An Eulerian-Eulerian Approach for Oil & Gas Separator ... - OnePetro

AN EULERIAN-EULERIAN APPROACH FOR OIL&GAS SEPARATOR DESIGN N. Scapin1, L. Cadei2, M. Montini2, G. Montenegro1, A. Bianco2, S. Masi2 1 Politecnico di Milano, 2ENI SpA. This paper was presented at the 13th Offshore Mediterranean Conference and Exhibition in Ravenna, Italy, March 29-31, 2017. It was selected for presentation by OMC 2017 Programme Committee following review of information contained in the abstract submitted by the author(s). The Paper as presented at OMC 2017 has not been reviewed by the Programme Committee.

ABSTRACT This paper presents a novel CFD analysis of an Oil& Gas separator, based on a multi-fluid EulerianEulerian model of the Navier-Stokes equations, implemented in OpenFOAM®. The simulation of a three-phase separator poses a particular challenge to the numerical modeling of transport phenomena since the three-phase flow can span across multiple flow regimes from disperse to separate. To handle such complex behavior, a new three-phase Eulerian-Eulerian solver has been implemented in OpenFOAM with a fully implicit treatment of drag terms and with the capability to describe both disperse and separate flow at high, fully coupled phase fractions. Furthermore, the mixture turbulence model implemented in OpenFOAM for bubble flows has been improved. Firstly, the source term of the turbulent kinetic energy has been modified with a more regime-independent formulation derived from the literature. Then, the derivation of the same model has been extended in order to manage the three phases. The work represents an improvement both from an academic and industrial perspective: it provides a consistent numerical framework for a multiphase flow involving a number of phases higher than two; it replaces the traditional Eulerian-Lagrangian approach with the more appropriate EulerianEulerian one for the analysis of industrial production facilities. These two aspects allow to describe more accurately the flow pattern transitions and to numerically capture the separation and phase inversion phenomena inherent to the system. LIST of SYMBOLS - 𝛼𝑘 : phase fraction; ̅𝒌 : phase-average velocity; - 𝒗 - 𝜌𝑘 : phase density; - 𝑝̅ and 𝑝̅𝑟𝑔ℎ : phase-average pressure and phase-average modified pressure; - 𝑴𝑰,𝒌𝒋 : momentum exchange across interface between phase k and phase j; -

𝒆𝒇𝒇

𝑹𝒌 : combined Reynolds (turbulent) and viscous stress; 𝐶𝑑 : drag coefficient; 𝐶𝑑𝑅𝑒: product of drag coefficient and particle Reynolds number; 𝑅𝑒𝑝 : particle Reynolds number; 𝑑𝑘 : particle diameter; 𝜈𝑘 : phase kinematic viscosity; 𝑘𝑘 : turbulent kinetic energy; 𝜀𝑘 : dissipation rate; 𝜈𝑘𝑡 : turbulent kinematic viscosity; 𝐶𝑡,𝑘𝑗 : turbulent response coefficient of phase k with respect to phase j; 𝑉𝑡,𝑘𝑗 : terminal velocity; 𝐠: acceleration field;

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INTRODUCTION Nowadays, multiphase flows can be considered as the standard condition for the majority of the Oil&Gas production fields. The reasons for this peculiarity are numerous and can be related to different parts of the hydrocarbons production chain. As an example, injection of fluids to sustain production as well as the presence of an active gas cap or an aquifer determine the existence of at least two different phases in the upcoming flow streams. Moreover, the difference in pressure between the wellbore and the production tubing determines the production of a certain (and in many cases non negligible) amount of gas (if the bubble point condition is reached) and the possible arising of non-ideal volumetric behaviors (e.g., retrograde condensation). Among the different challenges deriving from the inherent multiphase nature of the upcoming reservoir flows, a key issue is represented from the separation process that the fluids must undertake. In the present paper, a consistent CFD framework for handling the separation process that occurs in an oil&gas separator is presented. This framework can be essentially divided in two parts: 1. A simplified geometry for an oil and gas separator that find a compromise between the requirements in term of mesh generation and computational analysis and the possibility to provide suitable information concerning the internal fluid-dynamics; 2. An Eulerian-Eulerian three-phase solver able to capture the separation and assess its efficiency. Moreover, a proposal of a three-phase turbulence model has been included to close the system of equations without exceeding the computational cost; The principle mean of investigation is represented by the OpenFOAM, a C++ software based on the object-oriented programming. The open-source nature of this tool alongside with the vast multiphase library already implemented makes it ideal for the development, customization and optimization of new solvers.

OIL and GAS SEPARATORS Gas/liquid separators are process equipment used for the bulk gravity-based separation of gas, oil and water from a multiphase flow stream. Even though the major components are oil, water and gas, it is often present a small amount of solid particles; however, the term three-phase separator remains. The goal of a well-designed separator or group of separators is to provide an almost pure stream of three main components of the inlet mixture. The effectiveness of this process is of tremendous importance in the field treatment plant for the following reasons:  Liquid in gas carryover can damage compressors downstream and waste valuable products;  Oil in water carry under can overload water treatment plant and water in oil carry under can overload oil treatment plant;  Gas transported in liquids can damage pumping equipment downstream and create hazardous conditions upon gas release;  Solids accumulated in liquids can impair emulsion separation downstream and create accumulations in electrostatic components; Oil& Gas industry uses a great variety of separators that are often divided in the following four broad categories [1]: 1. Conventional separators: conventional gas-liquids separators are usually characterized as vertical, horizontal or spherical. Horizontal separators can be single or double barrel and can be equipped with sumps or boots. They may be further classified as two or three-phase, whereas vertical separators handles a two-phase mixture. Regardless of shape, separation vessels usually contain four major sections plus the necessary controls: inlet device, gas gravity separation, liquid-gravity separation and mist extraction. 2. Centrifugal gas-liquid separators: centrifugal separators utilize centrifugal action for the separation of materials of different densities and phases. They are built in stationary and rotary types. Various modifications units are used more than any other kind. Centrifugal separators are generally divided into three types: stationary vane separators, cyclone separators, inertial centrifugal separators. 3. Flare knock-out drums: flare knock-out (KO) drums are one type of gas-liquid separators that are used specifically for separation of liquids carried with gas streams flowing to the flares in an Oil and Gas plants. 2

4. Filters separators: gas-liquid filter separator (usually called filter separator) is used in separation of liquid and solid particles from a gas stream. A gas filter has a higher separation efficiency than the centrifugal separator, but is uses filter elements that must periodically be replaced. In the present paper, the focus is on a horizontal, three-phase conventional separator (Figure 1). It is worth saying that the mathematical model employed to describe the separation process and the numerical algorithm to solve the associated equations are general and can be rigorously applied to other configurations and geometry.

Figure 1: Oil and gas separator

MATHEMATICAL MODEL In the field of the multiphase CFD analysis, the main classes of numerical techniques are usually divided in the following three groups: 1. Discrete phase elements methods: the discrete phase elements method assumes that the topology of the two-phase is dispersed. The two phases are therefore referred as the continuous and the dispersed phase with the former treated from an Eulerian perspective (phase properties regarded as a field and described in an absolute reference frame), while the latter under the Lagrangian framework (phase properties described in frame moving with the particle). This approach is considered appropriate when the phase-fraction of the dispersed phase does not overcome the limit of 6-10 %; 2. Multi-fluid methods: under the multi-fluid formulation, each phase is described with a set of mass, momentum and energy equation. These equations are derived from the instant local formulation with an appropriate averaging process that introduces additional terms for which a modelling formulation is required. This method is particularly appropriate for treating flow with an arbitrary number of phases that span the entire value of phase fractions, face separation and inversion phenomena without a clear interface among them; 3. Interface resolving methods or direct numerical simulation (DNS): unlike the previous classes that do all require a priori information on the flow regime and knowledge on the size of bubbles and drops, these methods are able to reconstruct the position and the shape of the interface and capture arbitrary topological changes in the flow. Unfortunately, the present computational capabilities restricts these methods to a limited range of applications. Moreover, a realistic reconstruction of the interface coupled with a model for the superficial tension requires finer grid with respect to the other model adding additional computational effort to simulations that involves this approach.

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For the aforementioned reasons combined with the available computational resources, the multifluid or Eulerian-Eulerian model has been employed in the present work. For this purpose, only two equations are needed, i.e. mass and momentum equation (for each phase): 𝜕(𝛼𝑘 𝜌𝑘 ) ̅𝑘 ) = 0 + ∇ ⋅ (𝛼𝑘 𝜌𝑘 𝒗 𝜕𝑡 ̅𝑘 ) 𝜕(𝛼𝑘 𝜌𝑘 𝒗 ̅ 𝑒𝑓𝑓 ̅𝑘 𝒗 ̅𝑘 ) + ∇ ⋅ (𝛼𝑘 𝑹 + ∇ ⋅ (𝛼𝑘 𝜌𝑘 𝒗 𝑘 ) = −𝛼𝑘 ∇𝑝̅ + 𝛼𝑘 𝜌𝑘 𝒈 + 𝑴𝑰,𝒌𝒋 𝜕𝑡 Two manipulations must be performed on the previous equations in order to provide a stable and robust discretization. Regarding the mass conservation, the convective term is expanded and expressed in terms of the total averaged velocity and the relative velocities of the generic phase k and the other phases. Assuming incompressible phases, the final expression reads as: 3

𝜕𝛼𝑘 ̅) + ∇ ⋅ ( ∑ 𝛼𝑗 𝛼𝑘 𝒗 ̅𝑟,𝑘𝑗 ) = 0 + ∇ ⋅ (𝛼𝑘 𝒗 𝜕𝑡 𝑗=1,𝑗≠𝑘

The analysis of the momentum equation shows that at the boundary, where a non-slip impermeable boundary condition is usually applied, it reduces to: ∇𝑝̅ = 𝜌𝑘 𝒈 In presence of a multiphase flow with potential large density variations among phases, the wall pressure gradient may assume different values. Since this possibility is unphysical, the pressure 𝑝 is replaced with a corrected pressure 𝑝̅𝑟𝑔ℎ defined as ̅𝑝𝑟𝑔ℎ = 𝑝̅ − 𝜌𝑔ℎ. As a result, the momentum equation becomes: ̅𝑘 ) 𝜕(𝛼𝑘 𝜌𝑘 𝒗 ̅ 𝑒𝑓𝑓 ̅𝑘 𝒗 ̅𝑘 ) + ∇ ⋅ (𝛼𝑘 𝑹 + ∇ ⋅ (𝛼𝑘 𝜌𝑘 𝒗 𝑘 ) = −𝛼𝑘 ∇𝑝̅𝑟𝑔ℎ − 𝛼𝑘 𝑔ℎ∇𝜌 + 𝛼𝑘 (𝜌𝑘 − 𝜌)𝒈 + 𝑴𝑰,𝒌𝒋 𝜕𝑡 Where 𝜌 is the mixture density. The momentum equation requires two additional closure to be solved: the momentum exchange ̅ 𝑒𝑓𝑓 source 𝑴𝑰,𝒌𝒋 and the turbulence term 𝑹 𝑘 . Momentum exchange closure The term 𝑴𝑰,𝒌𝒋 accounts for the momentum exchange contribution among phases across the interface. In the contest of the Eulerian-Eulerian model, the closure relation is generally reported as the sum of drag force, virtual mass force and lift force. Since the dispersed phase is modelled as a solid sphere of small diameter and without possibilities of distortion, the contribution of the lift force is negligible compared to the other momentum exchange source and, therefore, omitted. The final closure expression is, therefore: 𝐶𝑑𝑅𝑒𝑘𝑗 𝜌𝑗 𝜈𝑗 𝐶𝑑𝑅𝑒𝑗𝑘 𝜌𝑘 𝜈𝑘 3 ̅𝑟 𝑴𝑰,𝒌𝒋 = (𝛼𝑘 𝑓𝑘 + 𝛼𝑗 𝑓𝑗 )𝒗 2 4 𝑑𝑘 𝑑𝑗2 ̅𝑗 ̅𝑘 𝐷𝒗 𝐷𝒗 + (𝛼𝑘 𝑓𝑘 𝐶𝑣𝑚,𝑘 𝜌𝑘 + 𝛼𝑗 𝑓𝑗 𝐶𝑣𝑚,𝑗 𝜌𝑗 ) ( − ) 𝐷𝑡 𝐷𝑡 The factor 𝑓𝑘 and 𝑓𝑗 represents the blending parameters used to provide a closure over the full range of phase fractions. The factor 𝑓𝑘 and 𝑓𝑗 goes to 1 as 𝛼𝑘 and 𝛼𝑗 approach 1, respectively. The variation trend of the two factors for intermediate value of phase fraction must be modelled. In the present work, this variation follows a linear trend. Therefore, in case of phase inversion phenomena, the 4

algorithm automatically takes into account the phase changes from continuous to dispersed. Clearly, with this method it is not necessary to specify which phase is continuous and which phase is dispersed, since it treats all the phases in the same way without considering a predominant one. The calculation of the term drag coefficient 𝐶𝑑𝑅𝑒𝑘𝑗 and 𝐶𝑑𝑅𝑒𝑗𝑘 relies on the Schiller-Naumann formulation, whose original formulation is (blue curve in figure 2): 𝐶𝑑𝑅𝑒 = 𝑚𝑎𝑥[24(1 + 0.15𝑅𝑒𝑝0.687 ); 0.44𝑅𝑒𝑝 ] Where 𝑅𝑒𝑝 represents the particle Reynold’s number. In the present work, we found that this formulation was not suitable for liquid-liquid separation where the density of the two phases can be very close. The main reason for this behaviour the fact that the term 𝐶𝑑𝑅𝑒 goes to 24 as the particle Reynold’s number goes to zero. Mathematically, the separation process would end when the relative velocity between two phases goes to zero; nevertheless, the relative velocity is never strictly zero in the computational domain, providing a continuous obstacle to the separation. For this reason, a modification to the formulation for the drag coefficient was proposed so that as the particle Reynold number goes to zero also the term 𝐶𝑑𝑅𝑒 goes to zero. In order to do so, two alternatives have been considered: a jump formulation (red curve in figure 2) that zeros the term 𝐶𝑑𝑅𝑒 below a cut-off value of 𝑅𝑒𝑝 (𝑐𝑅𝑒𝑝 ) and a polynomial function (green curve in figure 2) to provide a smooth variation of 𝐶𝑑𝑅𝑒 between 0 and 𝑐𝑅𝑒𝑝 . In order to guarantee the continuity of the formulation, the latter approach was chosen with a 𝑐𝑅𝑒𝑝 equal to 1. Clearly, a full and rigorous explanation of this choice would require an experimental campaign to find a momentum consistent formulation of the drag coefficient. In absence of experimental data, this preliminary solution is proposed to test the numerical stability and convergence of the solver in presence of a three-phase flow.

Figure 2: CdRe as function of particle Reynolds number for three formulation (Original, Jump and Polynomial)

Turbulence closure In the present work, the multiphase turbulence is handled under the two-equation mixture 𝑘 − 𝜀 model. The governing equations are: 𝑡 𝜕 𝜇𝑚 (𝜌𝑚 𝑘𝑚 ) + ∇ ⋅ (𝜌𝑚 𝒗 ̅𝑚 𝑘𝑚 ) = ∇ ⋅ [ ∇𝑘𝑚 ] + 𝐺𝑚 − 𝜌𝑚 𝜀𝑚 + 𝑆𝑘𝑚 𝜕𝑡 𝜎𝑚 𝑡 𝜕 𝜇𝑚 𝜀𝑚 𝜀𝑚 𝑚 (𝜌𝑚 𝜀𝑚 ) + ∇ ⋅ (𝜌𝑚 𝒗 ̅𝑚 𝜀𝑚 ) = ∇ ⋅ [ ∇𝜀𝑚 ] + (𝐶1 𝐺𝑚 − 𝐶2 𝜌𝑚 𝜀𝑚 ) + 𝐶3 𝑆 𝜕𝑡 𝜎𝑚 𝑘𝑚 𝑘𝑚 𝑘 The derivation of this model is based on the summation of a set of 𝑘 − 𝜀 models written for the single phase. In order to relate the turbulent quantities of all phases with respect to one of them, the concept of the response coefficient is introduced. Rigorously, the response coefficient is defined as the ratio the dispersed phase velocity fluctuations to those of the continuous phase, and is defined as: 5

𝐶𝑡,𝑘𝑗 =

̅𝑗 ′ 𝒗 ̅𝑘 ′ 𝒗

As the phase fraction increases, the dominance of the continuous phase on turbulence gradually diminishes as that phase becomes confined to thin interstices between dispersed phase elements. In the limit of unity dispersed phase fraction, this phase becomes continuous and its turbulence becomes the dominant/sole factor. Moreover, experiments [2-3-4] have proved at as the phase fraction increases beyond a certain limit, which could be as small as 6 %, the ratio of the dispersed to continuous phase fluctuations, 𝐶𝑡,𝑘𝑗 , approaches a constant value close to unity. Based on the scaling approach, the turbulent quantities 𝑘2 and 𝑘3 can be computed from 𝑘1 . The same approach can be applied to the dissipation energy equation. The last aspect to deal with consists in finding a suitable closure for the turbulent kinetic energy source term 𝑆𝑘𝑚 . Issa, Behazadi and Rusche [2] derived a model for two-phase flow at high-phase fraction: 2

̅𝑟 ] 𝑆𝑘,𝑘𝑗 = −𝐴𝑑,𝑘𝑗 [2𝛼𝑗 (𝐶𝑡,𝑘𝑗 − 1) 𝑘𝑘 + 𝜂𝑘 ∇𝛼𝑗 ⋅ 𝒗 The previous equation is derived under the assumption that the only contribution to the generation of turbulent kinetic energy is represented from the drag force exchange across the interface. Based on this idea, our proposal is based on applying the superimposition principle for the drag turbulent contribution, exploiting the idea of binary momentum interaction between phases. In this way, the source term can be re-written as: 3

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𝑆𝑘 = − ∑ 𝐴𝑑,𝑘𝑗 [2𝛼𝑗 (𝐶𝑡,𝑘𝑗 − 1) 𝑘𝑘 +

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𝜂𝑘t ∇𝛼𝑗

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̅𝑗 − 𝒗 ̅𝑘 )] ≈ − ∑ 𝐴𝑑,𝑘𝑗 [2𝛼𝑗 (𝐶𝑡,𝑘𝑗 − 1) 𝑘𝑘 ] ⋅ (𝒗

𝑘=1,𝑗≠𝑘

𝑘=1,𝑗≠𝑘

One of the numerical problem that may arises from this formulation is concerned with the very high value that term ∇𝛼𝑗 may assume across the interface, leading to serious numerical instabilities. For this reason, the gradient of the phase fraction is neglected in the present simulations. This model is not claimed to be completely rigorous and is only intended to serve as a preliminary vehicle to test the numerical solution procedure until a better closure model becomes available.

NUMERICAL MODEL The discretization procedure of mass and momentum equation is done using the finite volume method in a collocated variable arrangement. Alongside the standard issues in the numerical solution of flow equations, the presence of multiple, fully coupled number of phases with arbitrary density difference determines three additional challenges: phase-fraction boundness, implicit pressurevelocity coupling and elimination algorithm for drag term contributions. Phase-fraction boundness In order to ensure the phase-fraction boundness, the continuity equation is discretized and solved in in the form [5]: 3

𝜕⌊𝛼𝑘 ⌋ ⌊ ⌋ + ⌊∇ ⋅ (𝜙⌊𝛼𝑘 ⌋𝑓 )⌋ + ⌊∇ ⋅ (⌊𝛼𝑘 ⌋𝑓 ∑ 𝛼𝑗 𝜙𝑟,𝑘𝑗 )⌋ = 0 𝜕𝑡 𝑘=1,𝑘≠𝑗

̅, the second term Since the volumetric mixture flux 𝜙 satisfied the continuity of the mixture velocity 𝒗 is bounded in [0;1]. Problems for the boundness of the variables arise from the third term. The bounding of the solution is achieved by using the relative face flux 𝜙𝑟,𝑘𝑗 to interpolate 𝛼𝑘 to the face while – 𝜙𝑟,𝑘𝑗 is used to interpolate 𝛼𝑗 to the face. The discretization is performed using bounded scheme and the solution is achieved in the domain [𝜀; 1 − 𝜀] (with 𝜀 ≪ 1) for the three-phases. Finally, each phase fraction is normalized to ensure that the overall continuity is always guaranteed.

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Implicit pressure-velocity coupling The numerical discretization of the Navier-Stokes equations in pressure-based solvers requires an additional equation to handle the coupling between pressure and velocity in the momentum equation. In the present work the derivation of the pressure equation follows the approach used in OpenFOAM to relate pressure and velocity in a collocated variable arrangement. This choice requires the use of the Rhie-Chow interpolation in order to avoid pressure oscillations and check-board pattern. Starting ̅ = 0 and the semi-discretized form of the momentum from the incompressibility constrain ∇ ⋅ 𝒗 equation for each phase, the pressure equation for a multiphase incompressible system is: 3

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𝛼𝑘,𝑓 𝛼𝑘 𝑯𝒌 ∇ ⋅ {[∑ 𝛼𝑘,𝑓 ( ) ] ∇𝑝𝑟𝑔ℎ } = ∇ ⋅ [∑ 𝛼𝑘,𝑓 ( ) ] −∇ ⋅ {∑ ( ) [𝒈 ⋅ 𝒉∇𝜌 − 𝛼𝑘 |𝒈|(𝜌𝑘 − 𝜌)]𝑓 } 𝑨𝒌 𝑓 𝑨 𝑨 𝒌 𝒌 𝑓 𝑓 ⏟ ⏟ 𝑘=1

𝑘=1 𝑇𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡 𝑓𝑙𝑢𝑥𝑒𝑠

𝑘=1

𝐺𝑟𝑎𝑣𝑖𝑡𝑦−𝑏𝑜𝑢𝑦𝑎𝑛𝑐𝑦 𝑓𝑙𝑢𝑥𝑒𝑠

The term 𝑨𝒌 and 𝑯𝒌 represent the diagonal part and off-diagonal part of the linear system system associated to the generic k-phase velocity, respectively. It is worth noticing that the drag terms are absent in the pressure equation since their overall contribution is zero (in absence of capillary and surface tension contribution). This allows to reconstruct firstly the pressure field and then the slip fluxes and slip velocities, two quantities that enter in the elimination algorithm used to treat the nonlinear drag terms. Elimination algorithm for drag terms The direct solution of the momentum equation for each phase is not possible since each equation is coupled with the others due to the drag terms. In the present work a direct algorithm to “partialeliminate” the coupled term is implemented. The drag terms are therefore split in two contributions: the first is treated in the discretization of the momentum equation without pressure and gravity terms. The second one is treated in the pressure equation in the following way: 3

𝜙𝑘𝑛

𝑛 𝑛 ̅𝑛−1 − ∑ 𝐷𝑗𝑘 (𝒗 𝑘 )𝜙𝑗 = 𝜙𝑘,𝑠 𝑗=1,𝑗≠𝑘

𝑛 where 𝜙𝑘,𝑠 represents the flux associated to the k-phase taking into account of all contributions apart from drag terms. The previous equation can be written for each phase, yielding, in presence of three phases, to a system of three equations with three unknowns. Moreover, the system is linear and can be easily solved by substitution since drag coefficient terms 𝐷𝑗𝑘 is evaluated with the velocity at the previous iteration and, therefore, known. This treatment of drag terms is purely implicit and aims at reproducing numerically the physical coupling among phases. Finally, the same algorithm is used for the calculation of the velocity field: 3

̅𝑛𝑘 𝒗

̅𝑛−1 ̅𝑗𝑛 = 𝒗 ̅𝑛𝑘,𝑠 − ∑ 𝐷𝑗𝑘 (𝒗 𝑘 )𝒗 𝑗=1,𝑗≠𝑘

̅𝑛𝑘,𝑠 represents the velocity associated to k-phase taking into account of all contributions apart where 𝒗 from drag terms. This is computed with the “reconstruct algorithm” implemented in OpenFOAM to derive from the volumetric fluxes the corresponding velocity fields.

MODELLING ASSUMPTION The challenges arising from a CFD analysis of an Oil& Gas separator requires certain simplifying assumption that allows to treat it mathematically and numerically. Our assumptions can be divided in two groups: assumptions on the geometry and assumptions on the internal fluid-dynamics.

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Assumption on the geometry Our simplified geometry consists in:  One inlet and three outlets, one for each phase;  One weir in the middle of the two heavy liquid outlets;  Two different inlet pipe shapes: elbow and double elbow;  No internals;

Figure 3: Standard geometry, simplified geometry

It is worth highlighting that the previous assumptions provide a very simplified model of an oil and gas separator. This is visible by comparing a standard geometry with a simplified one as it is reported in figure 3. Today, all separators in use are characterized of at least more complex inlet devices, internals to promote coalescence and in some cases porous set in the gas outlet to avoid that even the small liquid droplets goes outside and reaches the downstream equipment. Nevertheless, the industrial validation procedure is essentially based on the same assumption we have made in the present work: since all the internals and other geometry complexities are designed to promote a more efficient separation, neglecting them means to consider the worse case and, therefore, have a safer margin when a real separator will be employed. Finally, a change in the shape of liquids’ outlet has been done in order to face the problem of setting the correct boundary conditions. In fact, the difficulties are related to two aspects: 1. The flow is multiphase and, therefore, the usual assumption of fully-developed flow at outlets cannot be considered strictly corrected; 2. Liquids outlet are located at different heights and a careful choice must be done in order to avoid pressure unbalances; In order to provide a solution for both issues, the proposed approach is to relate the gas and liquid outlet pressure with the hydrostatic law, as we can observe from figure 4. In this way, the phases at the outlets are completely separated, making the assumption of fully developed flow more reasonable and at the same time the pressure differences are respected.

Figure 4: set-up of b.c.

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Moreover, this choice allows simulating one of the control system method used in separators, known as level control techniques. In the design phase, each separator is characterized by at least two levels that can be reached from the liquid column: the normal liquid level (NLL) and the high liquid level (HLL). Of course, these two values must be provided for each liquid phase. When the upper limit is reached the operating conditions must change e.g., with a reduction of inlet flow rate or with an increase in the outflow liquid velocities. In our case, setting a pressure difference proportional to air density makes the gas height almost constant. If during the separation process the liquid height goes beyond the limit, the boundary condition at the outlets promotes the “elimination” of the liquid in excess, restoring the equilibrium between phases to fulfil the pressure constrain. Results of these choice are shown in figure 5a and figure 5b. We notice that in case the shape of liquid outlet is not modified, the complete separation is not occurring even after 700 s.

Figure 5: alpha_air and |v_air| arrows (t=700s) in air-water mixture

Assumption on the fluid-dynamics Regarding the fluid-dynamics aspects, our assumptions deal with two aspects: 1. Phase physical properties: the mixture is composed of three phases (gas, oil and water), which are pure and with constant properties; 2. Flow characteristic: the flow is incompressible and adiabatic with absence of physical reactions. Moreover, it can span only the disperse and separate flow regime;

PRELIMINARY RESULTS Before testing the code in a separator, the code has been validated in order to ensure the correctness of the algorithm in addressing the previously explained numerical aspects. For this purpose, two analytical benchmarks have been used: water faucet problem and phase-redistribution problem. Water Faucet problem The Water faucet problem [8] represents a standard benchmark for two-phase flow solvers. Even though the present code has been implemented with the capabilities of handling three independent phases, if two of them represent the same phase with phase fractions equally divided, the overall code should degenerates to a two phase equivalent solver. This test consists of a vertical tube of 12 m length and 1 m in diameter. The tube is initially filled with an air-water homogeneous mixture with a liquid phase fraction 𝛼𝑙0 = 0.8. At the inlet a mixture of the same composition enters the tube with a liquid velocity of 𝑣𝑙 = 10 𝑚/𝑠, whereas the gas is at rest (i.e., 𝑣𝑔 = 0 𝑚/𝑠). Due to gravity acceleration and mass conservation, the liquid vein diameter decreases and a phase fraction discontinuities propagates downward. Here the momentum transfer term is negligible (i.e., drag and virtual mass terms absent) and the dominant effect is the gravity force. The analytical solution for 𝛼𝑙 is given by:

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𝛼𝑙 (𝑥, 𝑡) =

𝛼𝑙0 𝑣𝑙0 √2𝑔𝑥 + (𝑣𝑙0 )2

𝑖𝑓 𝑥 ≤ 𝑢𝑙0 𝑡 +

𝑔𝑡 2 2

In figure 6, we plotted the analytical solution against the numerical one, computed with two discretization schemes: Upwind (1st order), Van Leer (2nd order). Due to time-dependent nature of the solution, we considered the results at 𝑡 = 0.5 s. As expected, 2nd order scheme produces better results in shock-capturing abilities reducing the numerical diffusion across the interface. On the other hand, 1st order schemes show the highest tendency to smear the discontinuities. Alongside with the good agreement with the analytical benchmark, this test proves that the first two numerical aspects previously explained are correctly implemented in the algorithm.

Figure 6: Water Faucet problem

Phase redistribution problem As a three-phase benchmark, phase redistribution [9] represents a test to verify the numerical convergence of multiphase codes. In this case, we consider a closed column of 2.5 m height and 0.1 m in diameter. The system is uniformly filled with a three-phase mixture composed of 0.2, 0.4 and 0.4 gas, oil and water volume fraction, respectively. We start from this an unphysical initial condition and let the system reach the equilibrium condition and separate the light and heavy phases. It is interesting to visualize the time-evolution of the phase fraction field and pressure field.

Figure 7: pressure distribution (left) and gas distribution profile (right)

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Figure 8: oil (left) and water (right) fraction profile

From figure 7 and 8, we can conclude that the algorithm correctly converges in 400 s to the solution of perfect phase separation. This is visible not only analysing the phase distribution at the latest instant but also considering the pressure distribution that progressively converges to the well known static solution (figure 7a). Regarding the phase fraction distribution, we can notice that gas fraction, due to the limited density compared to liquids, tend to separate immediately by forming a singlephase domain within 10 s of simulation (figure 7b). For the remaining part of the simulation, the threephase settling reduces to a two-phase separation process between oil and water. This is quite visible observing the specular nature of phase fraction profile for water and oil at t=250 s (figure 8). Finally, at t=400 s, each phase forms the correct hold-up according to its own initial composition.

SEPARATOR TESTS Finally, we present the results of the simulation for a geometry that reproduces more realistically the separator layout. In this case, a mixture composed of gas, oil and water is provided from the inlet nozzle located at the top of the separator (phase properties are reported in table 1). Phase Air Oil Water

Density [kg/m3] 1.2 800 1000

Viscosity [Pa⋅s] 1.84 ⋅ 10−5 3.64 ⋅ 10−2 3.94 ⋅ 10−4

𝛼𝑖𝑛𝑙𝑒𝑡 [-] 0.90 0.05 0.05

𝑣𝑖𝑛𝑙𝑒𝑡 [m/s] 2 2 2

Table 1: Phase properties

Three outlets for three phases are located at different heights with prescribe boundary conditions for the modified pressure in order simulate the control level techniques. The whole domain is initially filled with the gas phase. Since the gas separates immediately from the liquids stream, the focus is on the effectiveness of separation of the mixture oil-water. We report the phase distribution of oil at different time-instant (in figure 9):

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Figure 9: the first three figures represent the oil distribution at 50, 100, 200 sec. The last one represent all three-phase completely separated at 500 s (alphas field) with water (red), blue (oil), grey (air)

The inlet mixture separates in the three main components with an efficiency of 99 % within 500 s (around 8 min). Following the simple rule of thumb of API standard [10] (in this case, the oil gravity is 45°API), the retention time is in the correct range for effective separation.

Figure 10: Sketch of the separator vessel to highlight the zone where liquids separate from gas

In figure 10, it is reported the zone where it is assumed that occurs most of liquids separation from the gas stream. According to the industrial validation process, this area is sized using the Stokes law that provides the terminal velocity of single droplet immersed in an infinite medium at rest: 4𝑔𝑑𝑃 (𝜌𝑘 − 𝜌𝑗 ) 𝑉𝑡,𝑘𝑗 = √ 3𝜌𝑘 𝐶𝑑,𝑘𝑗 Clearly, the assumptions behind the Stokes law are generally not satisfied inside the separator where the convection processes represents a continuous obstacle to separation. Moreover, the terminal velocity is generally not uniform along the vessel. For this reason, it is worth analysing the vertical components of the two liquids 𝑣𝑤,𝑥 and 𝑣𝑜,𝑥 computed with a three-dimensional approach and evaluate the deviation from their ideal values provided from the previous equation. To make a significant comparison, we selected a zone of the separator sufficiently distant both from the inlet section and the gas-liquid interface where local phenomena render the assumptions behind the Stokes-law not valid. In figure 11, we report the comparison between the vertical components of liquids velocity and the terminal velocity. We notice that the local values of the liquids velocity may be significantly different from those provided using the simplified analytical model. Moreover, it is possible to see the inherent variation of the local settling velocities of liquid phases with respect to the gas stream. Nevertheless, we can observe that the average values of 𝑣𝑤,𝑥 and 𝑣𝑜,𝑥 are much more consistent with the respective terminal velocities, confirming that this value represents a useful average quantity for the sizing of oil and gas separators.

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Figure 11: water velocity vertical components (left), oil velocity vertical components (right)

CONCLUSIONS and FURTHER DEVELOPMENT A new Eulerian-Eulerian three-phase solver was proposed for the preliminary analysis of the separation process inside an Oil&Gas separator. The mathematical formulation of the governing equations and the closure relations were specifically chosen to numerically capture phase separations and phase inversion phenomena. The numerical algorithm was specifically designed to ensure the boundness of the solutions and to enhance the implicit treatment of source terms. The solver shows good agreement with the analytical multiphase benchmarks. Moreover, it provides consistent results in terms of separation effectiveness in the simplified geometry of industrial oil and gas separators. As further developments, we are currently working on three aspects: 1. Perform additional simulations a real-case geometry using different inlet nozzle shapes; 2. Compare the CFD results with the industrial validation procedure; 3. Study the reduction of the computational cost of the code introducing a simplifying solver for describing the separation process. REFERENCES 1. Bahadori, Alireza. Natural gas processing: technology and engineering design. Gulf Professional Publishing, 2014; 2. Behzadi, A., R.I. Issa, and H.Rusche. “Modelling of dispersed bubble and droplet flow at high phase fractions.” Chemical Engineering Science 59.4 (2004): 759-770; 3. Garnier, C., M. Lance, and J.L. Marié. “Measurement of local flow characteristics in buoyancy-driven bubbly flow at high void fraction.” Experimental Thermal and Fluid Science 26.6 (2002): 811-815. 4. Laure A. de Tournemine, V. Roig, C. Suzanne. “Experimental study of the turbulent in bubbly flows at high void fraction”. Institut de Mecanique des fluids de Toulouse. 5. Weller H., Derivation, modelling and solution of the conditionally averaged two-phase flow equations. Tech. rep., OpenCFD Ltd, United Kingdom, 2002; 6. Computational Fluid Dynamics of Dispersed Two-Phase at High Phase Fractions, H. Rusche, PhD Thesis, Imperial College, London (2003); 7. Passalacqua A. and Fox R. Implementation of an iterative solution procedure for multi-fluid gas-particle flow models on unstructured grids. Powder Technology, 213:174-187, 2011; 8. Ramson V. and Mousseau V. Convergence and accuracy of the RELAP5 two-phase flow model. Proceedings of the ANS International Topical Meeting on Advences in Mathematics, Computations, and Reactor Physics, 1, 1991; 9. Stadke, Herbert. Gasdynamic aspects of two-phase flow: Hyperbolicity, wave propagation phenomena and related numerical methods. John Wiley & Sons. 2006; 10. Manning, Francis S., and Richard E. Thompson. Oilfield processing of petroleum: Crude oil. Vol 2. Pennwell books, 1995.

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