation imposes additional momentum and energy transfer. Therefore, turbulent models are deployed to simulate momentum and energy transfer phenomena. The LRN k-ε model is introduced to account for the near-wall treatment by the addition of damping functions into the turbulent viscosity. The turbulent kinetic energy and dissipation are defined by: ( ) + 𝛻 . (𝜌𝑘𝑈 ) = 𝛤 + 𝐺 + 𝐺 -𝜌𝜀 -𝑌 + 𝑆 (4) ( ) + 𝛻 . (𝜌𝜀Ũ ) = 𝛤 + 𝐶 𝑓 (𝐺 + 𝐺 ) -𝐶 𝑓 𝜌 + 𝐸 + 𝐷 + 𝑆 (5) where Gk and Gb are the productions of turbulence kinetic energy, Ym is the fluctuation dilation incompressible turbulence, and turbulent viscosity is 𝜇 = , 𝛤 = 𝜇 + , 𝛤 = 𝜇 + . For the k-ε model, Cµ = 0.09, C1= 1,92, and C2= 1.3 are the model constants. fµ, f1, and f2 are the functions specially employed for the near-wall treatment. E and D are additional terms to further stabilize dissipation and diffusion in the area of the walls, Sk and Sε are the source terms. Energy conservation equation. The heat-transfer process especially with the turbulent flow is valuable for both engineering and due to its occurrence in numerous industrial applications. Thus, the energy equation is added to the governing equations. Assuming a quasi-transient state in which axial heat conductive heat effect is neglected, no added source term in the energy equation. The energy equation is shown as: + 𝛻 . 𝜌𝐶 𝑈𝑇 = 𝛻 . (𝛻 𝑇) + 𝛻 . 𝛻 𝑇 (8) where T is the temperature of the two-phase fluid, Cp is the specific heat capacity of the two-phase fluid, 𝑃𝑟 = 𝐶 is the Prandtl number, and λ is the thermal conductivity of the two-phase fluid.

Figure 2 .

 2 Figure 2. Computational half-pipe domain for the axisymmetric two-phase oil-water flow. Models Validation Prediction of Flow Patterns at different Flowline Inclinations The proposed model is validated by the experimental investigation of [8]. Boostani et al. studied the heat transfer coefficient of oil-water flow in horizontal and inclined pipes. The superficial Reynolds number of the water-phase, Resw was maintained at 6772, while the superficial Reynolds numbers of the oil-phase Reso varied at intervals. To validate the studied results, the numerical results of flow patterns and HTCs for oilwater two-phase flow at different flowline inclinations are compared with experimental data. Therefore, the thermohydraulic calculations are both tested and established. The oil-water flow pattern maps for different flowline orientations are classified in line with the definitions of [8] and [1]. The observed flow patterns are stratified flow (ST), stratified flow with a mixed interface (ST & MI), plugs of oil in water (Po/w), plugs of water in oil (Pw/o), dispersion of water in oil & oil in water (Dw/o & o/w), and dispersion of oil in water (Dw/o) flows. Table 2 illustrates a comparison between numerical and measured flow patterns visualized at different Reso (≈ 450 -3750) and Resw kept at 6772. The 3D non-isothermal two-phase flow model predicted the observed flow patterns of oilwater flow very reliably. For instance, in horizontal flow, the performance of the model was remarkably good for predicting Po/w, ST & MI, Dw/o & o/w, and Dw/o flows. Also, for flowline inclinations of -10 o , -4 o , and +10 o , the flow patterns visualized using the proposed model are mostly ST & MI, with Po/w, Dw/o & o/w, and Dw/o only observed at low and very high Reso. Varying the inclination angles downward reduces the gravitational forces, which are most pronounced at upward inclinations, and tends to decelerate the denser water phase in the upward direction. Water is the fastest-moving phase during the downward flow and it reduces the hold-up. Effect of flowline inclinations and Flow pattern on HTC Figure 2 describes the comparison between the numerical and measured HTC values under various Reso of about 450 -3750 when the Resw is 6772 at different flow inclinations (0 o , +4 o ). The numerical HTC results are slightly different from the experimental values as the flow patterns changes from Po/w to Dw/o. The relative errors

Figure 3 .

 3 Figure 3. The HTC variation at different 3D flowline inclinations as a function of Reso.

  Liquid holdup, HL The average HL at the flowline cross-section is expressed in Fig.4for flow having Reso values of about 450 -2750 and Resw = 6772. Po/w is observed at Reso of 450 with a higher water HL than oil and in the continuous water phase, SI & MI is noticed as Reso increases above 450 showing almost equal HL of oil and water in the continuous. oil-water phases. At Reso lower than 2750, Dw/o & o/w flow regime is observed that the HL of water decreases during the transition, which indicated that the dispersed water continuous flow moved much faster than the dispersed oil continuous phase. Additional increase in Reso to ≈2750 results in Dw/o where HL of water further reduces, which demonstrated that the dispersed water droplets moved faster than the continuous oil phase. it can be seen that the water droplet concentration increased towards the flowline midpoint where the velocity is highest. Lift forces were responsible for the increased droplet concentration at the midpoint.

Figure 4 .

 4 Figure 4. The liquid fraction in the flowline length of 0.9m Pressure along flowline length The value of the pressure gradient along the flowline length was investigated at different Reso values (450 -2750) while maintaining Resw = 6772 as shown in Fig. 5. It is observed in horizontal flow that the frictional pressure gradient increases in the transition from Po/w to Dw/o flow due to increased turbulence forces. Based on Po/w, when the inlet oil flow rate increases, the oil-water flow pattern transition to ST & MI. The pressure variation rose as the water continuous phase in Po/w was replaced with an oilwater continuous phase, which increases the wall friction and frictional pressure drop. Increasing the oil flow rates, the oil-water flow changes to Dw/o & o/w flow regime, and the pressure variation increases because of the high inlet velocity and the amount of continuous oil-water phases in the flowline. Based upon Dw/o & o/w, a further increase in the inlet oil flow rates, the oil-water flow changes to Dw/o. The pressure gradient further increases as the continuous oi-water phases are substituted with an oil continuous phase, which increases the frictional pressure drop. However, due to the influence of the hilly-terrain flowline as seen in this study. The frictional pressure gradients at the different flow transitions from Po/w to Dw/o are in general noticed to be lower than horizontal flow. This was ascribed to a higher in-situ water fraction or holdup in the upward and reduced

Figure 5 .Figure 6 .

 56 Figure 5. Pressure gradient at the flowline length of 1.8m

Table 1 .

 1 Boundary Conditions.

Acknowledgments

The author expresses heartfelt appreciation to the Petroleum Technology Development Fund (PTDF) for funding the Ph.D. thesis.

Responsibility Notice

The authors are responsible for the paper content.