# A mechanistic model of heat transfer for gas–liquid flow in vertical wellbore annuli

## Abstract

The most prominent aspect of multiphase flow is the variation in the physical distribution of the phases in the flow conduit known as the flow pattern. Several different flow patterns can exist under different flow conditions which have significant effects on liquid holdup, pressure gradient and heat transfer. Gas–liquid two-phase flow in an annulus can be found in a variety of practical situations. In high rate oil and gas production, it may be beneficial to flow fluids vertically through the annulus configuration between well tubing and casing. The flow patterns in annuli are different from pipe flow. There are both casing and tubing liquid films in slug flow and annular flow in the annulus. Multiphase heat transfer depends on the hydrodynamic behavior of the flow. There are very limited research results that can be found in the open literature for multiphase heat transfer in wellbore annuli. A mechanistic model of multiphase heat transfer is developed for different flow patterns of upward gas–liquid flow in vertical annuli. The required local flow parameters are predicted by use of the hydraulic model of steady-state multiphase flow in wellbore annuli recently developed by Yin et al. The modified heat-transfer model for single gas or liquid flow is verified by comparison with Manabe’s experimental results. For different flow patterns, it is compared with modified unified Zhang et al. model based on representative diameters.

## Keywords

Gas–liquid flow Vertical annuli Heat transfer Tubing liquid film Casing liquid film## 1 Introduction

As oil and gas development moves from land or shallow water to deep and ultradeep waters, multiphase flow occurs during production and transportation (Chen 2011). The flow normally occurs in horizontal, inclined, or vertical pipes and wells. Gas–liquid two-phase flow in an annulus can be found in a variety of practical situations. In high rate oil and gas production, it may be beneficial to flow fluids vertically through the annulus configuration between well tubing and casing. For surface facilities, some large, under-utilized flow lines can be converted to dual-service by putting a second pipe through the large line (i.e., flowing produced water in the inner line and gas in the annulus). During gas production, liquids may accumulate at the bottom of the gas wells during their later life. In order to remove or “unload” the undesirable liquids, a siphon tube is often installed inside the tubing string, which would form a gas–liquid two-phase flow in the annulus. Flow-assurance problems, such as hydrate blocking (Jamaluddin et al. 1991; Li et al. 2013; Wang et al. 2014, 2016) and wax deposition (Zhang et al. 2013; Bryan 2016; Theyab and Diaz 2016), are strongly associated with both the hydraulic and thermal behavior. For example, they are related to the fluid velocity, liquid fraction, slug characteristics, pressure gradient and convective-heat-transfer coefficients of different phase and flow patterns in multiphase flow. Therefore, multiphase hydrodynamics and heat transfer in an annulus need to be modeled properly to guide the design and operation of flow systems.

Compared to theoretical studies of multiphase hydrodynamics (Liu et al. 2007; Wang and Sun 2009), and multiphase heat transfer in pipe flow (Zheng et al. 2016; Gao et al. 2016; Karimi and Boostani 2016; Rushd and Sanders 2017), there are very limited research results in the open literature for multiphase heat transfer in wellbore annuli. Davis et al. (1979) obtained a model for calculating the local Nusselt numbers of stratified gas/liquid flow in turbulent liquid/turbulent gas conditions. The model was tested with heat-transfer experiments for air/water flow in a 63.5-mm inside diameter (ID) tube. Guo et al. (2017) established a mathematical model of heat transfer in a gas-drilling system, considering the flowing gas, formation fluid influx, Joule–Thomson cooling and entrained cuttings in the annular space. However, the multiphase flow effect on the heat transfer was not considered.

Shoham, et al. (1982) undertook experiments on heat-transfer for slug flow in a horizontal pipe. He found a substantial difference in heat-transfer coefficient existed between the top and bottom of the slug. Developing heat-transfer correlations of different flow patterns was the aim of most previous studies (Shah 1981). Twenty heat-transfer correlations were compared in Kim’s study (Kim et al. 1997). He collected the experimental data from the open literature and recommended the correlations for different flow patterns. However, the errors with the experimental results by Matzain (1999) were large. Later, a comprehensive mechanistic model about heat transfer in gas–liquid pipe flow was obtained (Manabe 2001). It was compared with the experimental data, and the performance was better. However, there were some inconsistencies in annular and slug flow. It needed to be modified.

A hydraulic model was developed to predict flow patterns, liquid holdup and pressure gradients for steady-state gas–liquid flow in wellbore annuli (Yin et al. 2014). The major advantage of this model compared with previous mechanistic models is that it is developed based on the dynamics of slug flow, and the liquid-film zone is used as the control volume. The effects of the tubing liquid film, casing liquid film and the droplets in the gas core area on the mass and momentum transfers are considered. Multiphase heat transfer depends on the hydrodynamic behavior of the flow. The objective of this study is to develop a heat-transfer model for gas–liquid flow that is consistent with the hydrodynamic model in vertical wellbore annuli.

## 2 Modeling

### 2.1 Single-phase flow

*ρ*

_{l}is the density of liquids, kg/cm

^{3};

*v*

_{l}is the velocity of liquids in the annulus, m/s; d

*l*is the length of the element.

*p*is the unit pressure, MPa;

*g*is the gravity acceleration, m/s

^{2};

*θ*is the inclination angle, °;

*τ*is the friction;

*d*is the annulus diameter, m;

*A*is the cross area of the annulus, m

^{2}.

*e*is the internal energy of the unit, J/kg;

*q*

_{1}is the heat loss from the annulus to the formation, W.

*h*is the enthalpy, J/kg;

*w*

_{l}is the mass flow rate of liquids in the annulus, kg/s.

*T*

_{D}proposed by Hasan and Kabir (1991), we may write an expression for heat transfer from the wellbore/formation interface to the formation as

*T*

_{e}is the formation temperature, °C;

*T*

_{wb}is the wellbore temperature, °C;

*k*

_{e}is the conductivity of formation, W/(m °C); the dimensionless temperature, defined by Hasan and Kabir (1991),

*T*

_{D}, can be easily estimated from the following models.

*t*

_{D}is the dimensionless flowing time,

*t*is the flowing time, s;

*r*

_{w}is the wellbore radius, m.

*q*

_{1}may be expressed in terms of an overall heat coefficient (Hansan and Kabir 1994),

*r*

_{co}is the outside casing radius, m;

*T*

_{a}is the fluid temperature in the annulus, °C;

*U*

_{a}is the overall heat-transfer coefficient of the annulus, W/(m

^{2}°C).

*r*

_{ci}is the inside casing radius, m;

*h*

_{a}is the convective-heat-transfer coefficient of fluids in the annulus, W/(m

^{2}°C);

*k*

_{cas}is the conductivity of the casing wall, W/(m °C);

*k*

_{cem}is the conductivity of cement, W/(m °C).

*c*

_{p}is the fluid heat capacity at the constant pressure in the annulus, J/(kg °C).

*η*

_{l}is the fluid Joule–Thomson coefficient in the annulus, °C/MPa.

*Φ*

_{a}, as

Equation (14) is the energy conservation equation of fluids in the annulus.

*T*

_{ei}is the temperature of formation at wellbore intake, °C;

*g*

_{e}is the formation thermal gradient, °C/100 m;

*L*is the depth, m.

If Eq. (15) is used for the riser of an offshore production well, then the surrounding sea temperature is not a linear function of depth. It will be calculated according to the actual environment (Wang and Sun 2009).

*U*

_{a},

*c*

_{p},

*η*

_{l},

*g*

_{e},

*θ*,

*v*

_{l}d

*v*

_{l}/d

*l*and d

*p*/d

*l*can be approximately constants, combining Eqs. (15) and (16) and integrating, yields an explicit equation for the temperature:

### 2.2 Bubbly flow and dispersed-bubble flow

*d*

_{R}is the hydraulic diameter, m;

*d*

_{ci}is the inside diameter of casing, m;

*d*

_{tuo}is the outside diameter of tubing, m.

*k*

_{m}is the conductivity of the mixture, W/(m °C),

*H*

_{l}is the liquid hold up;

*k*

_{l}is the conductivity of liquid, W/(m °C);

*k*

_{g}is the conductivity of gas, W/(m °C).

*f*

_{m}is the friction factor of the mixture;

*μ*

_{l}is the viscosity of liquid, mPa s;

*μ*

_{m}is the viscosity of the mixture, mPa s;

*μ*

_{g}is the viscosity of gas, mPa s,

*ρ*

_{m}is the density of the mixture, kg/m

^{3};

*v*

_{m}is the velocity of the mixture, m/s;

*c*

_{pm}is the specific heat of the mixture, J/(kg °C),

*c*

_{pl}is the specific heat of liquid, J/(kg °C);

*c*

_{pg}is the specific heat of gas, J/(kg °C).

Then Eq. (20) can be used in the heat transfer in bubble flow and dispersed-bubble flow based on the modified convective-heat-transfer coefficient *h* _{ab}.

### 2.3 Annular flow

*T*

_{gc}is the temperature of the gas core, °C;

*T*

_{cf}is the temperature of the casing film, °C;

*T*

_{tuf}is the temperature of the tubing film, °C;

*c*

_{pgc}is the specific heat of the gas core, J/(kg °C); \(A^{\prime}_{\text{gc}}\), \(B^{\prime}_{\text{gc}}\),

*Φ*

_{gc}are the local parameters defined as follows:

*w*

_{gc}is the mass flow rate of the gas core, kg/s;

*c*

_{pgc}is the specific heat of the gas core, J/(kg °C);

*v*

_{gc}is the velocity of the gas core, m/s;

*ρ*

_{gc}is the density of the gas core, kg/m

^{3};

*η*

_{gc}is the Joule–Thomson coefficient of the gas core, °C/MPa;

*U*

_{gc}is the overall heat-transfer coefficient of the gas core, W/(m

^{2}°C), and defined as

*r*

_{tuo}is the outside radius of the tubing, m;

*δ*

_{tu}is the thickness of the tubing film, m;

*δ*

_{c}is the thickness of the casing film, m;

*h*

_{gc}is the convective-heat-transfer coefficient of the gas core, W/(m

^{2}°C).

*U*

_{tuf}is the overall heat-transfer coefficient of the tubing film, W/(m

^{2}°C), and defined as

*h*

_{tuf}is the convective-heat-transfer coefficient of the tubing film, W/(m

^{2}°C);

*k*

_{tu}is the conductivity of the tubing wall, W/(m °C).

*w*

_{tuf}is the mass flow rate of the liquid tubing film, kg/s;

*ρ*

_{tuf}is the density of the liquid tubing film, kg/m

^{3};

*c*

_{ptuf}is the specific heat of the liquid tubing film, J/(kg °C);

*Φ*

_{tuf}is the local constant and defined as follows:

*v*

_{tuf}is the velocity of the liquid tubing film, m/s;

*η*

_{tuf}is the Joule–Thomson coefficient of the liquid tubing film, °C/MPa.

*w*

_{cf}is the mass flow rate of liquid casing film, kg/s;

*ρ*

_{cf}is the density of liquid casing film, kg/m

^{3};

*c*

_{pcf}is the specific heat of liquid casing film, J/(kg °C); \(C^{\prime}_{\text{cf}}\) is the local parameter defined in Eq. (23),

*U*

_{cf}is the overall heat-transfer coefficient of casing film, W/(m

^{2}°C), and defined as

*h*

_{cf}is the convective-heat-transfer coefficient of casing film, W/(m

^{2}°C).

*Φ*

_{cf}is the local parameter defined in Eq. (23),

*v*

_{cf}is the velocity of the liquid casing film, m/s;

*η*

_{cf}is the Joule–Thomson coefficient of the liquid casing film, °C/MPa.

*H*

_{tuf}is the holdup of the liquid tubing film;

*H*

_{cf}is the holdup of the liquid casing film.

*k*

_{lf},

*k*

_{gc}are the thermal conductivities of the liquid film and gas core, W/(m °C);

*d*

_{tufR},

*d*

_{cfR},

*d*

_{gcR}are the “hydraulic diameter” of the tubing film, the casing film and gas core, m.

*i*is tubing film or casing film,

*i*= tu or c.

*f*

_{if}is the friction factor at the wall in contact with the liquid film.

*δ*

_{c}and

*δ*

_{tu}, the hold up, liquid and gas physical properties .

### 2.4 Slug flow

#### 2.4.1 Film region

*U*

_{gc}to

*U*

_{T}. Then the heat transfer of the film region can be calculated.

*U*

_{T}is the overall heat-transfer coefficient of the Taylor bubble, W/(m

^{2}°C);

*h*

_{T}is the convective-heat-transfer coefficient of the Taylor bubble, W/(m

^{2}°C).

*k*

_{T}is the thermal conductivities of the Taylor bubble, W/(m °C);

*d*

_{TR}is the “representative diameter” of the Taylor bubble.

#### 2.4.2 Slug region

There are small discrete bubbles in a continuous liquid phase. The flow characters are similar to bubbly flow. So, the heat transfer can be calculated by the bubbly flow model.

#### 2.4.3 Slug unit

*T*

_{s}is the temperature of the fluid in slug flow, °C;

*T*

_{ls}is the temperature of the fluid in the slug unit, °C;

*T*

_{T}is the temperature of the Taylor bubble, °C;

*l*

_{F}is the length of the liquid film, m;

*l*

_{s}is the length of the slug unit, m;

*l*

_{U}is the length of the whole slug, m.

## 3 Solution procedure

## 4 Comparisons with the unified Zhang model

Composition of natural gas

Components | Molar fraction, % |
---|---|

N | 1.82 |

CO | 0.65 |

C | 93.83 |

C | 2.98 |

C | 0.59 |

| 0.08 |

| 0.05 |

Composition of oil

Basic parameters for the simulation

There are few experimental research results in the open literature for multiphase heat transfer in annuli. The unified Zhang et al. model (2006) is verified by comparison with Manabe’s experimental results for different flow patterns in a crude-oil/natural gas system, and good agreement has been observed in the comparison. So, the unified Zhang et al. model is modified to calculate the heat transfer of gas–liquid flow in annuli based on the “hydraulic diameter” of the annuli, Eq. (17), and the results are compared with the present mechanistic heat-transfer model for gas–liquid flow in annuli.

## 5 Conclusions and discussion

A heat-transfer model for gas–liquid flow in vertical annuli is developed in conjunction with the mechanistic hydrodynamic model of Yin et al. (2014), which can predict flow pattern transitions, liquid holdup, gas void fraction, pressure gradient, and slug characteristics in gas–liquid two-phase flow in vertical annuli. The heat-transfer modeling is based on energy-balance equations and analyses of the temperature differences and variations in the tubing liquid film, casing liquid film, gas core, Taylor bubble and slug body.

The heat-transfer model for single gas or liquid flow is verified by comparison with Manabe’s experimental results (2001). Good agreement has been observed in the comparison. For different flow patterns, it is compared with unified Zhang et al. model modified based on “hydraulic diameter”. For bubble or dispersed-bubble flow, the error is lower than 10%. With the gas void fraction and gas flow velocity increasing, the error will be larger but lower than 30%. In other words, the difference between the new model and modified unified Zhang et al. model will be small if the gas void fraction and velocity is small. The difference will be large when it changes. The modified method based on “hydraulic diameter” is no longer applicable for slug and annular flow in vertical annuli when the gas void fraction increases. It may be 1.3 times larger than the new model. Experimental investigations of heat transfer in vertical are required to improve the model performance.

## Notes

### Acknowledgements

This paper was sponsored by the National Natural Science Foundation of China (Grant No. 51504279), Shandong Provincial Natural Science Foundation, China (ZR2014EEQ021), Qingdao Science and Technology (15-9-1-96-jch), the Fundamental Research Funds for the Central Universities (17CX02073, 17CX02011A and R1502039A) and 973 Project (2015CB251206). We recognize the support of China University of Petroleum (East China) for the permission to publish this paper.

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