# Investigation on the Transition Criterion of Smooth Stratified Flow to Other Flow Patterns for Gas-Hydrate Slurry Flow

Authors: Bohui Shi; Jiaqi Wang; Yifan Yu; Lin Ding; Yang Liu; Haihao Wu
Journal: International Journal of Chemical Engineering
 (2017)

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## Abstract

A stability criterion for gas-hydrate slurry stratified flow was developed. The model was based on one-dimensional gas-liquid two-fluid model and perturbation method, considering unstable factors including shear stress, gravity, and surface tension. In addition, mass transfer between gas and liquid phase caused by hydrate formation was taken into account by implementing an inward and outward natural gas hydrates growth shell model for water-in-oil emulsion. A series of gas-hydrate slurry flow experiments were carried out in a high-pressure (>10 MPa) horizontal flow loop. The transition criterion of smooth stratified flow to other flow patterns for gas-hydrate slurry flow was established and validated and combined with experimental data at different water cuts. Meanwhile, parameters of this stability criterion were defined. This stability criterion was proved to be efficient for predicting the transition from smooth to nonsmooth stratified flow for gas-hydrate slurry.


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## Body

## 1. Introduction

Gas hydrates are ice-like crystals formed by inclosing gas molecules (guests) in clathrates of water molecules (host) under high pressure and low temperature [1]. Gas hydrates are known to cause blockage in oil and gas transportation pipelines [2]. With the tendency of offshore oil industry towards deeper water, gathering systems are confronting unprecedented flow assurance challenges on hydrate formation and plugging. Safety has become an unparalleled problem in offshore oil and gas facilities under undulating terrain condition, especially for high-pressure and low temperature operation conditions owing to the risk of hydrate blockage.Conventionally, chemical-based injection and insulation are two major techniques to prevent hydrate plugging in offshore production pipelines. However, these two techniques are of high capital expenditure and technical limitations [1, 3]. Recently, hydrate slurry technique has turned out to be a new option to avoid hydrate blocking. In this method, hydrates are not prevented from formation but are made transportable suspending in liquid phase under complex multiphase flow condition [4–9]. Obviously, multiphase flow analysis is an important factor for implementing hydrate slurry technique in deep water petroleum explorations and developments.Flow pattern is the key issue in characterizing multiphase flow. There are two generally recognized methods for flow pattern determination: plotting flow pattern map according to the experimental data [10–13] and establishing flow pattern transition criterion based on flow mechanism [14–16]. These two methods have been widely implemented in gas-liquid two-phase flow. However, the flow would be much too complicated, when hydrate forms in the liquid phase.Though lots of investigations have been made on thermodynamic and kinetic of gas-hydrate formation and decomposition [17–24], study on multiphase flow of gas-hydrate slurry is far from sufficient. Tajima et al. [25] observed CO2 hydrate slurry flow patterns in CO2-water two-phase flow using different types of static mixers. Zerpa et al. [26] and Joshi et al. [27] proposed a conceptual model for hydrate plug formation mechanism in water-dominated system, where gas-liquid and liquid-solid flow pattern were defined and characteristics of flow pattern transition with increasing of hydrates volume fraction were studied. Zhao [28] and Gong et al. [7, 8] numerically simulated gas-hydrate slurry stratified flow, and Kwon et al. [29] simulated the oil-gas-water-hydrate annular-mist flow. Ding et al. [30] reported the difference of the flow patterns with and without hydrates. And, it was concluded that hydrates formation resulted in the decrease of stratified smooth flow area on flow map and less stability of stratified smooth flow. However, little attention has been paid to the flow pattern transition criterion of gas-hydrate slurry flow, especially the stability criterion of stratified flow.In this paper, gas-hydrate slurry interface stability was analyzed by deriving dispersion function of the interface wave, and then the transition criterion from smooth stratified flow to other flow patterns was obtained. Thereafter, a set of gas-hydrate slurry flow experiments in high-pressure (>10 MPa) horizontal flow loop were carried out to investigate the flow pattern characteristics of gas-hydrate slurry flow. The transition criterion of smooth stratified flow to other flow patterns for gas-hydrate slurry flow was established and validated, combined with experimental data at different water cuts.


## 2. Model Development

A one-dimensional two-fluid model describing gas-hydrate slurry stratified flow was developed. The schematic description is shown in Figure1. In this work, the slurry phase was simplified as incompressible. Moreover, work exchange between the fluid and the environment as well as temperature change caused by hydrates formation was ignored.Figure 1
Sectional geometry of gas-hydrate slurry stratified flow.In Figure1, hG and hSL represent the height of gas phase and hydrate slurry phase, AG and ASL represent the cross section area of gas phase and hydrate slurry phase, UG and USL represent velocity of gas phase and hydrate slurry phase, τG, τSL, and τI are shear friction at interface of gas phase-pipe wall, hydrate slurry-pipe wall, and gas-hydrate slurry phase, SG, SSL, and SI represent wetted perimeter of gas phase, hydrate slurry phase, and gas-hydrate slurry interface, pG, pSL, and pI represent pressure of gas phase, hydrate slurry phase, and pressure at the interface, pIG and pISL represent relative pressure at the interface of gas phase and hydrate slurry phase, β represents angle of inclination, Δx represents distance to the infinitesimal segment, and x and y represent axial and radial direction of the pipe. In addition, the cross section geometry can be found in the supporting information document in Supplementary Material available online at https://doi.org/10.1155/2017/9846507 (cross section geometry).


### 2.1. Flow Analysis

#### 2.1.1. Continuity Equation

For the gas-hydrate slurry stratified flow shown in Figure1, continuity equation for each phase can be derived as follows:(1a)∂∂tρGAG+∂∂xρGAGUG=ΓA,(1b)∂∂tρSLASL+∂∂xρSLASLUSL=-ΓA,where ρG and ρSL are densities of gas and hydrate slurry phase, A represents overall cross section area, and Γ represents mass transfer rate, which is sectional average weight of gas transferred from gas to liquid per second for hydrate formation. The mass transfer rate, Γ, can be estimated using the inward and outward gas-hydrate growth shell model [23], since this model was developed to describe the hydrates formation in water-in-oil emulsion system. The process of hydrates nucleating, forming, and growing as shell covering the water droplets suspended in the continuous liquid phase can be simulated. And, this inward and outward gas-hydrate growth shell model is proved suitable and preferred to be applied in the hydrates slurry flow for the calculation of hydrates fraction and gas consumption. And, the application of this hydrate growth shell model is available in the supporting information (Hydrate Growth Model Implementation).By introducing slurry phase heighthSL, which is a key parameter for stratified flow, (1a) and (1b) can be transformed into (2a) and (2b) as function of hSL.(2a)∂ρGhSL∂t-AGASL′∂ρGUG∂x+ρGUG∂hSL∂x=-ΓAASL′,(2b)∂ρSLhSL∂t+ASLASL′∂ρSLUSL∂x+ρSLUSL∂hSL∂x=-ΓAASL′,where ASL′=dASL/hSL.


#### 2.1.2. Momentum Balance Equation

Momentum balance for each phase can be described as follows:(3a)∂ρGAGUG∂t+∂ρGAGUG2∂x-ΓAUG=-τGSG-τISI-ρGgAGsin⁡β-AG∂pIG∂x-AGρGgcos⁡β∂hSL∂x,(3b)∂ρSLASLUSL∂t+∂ρSLASLUSL2∂x+ΓAUG=-τSLSSL+τISI-ρSLgASLsin⁡β-ASL∂pISL∂x-ASLρSLgcos⁡β∂hSL∂x.Assuming the interface to be flat in radial direction [31] and ignoring the influence of shorter waves on flow stability [32], the pressure difference at the interface can be approximated as(4)PIG-PISL=σ∂2hSL∂x2,where σ represents interfacial tension and is considered constant in the control volume.Interfacial shear stressτI consists of smooth interface shear stress and instability contribution. The following method, (5a) and (5b), is used in this work [33] to determine interfacial shear stress τI, introducing a correlated coefficient Ch(5a)τI=fIρfUG-USLUG-USL2+ρfUG-USL2Ch∂hSL∂x,(5b)fI=ξISIπUfρfμf-ηI,where ρf, Uf, and μf refer to corresponding density, velocity, and viscosity of the phase with higher velocity and subscript f represents G or SL according to the specific situation. Interfacial frication factor fI can be calculated after obtaining empirical parameters ξI and ηI through experiments.Substituting (4) and (5a) and (5b) into (3a) and (3b), the key equation of momentum balance relationship can be obtained as follows:(6a)∂ρGUG∂t-∂ρSLUSL∂t+∂ρGUG2∂x-∂ρSLUSL2∂x-UGΓ1RSL+1RG+Π∂hSL∂x+σ∂3hSL∂x3=F,(6b)Π=ρfUG-USL2ChSI1AG+1ASL-ρSL-ρGgcos⁡β,(6c)F=τSLSSLASL-τGSGAG-fIρfUG-USLUG-USL2SI1AG+1ASL+ρSL-ρGgsin⁡β,where RSL=ASL/A and RG=AG/A refer to liquid holdup and void fraction, respectively. For steady state, F=0, (6a), (6b), and (6c) reduce to steady-state momentum equation.Gas and hydrate slurry shear stresses at the pipe wall can be expressed as (7a) and (7b) by introducing fanning friction factor. (7a)τG=12fGρGUGUG,(7b)τSL=12fSLρSLUSLUSL.For gas phase, the following methods as (8a) and (8b) are applied for friction coefficient fG.(8a)fG=16ReG,laminar  flow  where  ReG<2300,(8b)fG=0.07725log⁡ReG/72,turbulent  flow  whereReG≥2300.For hydrate slurry phase, a friction coefficient method, (9) [34], is used, considering the influence of liquid flow as well as hydrate formation and agglomeration.(9)fSL=4log⁡DSL2εf+3.48-2+47ρHρLψmaxψh1/3-1-2dpDSL2,where ρH and ρL are hydrate and liquid densities, ψh and ψmax are hydrate volume fraction and its maximum value, dp is hydrate particle diameter, DSL is wetted perimeter diameter of hydrate slurry phase, and εf is the absolute roughness of the pipe wall.Substituting (7a) and (7b) into (6c), (10) can be obtained.(10)F=±fSLρSLUSL22SSLARSL-fGρGUG22SGARG∓fIρfUG-USL22SI1ARG+1ARSL+ρSL-ρGgsin⁡β,where the first term is positive when USL>0 and third term is negative when UG-USL>0, and vice versa [35].


### 2.2. Stability Analysis

#### 2.2.1. Flow Stability Equation

Referring to the method used in interface stability analysis, where the steady-state contribution and perturbation contribution are separated,USL, UG, and hSL can be rewritten as combinations of steady-state part and perturbed part shown in(11a)USL=U-SL+U^SL,UG=U-G+U^G,hSL=h-SL+h^SL,(11b)∂U-SL∂x=∂U-G∂x=∂h-SL∂x=0,(11c)U^SLU-SL≪1,U^GU-G≪1,h^SLh-SL≪1.Based on (11a), (11b), and (11c), continuous equations (2a) and (2b) and momentum equations (6a), (6b), and (6c) can be linearized and simplified. The following fifth-order linear equation can be obtained to describe one-dimensional steady-state gas-hydrate slurry stratified flow. (12)σAASL′∂5h^SL∂x5+2ρ-GU-GRG+ρ-SLU-SLRSL∂3h^SL∂t∂x2+ρ-GU-G2RG+ρ-SLU-SL2RSL+ΠAASL′∂3h^SL∂x3+ρ-GRG+ρ-SLRSL∂3h^SL∂t2∂x-ΓGRG1RSL+1RG-1RSL∂F∂U-SLhSL,UG+1RG∂F∂U-GhSL,USL∂2h^SL∂t∂x-ΓGU-GRG1RSL+1RG+AASL′∂F∂h-LUSL,UG-U-SLRSL∂F∂U-SLhSL,UG+U-GRG∂F∂U-GhSL,USL∂2h^SL∂x2=0.Characterize the perturbation of liquid hold-up in exponential form as in the following [36]: (13)h^SL=εeiwt-kx,where w, k, and ε are parameters in wave equation.Substituting (13) into (12), (14a), (14b), (14c), (14d), (14e), (14f), (14g), (14h), (14i), and (14j) can be obtained to describe the stratified smooth flow stability of gas-hydrate slurry stratified flow.(14a)w2-2a∗k-ib∗w-d∗k4+c∗k2-ie∗k=0,where following parameters are defined to simplify expression (14a):(14b)a∗=1ρ∗ρ-GU-GRG+ρ-SLU-SLRSL,(14c)b∗=12ρ∗ΓRG1RSL+1RG-∂F∂U-SL,SRSL,UG,S+∂F∂U-G,SRSL,USL,S,(14d)c∗=1ρ∗ρ-GU-G2RG+ρ-SLU-SL2RSL+1ASL′ρfUG-USL2ChSI1RG+1RSL-AρSL-ρGgcos⁡β,(14e)d∗=σρ∗AASL′,(14f)e∗=1ρ∗ΓGU-GRG1RSL+1RG+∂F∂R-LUSL,UG,(14g)ρ∗=ρ-GRG+ρ-SLRSL,(14h)∂F∂U-SL,SRSL,UG,S=1RSL∂F∂U-SLhSL,UG,(14i)∂F∂U-G,SRSL,USL,S=1RG∂F∂U-GhSL,USL,(14j)∂F∂R-SLUSL,S,UG,S=AASL′∂F∂h-LUSL,UG-U-SLRSL∂F∂U-SLhSL,UG+U-GRG∂F∂U-GhSL,USL.


#### 2.2.2. Stability Criterion

Analytically solve quadratic equations (14a), (14b), (14c), (14d), (14e), (14f), (14g), (14h), (14i), and (14j), and roots can be expressed as follows:(15)w=a∗k-ib∗±a∗2-c∗k2-b∗2+d∗k4+ie∗k-2a∗b∗k.Substituting w=wR+wI, (14a) can be split into imaginary part and real part as follows:Imaginary part:(16a)-2a∗kwI+2b∗wR-e∗k+2wRwI=0Real part:(16b)wR2-2a∗kwR-d∗k4+c∗k2-2b∗wI-wI2=0It could be found in (16a) and (16b) that if wI<0, perturbation contribution of liquid holdup increases exponentially with time, resulting in flow instability. With wI approaching zero, stability criterion for gas-hydrate slurry stratified smooth flow can be derived as follows:(17)e∗2b∗-a∗2-a∗2-c∗-d∗k2<0.Analyzing (17), two critical wave velocities at the inception of instability are introduced as (18a) and (18b). CV is the critical wave velocity considering the viscous effect, while CIV is the one without the influence of viscosity [33, 37, 38].(18a)CV=wRk=e∗2b∗=ΓU-G/RG1/RSL+1/RG+∂F/∂R-LUSL,UGΓ/RG1/RSL+1/RG-∂F/∂U-SL,SRSL,UG,S+∂F/∂U-G,SRSL,USL,S,(18b)CIV=a∗=ρ-GU-GRSL+ρ-SLU-SLRGρ-GRSL+ρ-SLRG.Substituting (14a), (14b), (14c), (14d), (14e), (14f), (14g), (14h), (14i), and (14j) and (18a) and (18b) into (17), the stability criterion for gas-hydrate slurry stratified smooth flow can be obtained as follows:(19)CV-CIV2+ρ-Gρ-SLρ∗2RGRSLU-G-U-SL2-ρ-SL-ρ-Gρ∗gcos⁡βASI-σρ∗ASIk2+ρfUG-USL2Chρ∗1RG+1RSL<0.In (19), the first term corresponds to the viscous shear stress that amplifies any disturbance on the surface; the second and third terms are stability criterion disregarding viscous effects, also known as the one-dimensional interface IKH instability criterion; the fourth term stands for the surface tension contribution and depends mainly on wave variable k; the last term amplifies instability brought by other factors and is independent of the direction of relative velocity as well as shear stress between the two phases.Neglecting interfacial tension, viscous shear stress, and interface shear stress, (20) can be simplified.(20)U-G-U-SL<ρ-GRSL+ρ-SLRGρ-SLρ-L-ρ-Gρ-Ggcos⁡βASI0.5.Considering the large density difference between gas and hydrate slurry phases, (20) can be further simplified as follows, shown in (21). This equation matches the criterion presented by Taitel and Dukler [14] in 1976.(21)U-G-U-SL<ρ-L-ρ-Gρ-Ggcos⁡βAGSI0.5.Calculation procedure based on the criterion is presented in Scheme 1. For stratified flow boundary determination, superficial velocity of one phase can be calculated when that of the other phase is given. In implanting this model, bisection method is applied in the iteration calculation. Scheme 1 shows the flowchart to calculate liquid superficial velocity USL,S at a given gas superficial velocity UG,S, while UG,S at given USL,S can be obtained through similar procedure.Scheme 1
Algorithm to determine the stratified flow boundary by calculating liquid superficial velocityUSL,S at a given gas superficial velocity UG,S.


## 2.1. Flow Analysis

### 2.1.1. Continuity Equation

For the gas-hydrate slurry stratified flow shown in Figure1, continuity equation for each phase can be derived as follows:(1a)∂∂tρGAG+∂∂xρGAGUG=ΓA,(1b)∂∂tρSLASL+∂∂xρSLASLUSL=-ΓA,where ρG and ρSL are densities of gas and hydrate slurry phase, A represents overall cross section area, and Γ represents mass transfer rate, which is sectional average weight of gas transferred from gas to liquid per second for hydrate formation. The mass transfer rate, Γ, can be estimated using the inward and outward gas-hydrate growth shell model [23], since this model was developed to describe the hydrates formation in water-in-oil emulsion system. The process of hydrates nucleating, forming, and growing as shell covering the water droplets suspended in the continuous liquid phase can be simulated. And, this inward and outward gas-hydrate growth shell model is proved suitable and preferred to be applied in the hydrates slurry flow for the calculation of hydrates fraction and gas consumption. And, the application of this hydrate growth shell model is available in the supporting information (Hydrate Growth Model Implementation).By introducing slurry phase heighthSL, which is a key parameter for stratified flow, (1a) and (1b) can be transformed into (2a) and (2b) as function of hSL.(2a)∂ρGhSL∂t-AGASL′∂ρGUG∂x+ρGUG∂hSL∂x=-ΓAASL′,(2b)∂ρSLhSL∂t+ASLASL′∂ρSLUSL∂x+ρSLUSL∂hSL∂x=-ΓAASL′,where ASL′=dASL/hSL.


### 2.1.2. Momentum Balance Equation

Momentum balance for each phase can be described as follows:(3a)∂ρGAGUG∂t+∂ρGAGUG2∂x-ΓAUG=-τGSG-τISI-ρGgAGsin⁡β-AG∂pIG∂x-AGρGgcos⁡β∂hSL∂x,(3b)∂ρSLASLUSL∂t+∂ρSLASLUSL2∂x+ΓAUG=-τSLSSL+τISI-ρSLgASLsin⁡β-ASL∂pISL∂x-ASLρSLgcos⁡β∂hSL∂x.Assuming the interface to be flat in radial direction [31] and ignoring the influence of shorter waves on flow stability [32], the pressure difference at the interface can be approximated as(4)PIG-PISL=σ∂2hSL∂x2,where σ represents interfacial tension and is considered constant in the control volume.Interfacial shear stressτI consists of smooth interface shear stress and instability contribution. The following method, (5a) and (5b), is used in this work [33] to determine interfacial shear stress τI, introducing a correlated coefficient Ch(5a)τI=fIρfUG-USLUG-USL2+ρfUG-USL2Ch∂hSL∂x,(5b)fI=ξISIπUfρfμf-ηI,where ρf, Uf, and μf refer to corresponding density, velocity, and viscosity of the phase with higher velocity and subscript f represents G or SL according to the specific situation. Interfacial frication factor fI can be calculated after obtaining empirical parameters ξI and ηI through experiments.Substituting (4) and (5a) and (5b) into (3a) and (3b), the key equation of momentum balance relationship can be obtained as follows:(6a)∂ρGUG∂t-∂ρSLUSL∂t+∂ρGUG2∂x-∂ρSLUSL2∂x-UGΓ1RSL+1RG+Π∂hSL∂x+σ∂3hSL∂x3=F,(6b)Π=ρfUG-USL2ChSI1AG+1ASL-ρSL-ρGgcos⁡β,(6c)F=τSLSSLASL-τGSGAG-fIρfUG-USLUG-USL2SI1AG+1ASL+ρSL-ρGgsin⁡β,where RSL=ASL/A and RG=AG/A refer to liquid holdup and void fraction, respectively. For steady state, F=0, (6a), (6b), and (6c) reduce to steady-state momentum equation.Gas and hydrate slurry shear stresses at the pipe wall can be expressed as (7a) and (7b) by introducing fanning friction factor. (7a)τG=12fGρGUGUG,(7b)τSL=12fSLρSLUSLUSL.For gas phase, the following methods as (8a) and (8b) are applied for friction coefficient fG.(8a)fG=16ReG,laminar  flow  where  ReG<2300,(8b)fG=0.07725log⁡ReG/72,turbulent  flow  whereReG≥2300.For hydrate slurry phase, a friction coefficient method, (9) [34], is used, considering the influence of liquid flow as well as hydrate formation and agglomeration.(9)fSL=4log⁡DSL2εf+3.48-2+47ρHρLψmaxψh1/3-1-2dpDSL2,where ρH and ρL are hydrate and liquid densities, ψh and ψmax are hydrate volume fraction and its maximum value, dp is hydrate particle diameter, DSL is wetted perimeter diameter of hydrate slurry phase, and εf is the absolute roughness of the pipe wall.Substituting (7a) and (7b) into (6c), (10) can be obtained.(10)F=±fSLρSLUSL22SSLARSL-fGρGUG22SGARG∓fIρfUG-USL22SI1ARG+1ARSL+ρSL-ρGgsin⁡β,where the first term is positive when USL>0 and third term is negative when UG-USL>0, and vice versa [35].


## 2.1.1. Continuity Equation

For the gas-hydrate slurry stratified flow shown in Figure1, continuity equation for each phase can be derived as follows:(1a)∂∂tρGAG+∂∂xρGAGUG=ΓA,(1b)∂∂tρSLASL+∂∂xρSLASLUSL=-ΓA,where ρG and ρSL are densities of gas and hydrate slurry phase, A represents overall cross section area, and Γ represents mass transfer rate, which is sectional average weight of gas transferred from gas to liquid per second for hydrate formation. The mass transfer rate, Γ, can be estimated using the inward and outward gas-hydrate growth shell model [23], since this model was developed to describe the hydrates formation in water-in-oil emulsion system. The process of hydrates nucleating, forming, and growing as shell covering the water droplets suspended in the continuous liquid phase can be simulated. And, this inward and outward gas-hydrate growth shell model is proved suitable and preferred to be applied in the hydrates slurry flow for the calculation of hydrates fraction and gas consumption. And, the application of this hydrate growth shell model is available in the supporting information (Hydrate Growth Model Implementation).By introducing slurry phase heighthSL, which is a key parameter for stratified flow, (1a) and (1b) can be transformed into (2a) and (2b) as function of hSL.(2a)∂ρGhSL∂t-AGASL′∂ρGUG∂x+ρGUG∂hSL∂x=-ΓAASL′,(2b)∂ρSLhSL∂t+ASLASL′∂ρSLUSL∂x+ρSLUSL∂hSL∂x=-ΓAASL′,where ASL′=dASL/hSL.


## 2.1.2. Momentum Balance Equation

Momentum balance for each phase can be described as follows:(3a)∂ρGAGUG∂t+∂ρGAGUG2∂x-ΓAUG=-τGSG-τISI-ρGgAGsin⁡β-AG∂pIG∂x-AGρGgcos⁡β∂hSL∂x,(3b)∂ρSLASLUSL∂t+∂ρSLASLUSL2∂x+ΓAUG=-τSLSSL+τISI-ρSLgASLsin⁡β-ASL∂pISL∂x-ASLρSLgcos⁡β∂hSL∂x.Assuming the interface to be flat in radial direction [31] and ignoring the influence of shorter waves on flow stability [32], the pressure difference at the interface can be approximated as(4)PIG-PISL=σ∂2hSL∂x2,where σ represents interfacial tension and is considered constant in the control volume.Interfacial shear stressτI consists of smooth interface shear stress and instability contribution. The following method, (5a) and (5b), is used in this work [33] to determine interfacial shear stress τI, introducing a correlated coefficient Ch(5a)τI=fIρfUG-USLUG-USL2+ρfUG-USL2Ch∂hSL∂x,(5b)fI=ξISIπUfρfμf-ηI,where ρf, Uf, and μf refer to corresponding density, velocity, and viscosity of the phase with higher velocity and subscript f represents G or SL according to the specific situation. Interfacial frication factor fI can be calculated after obtaining empirical parameters ξI and ηI through experiments.Substituting (4) and (5a) and (5b) into (3a) and (3b), the key equation of momentum balance relationship can be obtained as follows:(6a)∂ρGUG∂t-∂ρSLUSL∂t+∂ρGUG2∂x-∂ρSLUSL2∂x-UGΓ1RSL+1RG+Π∂hSL∂x+σ∂3hSL∂x3=F,(6b)Π=ρfUG-USL2ChSI1AG+1ASL-ρSL-ρGgcos⁡β,(6c)F=τSLSSLASL-τGSGAG-fIρfUG-USLUG-USL2SI1AG+1ASL+ρSL-ρGgsin⁡β,where RSL=ASL/A and RG=AG/A refer to liquid holdup and void fraction, respectively. For steady state, F=0, (6a), (6b), and (6c) reduce to steady-state momentum equation.Gas and hydrate slurry shear stresses at the pipe wall can be expressed as (7a) and (7b) by introducing fanning friction factor. (7a)τG=12fGρGUGUG,(7b)τSL=12fSLρSLUSLUSL.For gas phase, the following methods as (8a) and (8b) are applied for friction coefficient fG.(8a)fG=16ReG,laminar  flow  where  ReG<2300,(8b)fG=0.07725log⁡ReG/72,turbulent  flow  whereReG≥2300.For hydrate slurry phase, a friction coefficient method, (9) [34], is used, considering the influence of liquid flow as well as hydrate formation and agglomeration.(9)fSL=4log⁡DSL2εf+3.48-2+47ρHρLψmaxψh1/3-1-2dpDSL2,where ρH and ρL are hydrate and liquid densities, ψh and ψmax are hydrate volume fraction and its maximum value, dp is hydrate particle diameter, DSL is wetted perimeter diameter of hydrate slurry phase, and εf is the absolute roughness of the pipe wall.Substituting (7a) and (7b) into (6c), (10) can be obtained.(10)F=±fSLρSLUSL22SSLARSL-fGρGUG22SGARG∓fIρfUG-USL22SI1ARG+1ARSL+ρSL-ρGgsin⁡β,where the first term is positive when USL>0 and third term is negative when UG-USL>0, and vice versa [35].


## 2.2. Stability Analysis

### 2.2.1. Flow Stability Equation

Referring to the method used in interface stability analysis, where the steady-state contribution and perturbation contribution are separated,USL, UG, and hSL can be rewritten as combinations of steady-state part and perturbed part shown in(11a)USL=U-SL+U^SL,UG=U-G+U^G,hSL=h-SL+h^SL,(11b)∂U-SL∂x=∂U-G∂x=∂h-SL∂x=0,(11c)U^SLU-SL≪1,U^GU-G≪1,h^SLh-SL≪1.Based on (11a), (11b), and (11c), continuous equations (2a) and (2b) and momentum equations (6a), (6b), and (6c) can be linearized and simplified. The following fifth-order linear equation can be obtained to describe one-dimensional steady-state gas-hydrate slurry stratified flow. (12)σAASL′∂5h^SL∂x5+2ρ-GU-GRG+ρ-SLU-SLRSL∂3h^SL∂t∂x2+ρ-GU-G2RG+ρ-SLU-SL2RSL+ΠAASL′∂3h^SL∂x3+ρ-GRG+ρ-SLRSL∂3h^SL∂t2∂x-ΓGRG1RSL+1RG-1RSL∂F∂U-SLhSL,UG+1RG∂F∂U-GhSL,USL∂2h^SL∂t∂x-ΓGU-GRG1RSL+1RG+AASL′∂F∂h-LUSL,UG-U-SLRSL∂F∂U-SLhSL,UG+U-GRG∂F∂U-GhSL,USL∂2h^SL∂x2=0.Characterize the perturbation of liquid hold-up in exponential form as in the following [36]: (13)h^SL=εeiwt-kx,where w, k, and ε are parameters in wave equation.Substituting (13) into (12), (14a), (14b), (14c), (14d), (14e), (14f), (14g), (14h), (14i), and (14j) can be obtained to describe the stratified smooth flow stability of gas-hydrate slurry stratified flow.(14a)w2-2a∗k-ib∗w-d∗k4+c∗k2-ie∗k=0,where following parameters are defined to simplify expression (14a):(14b)a∗=1ρ∗ρ-GU-GRG+ρ-SLU-SLRSL,(14c)b∗=12ρ∗ΓRG1RSL+1RG-∂F∂U-SL,SRSL,UG,S+∂F∂U-G,SRSL,USL,S,(14d)c∗=1ρ∗ρ-GU-G2RG+ρ-SLU-SL2RSL+1ASL′ρfUG-USL2ChSI1RG+1RSL-AρSL-ρGgcos⁡β,(14e)d∗=σρ∗AASL′,(14f)e∗=1ρ∗ΓGU-GRG1RSL+1RG+∂F∂R-LUSL,UG,(14g)ρ∗=ρ-GRG+ρ-SLRSL,(14h)∂F∂U-SL,SRSL,UG,S=1RSL∂F∂U-SLhSL,UG,(14i)∂F∂U-G,SRSL,USL,S=1RG∂F∂U-GhSL,USL,(14j)∂F∂R-SLUSL,S,UG,S=AASL′∂F∂h-LUSL,UG-U-SLRSL∂F∂U-SLhSL,UG+U-GRG∂F∂U-GhSL,USL.


### 2.2.2. Stability Criterion

Analytically solve quadratic equations (14a), (14b), (14c), (14d), (14e), (14f), (14g), (14h), (14i), and (14j), and roots can be expressed as follows:(15)w=a∗k-ib∗±a∗2-c∗k2-b∗2+d∗k4+ie∗k-2a∗b∗k.Substituting w=wR+wI, (14a) can be split into imaginary part and real part as follows:Imaginary part:(16a)-2a∗kwI+2b∗wR-e∗k+2wRwI=0Real part:(16b)wR2-2a∗kwR-d∗k4+c∗k2-2b∗wI-wI2=0It could be found in (16a) and (16b) that if wI<0, perturbation contribution of liquid holdup increases exponentially with time, resulting in flow instability. With wI approaching zero, stability criterion for gas-hydrate slurry stratified smooth flow can be derived as follows:(17)e∗2b∗-a∗2-a∗2-c∗-d∗k2<0.Analyzing (17), two critical wave velocities at the inception of instability are introduced as (18a) and (18b). CV is the critical wave velocity considering the viscous effect, while CIV is the one without the influence of viscosity [33, 37, 38].(18a)CV=wRk=e∗2b∗=ΓU-G/RG1/RSL+1/RG+∂F/∂R-LUSL,UGΓ/RG1/RSL+1/RG-∂F/∂U-SL,SRSL,UG,S+∂F/∂U-G,SRSL,USL,S,(18b)CIV=a∗=ρ-GU-GRSL+ρ-SLU-SLRGρ-GRSL+ρ-SLRG.Substituting (14a), (14b), (14c), (14d), (14e), (14f), (14g), (14h), (14i), and (14j) and (18a) and (18b) into (17), the stability criterion for gas-hydrate slurry stratified smooth flow can be obtained as follows:(19)CV-CIV2+ρ-Gρ-SLρ∗2RGRSLU-G-U-SL2-ρ-SL-ρ-Gρ∗gcos⁡βASI-σρ∗ASIk2+ρfUG-USL2Chρ∗1RG+1RSL<0.In (19), the first term corresponds to the viscous shear stress that amplifies any disturbance on the surface; the second and third terms are stability criterion disregarding viscous effects, also known as the one-dimensional interface IKH instability criterion; the fourth term stands for the surface tension contribution and depends mainly on wave variable k; the last term amplifies instability brought by other factors and is independent of the direction of relative velocity as well as shear stress between the two phases.Neglecting interfacial tension, viscous shear stress, and interface shear stress, (20) can be simplified.(20)U-G-U-SL<ρ-GRSL+ρ-SLRGρ-SLρ-L-ρ-Gρ-Ggcos⁡βASI0.5.Considering the large density difference between gas and hydrate slurry phases, (20) can be further simplified as follows, shown in (21). This equation matches the criterion presented by Taitel and Dukler [14] in 1976.(21)U-G-U-SL<ρ-L-ρ-Gρ-Ggcos⁡βAGSI0.5.Calculation procedure based on the criterion is presented in Scheme 1. For stratified flow boundary determination, superficial velocity of one phase can be calculated when that of the other phase is given. In implanting this model, bisection method is applied in the iteration calculation. Scheme 1 shows the flowchart to calculate liquid superficial velocity USL,S at a given gas superficial velocity UG,S, while UG,S at given USL,S can be obtained through similar procedure.Scheme 1
Algorithm to determine the stratified flow boundary by calculating liquid superficial velocityUSL,S at a given gas superficial velocity UG,S.


## 2.2.1. Flow Stability Equation

Referring to the method used in interface stability analysis, where the steady-state contribution and perturbation contribution are separated,USL, UG, and hSL can be rewritten as combinations of steady-state part and perturbed part shown in(11a)USL=U-SL+U^SL,UG=U-G+U^G,hSL=h-SL+h^SL,(11b)∂U-SL∂x=∂U-G∂x=∂h-SL∂x=0,(11c)U^SLU-SL≪1,U^GU-G≪1,h^SLh-SL≪1.Based on (11a), (11b), and (11c), continuous equations (2a) and (2b) and momentum equations (6a), (6b), and (6c) can be linearized and simplified. The following fifth-order linear equation can be obtained to describe one-dimensional steady-state gas-hydrate slurry stratified flow. (12)σAASL′∂5h^SL∂x5+2ρ-GU-GRG+ρ-SLU-SLRSL∂3h^SL∂t∂x2+ρ-GU-G2RG+ρ-SLU-SL2RSL+ΠAASL′∂3h^SL∂x3+ρ-GRG+ρ-SLRSL∂3h^SL∂t2∂x-ΓGRG1RSL+1RG-1RSL∂F∂U-SLhSL,UG+1RG∂F∂U-GhSL,USL∂2h^SL∂t∂x-ΓGU-GRG1RSL+1RG+AASL′∂F∂h-LUSL,UG-U-SLRSL∂F∂U-SLhSL,UG+U-GRG∂F∂U-GhSL,USL∂2h^SL∂x2=0.Characterize the perturbation of liquid hold-up in exponential form as in the following [36]: (13)h^SL=εeiwt-kx,where w, k, and ε are parameters in wave equation.Substituting (13) into (12), (14a), (14b), (14c), (14d), (14e), (14f), (14g), (14h), (14i), and (14j) can be obtained to describe the stratified smooth flow stability of gas-hydrate slurry stratified flow.(14a)w2-2a∗k-ib∗w-d∗k4+c∗k2-ie∗k=0,where following parameters are defined to simplify expression (14a):(14b)a∗=1ρ∗ρ-GU-GRG+ρ-SLU-SLRSL,(14c)b∗=12ρ∗ΓRG1RSL+1RG-∂F∂U-SL,SRSL,UG,S+∂F∂U-G,SRSL,USL,S,(14d)c∗=1ρ∗ρ-GU-G2RG+ρ-SLU-SL2RSL+1ASL′ρfUG-USL2ChSI1RG+1RSL-AρSL-ρGgcos⁡β,(14e)d∗=σρ∗AASL′,(14f)e∗=1ρ∗ΓGU-GRG1RSL+1RG+∂F∂R-LUSL,UG,(14g)ρ∗=ρ-GRG+ρ-SLRSL,(14h)∂F∂U-SL,SRSL,UG,S=1RSL∂F∂U-SLhSL,UG,(14i)∂F∂U-G,SRSL,USL,S=1RG∂F∂U-GhSL,USL,(14j)∂F∂R-SLUSL,S,UG,S=AASL′∂F∂h-LUSL,UG-U-SLRSL∂F∂U-SLhSL,UG+U-GRG∂F∂U-GhSL,USL.


## 2.2.2. Stability Criterion

Analytically solve quadratic equations (14a), (14b), (14c), (14d), (14e), (14f), (14g), (14h), (14i), and (14j), and roots can be expressed as follows:(15)w=a∗k-ib∗±a∗2-c∗k2-b∗2+d∗k4+ie∗k-2a∗b∗k.Substituting w=wR+wI, (14a) can be split into imaginary part and real part as follows:Imaginary part:(16a)-2a∗kwI+2b∗wR-e∗k+2wRwI=0Real part:(16b)wR2-2a∗kwR-d∗k4+c∗k2-2b∗wI-wI2=0It could be found in (16a) and (16b) that if wI<0, perturbation contribution of liquid holdup increases exponentially with time, resulting in flow instability. With wI approaching zero, stability criterion for gas-hydrate slurry stratified smooth flow can be derived as follows:(17)e∗2b∗-a∗2-a∗2-c∗-d∗k2<0.Analyzing (17), two critical wave velocities at the inception of instability are introduced as (18a) and (18b). CV is the critical wave velocity considering the viscous effect, while CIV is the one without the influence of viscosity [33, 37, 38].(18a)CV=wRk=e∗2b∗=ΓU-G/RG1/RSL+1/RG+∂F/∂R-LUSL,UGΓ/RG1/RSL+1/RG-∂F/∂U-SL,SRSL,UG,S+∂F/∂U-G,SRSL,USL,S,(18b)CIV=a∗=ρ-GU-GRSL+ρ-SLU-SLRGρ-GRSL+ρ-SLRG.Substituting (14a), (14b), (14c), (14d), (14e), (14f), (14g), (14h), (14i), and (14j) and (18a) and (18b) into (17), the stability criterion for gas-hydrate slurry stratified smooth flow can be obtained as follows:(19)CV-CIV2+ρ-Gρ-SLρ∗2RGRSLU-G-U-SL2-ρ-SL-ρ-Gρ∗gcos⁡βASI-σρ∗ASIk2+ρfUG-USL2Chρ∗1RG+1RSL<0.In (19), the first term corresponds to the viscous shear stress that amplifies any disturbance on the surface; the second and third terms are stability criterion disregarding viscous effects, also known as the one-dimensional interface IKH instability criterion; the fourth term stands for the surface tension contribution and depends mainly on wave variable k; the last term amplifies instability brought by other factors and is independent of the direction of relative velocity as well as shear stress between the two phases.Neglecting interfacial tension, viscous shear stress, and interface shear stress, (20) can be simplified.(20)U-G-U-SL<ρ-GRSL+ρ-SLRGρ-SLρ-L-ρ-Gρ-Ggcos⁡βASI0.5.Considering the large density difference between gas and hydrate slurry phases, (20) can be further simplified as follows, shown in (21). This equation matches the criterion presented by Taitel and Dukler [14] in 1976.(21)U-G-U-SL<ρ-L-ρ-Gρ-Ggcos⁡βAGSI0.5.Calculation procedure based on the criterion is presented in Scheme 1. For stratified flow boundary determination, superficial velocity of one phase can be calculated when that of the other phase is given. In implanting this model, bisection method is applied in the iteration calculation. Scheme 1 shows the flowchart to calculate liquid superficial velocity USL,S at a given gas superficial velocity UG,S, while UG,S at given USL,S can be obtained through similar procedure.Scheme 1
Algorithm to determine the stratified flow boundary by calculating liquid superficial velocityUSL,S at a given gas superficial velocity UG,S.


## 3. Experiments

To analyze stability transition criterion for stratified smooth flow using (19), parameters including ξI, ηI, Ch, and k need to be determined. In this paper, these parameters were obtained based on the experiments at different water cuts using high-pressure hydrates slurry flow loop.


### 3.1. Material and Facility

A double-pass high-pressure horizontal flow loop shown in Figure2 was designed and established in China University of Petroleum, Beijing, supported by CNOOC [24, 39–42]. This flow loop is 30 m long, with inner diameter of 1 inch and pressure exceeding 10 MPa. Test section temperature can be maintained from −20°C to 80°C by circulating water-glycol blend with the countercurrent flowing surrounding it in a 2-inch jacket. Fluid inside the loop can be observed through high-pressure windows at two different locations. Gas and liquid can be injected separately by a plunge compressor (2200 m3/h) and a magnetic centrifugal pump (12.0 m3/h). Gas is injected at the inlet of the test section. At the outlet of the test section, gas and liquid flow into an insulated separator and are redirected towards the test section after pressurization.Figure 2
Sketch of high-pressure hydrate flow loop.Thermocouples are placed along the pipe, inside the separator, inside the water-glycol system, and on different gas utilities. A Coriolis flow meter is stalled to measure liquid mixture density and flow rate. Two FM1000 gamma ray densitometers are available to measure the mean density of the multiphase fluid. Differential pressure sensors are installed to follow the evolution of pressure. Rapid data acquisition system permits the detection of quickly occurring events. A Focused Beam Reflectance Measurements (FBRM) probe and Particle Video Microscope (PVM) are installed to capture the evolution of the droplets, bubbles, or solid particles in fluid.Deionized water, civil natural gas, and −20# diesel (compositions listed in Table1) were used for the experiments. The dosage of added antiagglomerate was 3 wt% (measuring error ± 0.01 g) to prevent the hydrate crystals from aggregating [43].Table 1
Compositions of civil natural gas and −20# diesel oil.
Comp Mol% Civil natural gas N2 1.5603 CO 2.0911 CO2 0.9129 C1 90.6061 C2 3.1207 C3 3.1207 iC4 0.3291 iC5 0.0425 nC6+ 0.0106  −20# diesel oil C 11 0.8863 C 12 3.3586 C 13 5.3886 C 14 6.1990 C 15 6.7780 C16 6.8310 C17 7.9890 C18 7.4618 C19 6.3752 C 20+ 48.7325


### 3.2. Experiment Procedure

Four groups of experiments at different water cuts (15%, 20%, 25%, and 30%) were carried out with gas, −20# diesel oil, and deionized water. Experimental conditions are listed in Table2. Specific experimental procedure is detailed as follows:(1)
Vacuum the experimental system one hour before injecting the required amount of −20# diesel oil, deionized water, and antiagglomerates.(2)
Start and set the temperature controller at 18°C, the magnetic centrifugal pump at 40 Hz, and the control valve at 100%, get the liquid phase well circulated for least 5 hours to form water-in-oil emulation, and then open FBRM to monitor changes of the particles in the fluid.(3)
Open the gas injection valve and pressure the system up to experimental set point, after the temperature of the system stays constant at 18°C and particles size observed using FBRM becomes stable.(4)
Set the system temperature to experimental set point and cool the flow loop; hydrate would form as the temperature drops below the hydrate equilibrium temperate at the system pressure.(5)
Maintain temperature and pressure at experimental set point for at least 5 hours to ensure the fully hydrate formation, adjust compressor inlet valve and pump speed to attain a gas flow rate within 20~195 kg/h and liquid flow rate within 75~860 kg/h, and then observe the flow pattern and record the data; for each data point presented in this work, flow rate of both gas and slurry phase were controlled and flow pattern was observed when the flow rates, pressure, and temperature were stabilized.(6)
Increase the system temperature, stop the compressor and pump, evacuate the residual gas, discharge the experiment liquid, clean and flush the flow loop with compressed air, and replace it with nitrogen.(7)
Repeat the procedure at different water cuts.Table 2
Experimental conditions of gas-hydrate slurry flow.
Water cut(%) P  (MPa) T  (K) V L  (m3) N e x p ∗  (—) 15 4.13 277.33 0.0777 34 20 4.26 279.82 0.0875 75 25 3.85 278.99 0.0933 96 30 4.20 278.99 0.1000 88 N e x p   ∗: number of experimental data points.


## 3.1. Material and Facility

A double-pass high-pressure horizontal flow loop shown in Figure2 was designed and established in China University of Petroleum, Beijing, supported by CNOOC [24, 39–42]. This flow loop is 30 m long, with inner diameter of 1 inch and pressure exceeding 10 MPa. Test section temperature can be maintained from −20°C to 80°C by circulating water-glycol blend with the countercurrent flowing surrounding it in a 2-inch jacket. Fluid inside the loop can be observed through high-pressure windows at two different locations. Gas and liquid can be injected separately by a plunge compressor (2200 m3/h) and a magnetic centrifugal pump (12.0 m3/h). Gas is injected at the inlet of the test section. At the outlet of the test section, gas and liquid flow into an insulated separator and are redirected towards the test section after pressurization.Figure 2
Sketch of high-pressure hydrate flow loop.Thermocouples are placed along the pipe, inside the separator, inside the water-glycol system, and on different gas utilities. A Coriolis flow meter is stalled to measure liquid mixture density and flow rate. Two FM1000 gamma ray densitometers are available to measure the mean density of the multiphase fluid. Differential pressure sensors are installed to follow the evolution of pressure. Rapid data acquisition system permits the detection of quickly occurring events. A Focused Beam Reflectance Measurements (FBRM) probe and Particle Video Microscope (PVM) are installed to capture the evolution of the droplets, bubbles, or solid particles in fluid.Deionized water, civil natural gas, and −20# diesel (compositions listed in Table1) were used for the experiments. The dosage of added antiagglomerate was 3 wt% (measuring error ± 0.01 g) to prevent the hydrate crystals from aggregating [43].Table 1
Compositions of civil natural gas and −20# diesel oil.
Comp Mol% Civil natural gas N2 1.5603 CO 2.0911 CO2 0.9129 C1 90.6061 C2 3.1207 C3 3.1207 iC4 0.3291 iC5 0.0425 nC6+ 0.0106  −20# diesel oil C 11 0.8863 C 12 3.3586 C 13 5.3886 C 14 6.1990 C 15 6.7780 C16 6.8310 C17 7.9890 C18 7.4618 C19 6.3752 C 20+ 48.7325


## 3.2. Experiment Procedure

Four groups of experiments at different water cuts (15%, 20%, 25%, and 30%) were carried out with gas, −20# diesel oil, and deionized water. Experimental conditions are listed in Table2. Specific experimental procedure is detailed as follows:(1)
Vacuum the experimental system one hour before injecting the required amount of −20# diesel oil, deionized water, and antiagglomerates.(2)
Start and set the temperature controller at 18°C, the magnetic centrifugal pump at 40 Hz, and the control valve at 100%, get the liquid phase well circulated for least 5 hours to form water-in-oil emulation, and then open FBRM to monitor changes of the particles in the fluid.(3)
Open the gas injection valve and pressure the system up to experimental set point, after the temperature of the system stays constant at 18°C and particles size observed using FBRM becomes stable.(4)
Set the system temperature to experimental set point and cool the flow loop; hydrate would form as the temperature drops below the hydrate equilibrium temperate at the system pressure.(5)
Maintain temperature and pressure at experimental set point for at least 5 hours to ensure the fully hydrate formation, adjust compressor inlet valve and pump speed to attain a gas flow rate within 20~195 kg/h and liquid flow rate within 75~860 kg/h, and then observe the flow pattern and record the data; for each data point presented in this work, flow rate of both gas and slurry phase were controlled and flow pattern was observed when the flow rates, pressure, and temperature were stabilized.(6)
Increase the system temperature, stop the compressor and pump, evacuate the residual gas, discharge the experiment liquid, clean and flush the flow loop with compressed air, and replace it with nitrogen.(7)
Repeat the procedure at different water cuts.Table 2
Experimental conditions of gas-hydrate slurry flow.
Water cut(%) P  (MPa) T  (K) V L  (m3) N e x p ∗  (—) 15 4.13 277.33 0.0777 34 20 4.26 279.82 0.0875 75 25 3.85 278.99 0.0933 96 30 4.20 278.99 0.1000 88 N e x p   ∗: number of experimental data points.


## 4. Results and Discussions

### 4.1. Experiment Results

In the experiments, 293 smooth and nonsmooth stratified flow pattern data points at different water cuts were obtained under steady-state conditions (Tables S.1–S.4), which was visually observed through the sight glass of the loop. Four types of flow patterns were found, including two typical flow patterns (stratified smooth flow and slug flow) and two transitional flow patterns (stratified wavy flow and short slug flow). Distinct gas-liquid interface can be observed, and the interface was flat for the stratified smooth flow (Figure3(a)) and was wavy for stratified wave flow (Figure 3(b)). A liquid film zone and a liquid slug were found intermittently for slug flow (Figure 4), while a relatively longer liquid film and shorter slug length were found for short slug flow.Figure 3
Photos of stratified flow: (a) stratified smooth flow and (b) stratified wave flow.
(a) (b)Figure 4
Photos of slug flow: (a) liquid film zone before liquid slug, (b) liquid slug, and (c) liquid film zone after liquid slug.
(a) (b) (c)The morphologies, sizes, and distributions of fully formed hydrate particles in the slurry were recorded by PVM as shown in Figure5. The chord length distribution before and after hydrates formation is displayed in Figure 6. According to experimental observations, it is can be concluded that hydrates particles form as shells covering water droplets in the water-in-oil emulsion, and the size of hydrates particles will be enlarged compared with water droplet. In addition, hydrate particle morphologies, sizes, and distributions showed no distinguishable difference at different flow patterns, implying that their influences on flow pattern transition were not signification.Figure 5
Hydrate particles (marked in the red circles) in gas-liquid multiphase system.Figure 6
Chord length distribution before and after hydrates formation.


### 4.2. Model Implementation

Based on experimental conditions listed in Table2, hydrate growth parameters and gas consumption rate at different water cuts were calculated using the inward and outward hydrate growth shell model [23]. Results presented in Table 3 showed that gas consumption rate increased with water cut. Detailed calculation procedure including water consumption and hydrate fraction is presented in the supporting information (Hydrate Growth Model Implementation). Gas-slurry stratified smooth flow region transition boundary at different water cuts was obtained by implementing the calculation procedure presented in Scheme 1. Parameters in the model are obtained, where ξI is 0.01, ηI is −0.2, Ch is 0.01, and k is 10.Table 3
Hydrate growth parameters of different water cuts and gas consumed rate.
Water cut% K ∗ × 10 - 7  mol/(m2⋅Mpa⋅s) D f 0 × 10 - 9   m2/s ξ ε H 0 × 10 - 32  m3 ζ Γ G   ×   10 - 4  kg/m3⋅s 15 4.364 1.149 1.489 14.66 0.2527 7.188 20 7.606 13.82 1.527 6.339 0.2679 9.575 25 8.980 30.62 1.546 4.891 0.2767 11.72 30 16.16 321.5 1.564 2.994 0.2820 13.88


### 4.3. Model Evaluation

To compare the transition criterion of smooth stratified flow to other flow patterns for gas-hydrate slurry flow developed in this work with the classical flow pattern distribution models, flow region boundaries calculated using this model as well as that calculated using Taitel-Dukler [14] and Mandhane [13] methods are presented in Figure 7.Figure 7
Flow pattern map and boundary for stratified flow of gas-slurry flow at different water cuts: (a) 15%, (b) 20%, (c) 25%, and (d) 30%.
(a) (b) (c) (d)As is shown in Figure7, Mandhane’s model can roughly divide stratified flow from nonstratified flow at four water cuts, yet the results lead to a larger smooth stratified flow region. Taitel-Dukler model cannot give a satisfying stratified flow transition boundary, due to not considering the effect of hydrate formation on flow stability. Transition boundaries obtained based on the model developed in this work show good agreement with experimental data.All data points are summarized in Figure8. It can be seen that no distinctive effect of water cut on flow pattern division is observed, though gas consumption rate differs evidently according to Table 3. This may be due to the similar water consumption rates and hydrate fractions as shown in Figure 9. As calculated flow pattern boundary is primarily affected by fluid viscosity, and hydrate slurry viscosity is further determined by hydrate volume fraction, proximal transition lines are obtained under similar hydrate fraction though at different water cuts.Figure 8
Flow pattern distribution for gas-slurry stratified flow at different water cuts.Figure 9
Hydrate fraction and water conversion rate at different water cuts.The pattern experiment data of gas-hydrate slurry multiphase flow pattern in this work are not effectively enough to obtain more precise correlation parameters or investigate the flow pattern transition mechanism deeply. More experimental and theoretical research should be carried out in the future. Nevertheless, the good agreement with experimental data proved this work to be both feasible and significant.


## 4.1. Experiment Results

In the experiments, 293 smooth and nonsmooth stratified flow pattern data points at different water cuts were obtained under steady-state conditions (Tables S.1–S.4), which was visually observed through the sight glass of the loop. Four types of flow patterns were found, including two typical flow patterns (stratified smooth flow and slug flow) and two transitional flow patterns (stratified wavy flow and short slug flow). Distinct gas-liquid interface can be observed, and the interface was flat for the stratified smooth flow (Figure3(a)) and was wavy for stratified wave flow (Figure 3(b)). A liquid film zone and a liquid slug were found intermittently for slug flow (Figure 4), while a relatively longer liquid film and shorter slug length were found for short slug flow.Figure 3
Photos of stratified flow: (a) stratified smooth flow and (b) stratified wave flow.
(a) (b)Figure 4
Photos of slug flow: (a) liquid film zone before liquid slug, (b) liquid slug, and (c) liquid film zone after liquid slug.
(a) (b) (c)The morphologies, sizes, and distributions of fully formed hydrate particles in the slurry were recorded by PVM as shown in Figure5. The chord length distribution before and after hydrates formation is displayed in Figure 6. According to experimental observations, it is can be concluded that hydrates particles form as shells covering water droplets in the water-in-oil emulsion, and the size of hydrates particles will be enlarged compared with water droplet. In addition, hydrate particle morphologies, sizes, and distributions showed no distinguishable difference at different flow patterns, implying that their influences on flow pattern transition were not signification.Figure 5
Hydrate particles (marked in the red circles) in gas-liquid multiphase system.Figure 6
Chord length distribution before and after hydrates formation.


## 4.2. Model Implementation

Based on experimental conditions listed in Table2, hydrate growth parameters and gas consumption rate at different water cuts were calculated using the inward and outward hydrate growth shell model [23]. Results presented in Table 3 showed that gas consumption rate increased with water cut. Detailed calculation procedure including water consumption and hydrate fraction is presented in the supporting information (Hydrate Growth Model Implementation). Gas-slurry stratified smooth flow region transition boundary at different water cuts was obtained by implementing the calculation procedure presented in Scheme 1. Parameters in the model are obtained, where ξI is 0.01, ηI is −0.2, Ch is 0.01, and k is 10.Table 3
Hydrate growth parameters of different water cuts and gas consumed rate.
Water cut% K ∗ × 10 - 7  mol/(m2⋅Mpa⋅s) D f 0 × 10 - 9   m2/s ξ ε H 0 × 10 - 32  m3 ζ Γ G   ×   10 - 4  kg/m3⋅s 15 4.364 1.149 1.489 14.66 0.2527 7.188 20 7.606 13.82 1.527 6.339 0.2679 9.575 25 8.980 30.62 1.546 4.891 0.2767 11.72 30 16.16 321.5 1.564 2.994 0.2820 13.88


## 4.3. Model Evaluation

To compare the transition criterion of smooth stratified flow to other flow patterns for gas-hydrate slurry flow developed in this work with the classical flow pattern distribution models, flow region boundaries calculated using this model as well as that calculated using Taitel-Dukler [14] and Mandhane [13] methods are presented in Figure 7.Figure 7
Flow pattern map and boundary for stratified flow of gas-slurry flow at different water cuts: (a) 15%, (b) 20%, (c) 25%, and (d) 30%.
(a) (b) (c) (d)As is shown in Figure7, Mandhane’s model can roughly divide stratified flow from nonstratified flow at four water cuts, yet the results lead to a larger smooth stratified flow region. Taitel-Dukler model cannot give a satisfying stratified flow transition boundary, due to not considering the effect of hydrate formation on flow stability. Transition boundaries obtained based on the model developed in this work show good agreement with experimental data.All data points are summarized in Figure8. It can be seen that no distinctive effect of water cut on flow pattern division is observed, though gas consumption rate differs evidently according to Table 3. This may be due to the similar water consumption rates and hydrate fractions as shown in Figure 9. As calculated flow pattern boundary is primarily affected by fluid viscosity, and hydrate slurry viscosity is further determined by hydrate volume fraction, proximal transition lines are obtained under similar hydrate fraction though at different water cuts.Figure 8
Flow pattern distribution for gas-slurry stratified flow at different water cuts.Figure 9
Hydrate fraction and water conversion rate at different water cuts.The pattern experiment data of gas-hydrate slurry multiphase flow pattern in this work are not effectively enough to obtain more precise correlation parameters or investigate the flow pattern transition mechanism deeply. More experimental and theoretical research should be carried out in the future. Nevertheless, the good agreement with experimental data proved this work to be both feasible and significant.


## 5. Conclusion

In this work, a one-dimension two-fluid model for gas-hydrate slurry stratified flow was developed based on perturbation method, and a stability criterion for smooth stratified flow was proposed. In establishing the criterion, mass transfer between gas and slurry phase caused by hydrate formation was considered, and various mathematical techniques were applied in linearizing the equation sets. As the formation of hydrate would evidently complicate the flow, the influences of shear stress, gravity, surface tension, hydrate formation, and other unstable factors were considered. Compared with the classical two-phase gas-liquid stratified flow stability criteria proposed by Taitel-Dukler [14] and Mandhane [13], transition boundary based on this study showed better agreement with experiment results.Groups of gas-hydrate slurry multiphase flow experiments were carried out on a most advanced high-pressure (>10 MPa) hydrate slurry flow loop in China, and 293 experimental data points of both smooth and nonsmooth stratified flow were obtained. Model parameters were generated in combining the experiment data with the gas-hydrate slurry stratified flow stability creation developed in this work, and the model was proved applicable in gas-hydrate slurry multiphase flow numerical simulation and characteristic study.



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