Modelling of Transient CO2/Water Flow in Wellbore considering Multiple Mass and Heat Transfer

A transient fully coupled model is proposed to investigate the two-phase flow of CO2 and water-based fluid in a wellbore, considering the complex mass and heat transfer in different flow patterns and dynamic coupling between the wellbore and reservoir. Based on mass conservation, momentum, and energy balance, the model employs a state-of-the-art equation of state and transport models to analyze the variations of multiphase flow behaviors and CO2 properties in a wellbore. Applied in the scenario of a drilled gas kick, the proposed model is used to simulate the processes of gas migration and two-phase flow in the wellbore. The results indicate that the CO2 solubility increases gradually with the increment of depth, the trend of which shows an abrupt change in 500-1000m due to the phase transition of CO2. During kick development, the fronts of free gas and dissolved gas increase almost linearly with time. Through a comparison of CO2 and CH4 kicks, gas dissolution is found to significantly suppress the development process of CO2 kick. The error in kick prediction can reach 42% if the effect of gas dissolution is neglected. However, it can be neglected for CH4 kick.


Introduction
As an important greenhouse gas, carbon dioxide has aroused wide concern in the field of energy and environment. Its use in energy development is one of the hot researches recently. Compared to the conventional working fluids, liquid or supercritical CO 2 has good properties (large density and heat capacity, low viscosity, and surface tension, etc.) for heat transfer and fluid flow and is widely used in the operations of drilling, fracturing, enhanced oil recovery (EOR), and geothermal exploitation [1][2][3][4][5]. Generally, affected by variations of the temperature and pressure in the wellbore, the thermophysical properties of CO 2 change significantly in the temporal and spatial scales, as shown in Figure 1. Particularly, there exist complicated flow patterns and mass and heat transfer processes while CO 2 and water-based fluid coexisting in the wellbore. Therefore, it is necessary to develop a reliable model for transient CO 2 /water flow, which can be of important significance for accurate prediction and control of multiphase flow parameters in the actual wellbore/reservoir systems.
At present, a lot of research works have been done to model the single CO 2 flow in the wellbore during the processes of CO 2 drilling, injection, and production [6][7][8]. In actual operations, the wellbore two-phase flow of CO 2 and water can be generated due to the influx of formation fluids (such as a drilled gas kick). As the flow of different fluid components, the phase interface may exhibit various flow patterns, accompanied with complicated mass and heat transfer phenomena. Zha et al. [9] proposed an equivalent single-phase flow model to interpret the oil-water two-phase flow in a wellbore during the well test period. Shang et al. [10] developed a mathematical model coupling fluid flow in the horizontal wellbore and reservoir, based on the principle of mirror reflection and mass conservation. Regarding the nonisothermal multiphase flow process in the wellbore, Pan and Oldenburg [11] developed an integrated simulator based on the drift-flux model, which can be applied in the process of geothermal exploitation using supercritical CO 2 . Considering the migration characteristics of gas kick in the wellbore, He et al. [12] built a kick simulation model and analyzed the effect of sour gas dissolution on well control operations. Lu and Connell [13] developed a quasisteady wellbore two-phase flow model to simulate the nonisothermal flow of carbon dioxide in injection wells during geological storage. Recently, Wang et al. [14][15][16] proposed a series of calculation models of CO 2 density, friction coefficient, Joule-Thomson coefficient, and so on, which can significantly improve the simulation accuracy of CO 2 thermophysical parameters in the wellbore in the drilling scenario. Based on that, they developed the wellbore temperature and pressure models considering the phase transition of sour gases. These models can perform more accurate estimations of wellbore temperature and pressure fields [17][18][19]. Their simulation results indicated that the gas kicks can be "hidden" and "abrupt" successively, affected by the phase transition of the fluids in the dynamically changing temperature and pressure environment. Furthermore, the critical wellhead back pressures for suppressing the abrupt expansion of sour gases were obtained [20]. These research and findings can provide a solid theoretical basis for early kick detection and wellbore pressure control.
In this study, we developed a transient fully coupled model for wellbore CO 2 /water flow, which considers the complicated mass and heat transfer mechanisms in different flow patterns and the dynamical coupling between wellbore and reservoir. Subsequently, the proposed model is applied to analyze the multiphase flow process during a drilled CO 2 kick.

Mass Conservation Equation.
Considering the onedimensional unsteady flow [19], the equations expressing the mass conservation laws among the free CO 2 phase, con-tinuous water phase, and dissolved CO 2 phase can be presented as follows: where t is the time, s; s is the distance, m; A is the crosssectional area, m 2 ; α CO2 is the void fraction of free CO 2 phase; ρ CO2 is the CO 2 density, kg/m 3 ; v CO2 is the velocity of free CO 2 phase, m/s; _ m L is the mass transfer rate of CO 2 dissolution, kg/(m•s); and q g is the mass transfer rate of CO 2 between the wellbore and reservoir, kg/(m•s). The rate of CO 2 influx from reservoir to wellbore is mainly dominated by the pressure underbalance and reservoir properties, which can be estimated using the model of Sun et al. [21].
(2) Continuous water phase where α l is the void fraction of continuous water phase; ρ l is the density of water, kg/m 3 ; and v l is the velocity of water, m/s.
where x sol is the mass fraction of dissolved gas, kg/kg. In equations (1)-(3), the relation for velocities of different phases can be described by the drift-flux model: where C 0 is distribution coefficient, dimensionless and v ∞ is the drift velocity of gas, m/s. Commonly, the distribution coefficient and drift velocity are closely related to the flow pattern, phase distributions, pipeline configuration, etc. Here, the slip relation developed by Bhagwat and Ghajar [22] is employed.

Momentum Conservation Equation.
∂ ∂t where ρ m is the density of fluid mixture, kg/m 3 ; v m is the velocity of fluid mixture, m/s; p is pressure, Pa; d c is the hydraulic diameter, m; θ is the inclination angle, rad; and g is gravitational acceleration, m/s 2 . In equation (5), f is the  Figure 1: Phase diagram of carbon dioxide. The blue curve represents a typical temperature-pressure distribution in the wellbore as CO 2 flow. As seen, the phase and thermophysical properties of CO 2 can vary significantly.
2 Geofluids friction coefficient, which can be described using the model of Wang et al. [15].
where Re is the Reynolds number, dimensionless and ε is roughness, m.

Energy Conservation Equation.
Considering the phase transition, Joule-Thomson cooling, heat transfer between the wellbore and formations, and reservoir coupling, the temperature model for wellbore CO 2 /water flow can be obtained based on energy conservation [23]: where T is the fluid temperature,°C; C p,CO2 is specific heat capacity of CO 2 at constant pressure, J/(kg•°C); C p,l is specific heat capacity of the liquid phase at constant pressure, J/(kg•°C); C J is the Joule-Thomson coefficient of gas,°C/Pa; h e is the enthalpy of the influx gas at reservoir condition, J/kg; h g is the enthalpy of CO 2 in the wellbore, J/kg; ΔH sol is the dissolution heat of gas in water, J/kg; Q transfer is the rate of heat exchange between the wellbore and formations, Q transfer = Qðh,ΔT, r i , λ i , tÞ; h is the convective heat transfer coefficient, W/(m 2 •°C); ΔT is the temperature difference,°C ; r i represents the geometric size of wellbore system, m; and λ i is the thermal conductivity of wellbore system, W/(m•°C).
Generally, the Joule-Thomson effect of water can be negligible compared to that of CO 2 . According to the PVT properties of gas, the Joule-Thomson coefficient of CO 2 is defined as follows: Wang et al. [16] revealed that the absolute average errors of the CO 2 Joule-Thomson coefficient predicted by the state equations are relatively low in vapor and supercritical states, but larger errors appear near the CO 2 critical point and liquid state. Therefore, they built a religious empirical model, in which the absolute average errors at the vapor, liquid, and supercritical states are 1.52%, 4.59%, and 3.08%, respectively.

Mass and Heat Transfer for Different Flow Patterns
3.1. Flow Pattern Transition. The mass and heat transfer characteristics in the two-phase flow are significantly governed by the flow patterns. In this study, the model developed by Hasan and Kabir [24] is used to flow pattern identification, as shown in Table 1.

Interphase Mass Transfer
Model. The dissolution of sour gas in the wellbore two-phase flow is a diffusion process governed by concentration difference. Therefore, the gas dissolution rate is a function of gas concentration and mass transfer 3 Geofluids coefficient: where c is the gas concentration, mol/m 3 ; c sat is the gas concentration at saturation, mol/m 3 ; M g is the molecular mass of CO 2 , kg/mol; k M is the interphase mass transfer coefficient between CO 2 and liquid phase, m/s; and S int is the contact area of CO 2 and liquid phase, m 2 . In this study, the contact area in different flow patterns is estimated using the model proposed by Sun et al. [23]. The determination of the mass transfer coefficient, which is related to the fluid properties (such as density, viscosity, and diffusivity), flow velocity, and annulus size, is challenging. Considering the laminar flow and turbulent flow conditions, the expression presented by Cussler [25] is employed.
where L is the pipe length, m; ν is the kinematic viscosity, m 2 /s; and D g is the gas diffusivity coefficient, m 2 /s. Furthermore, we use the model of Duan and Sun [26] to calculate the CO 2 solubility in water-based fluid in this study.
where φ CO2 is the fugacity coefficient of CO 2 and y CO2 is the Churn flow v gs < 3:1 σg ρ L − ρ g /ρ 2 g h i 0: 25 and v 2 gs ρ g < 17:1 log 10 v 2 Note that v gs is the superficial velocity of gas phase, m/s and v Ls is the superficial velocity of liquid phase, m/s.  4 Geofluids mole fraction of CO 2 in vapor phase, λ CO2−c is the binary interaction parameter of CO 2 and cation, λ CO2−a is the binary interaction parameter of CO 2 and anion, and ζ CO2−c−a is the ternary interaction parameter of CO 2 , cation, and anion.

Convection Heat Transfer Model in Different Flow
Patterns. As wellbore multiphase flow, convection heat transfer mainly occurs in the heat transfer boundary layer. Therefore, the flow pattern can have an important influence on the heat transfer mechanisms. Generally, the convection heat transfer coefficient can be written as follows: where λ is the thermal conductivity of CO 2 , W/(m•°C).
Through the experiments of heat transfer in wellbore multiphase flow, Gao et al. [27] proposed the model of convection heat transfer coefficient in different flow patterns. Furthermore, the thermophysical properties of CO 2 vary greatly accompanied with the complicated mass and heat transfer processes. The detailed calculation models for thermophysical parameters of CO 2 are presented in Appendix A.

Model
Simulation. The integrated model is solved using the simulation method proposed by Sun et al. [23], which employs a fully implicit scheme, constant space steps, and varying time steps. The overall simulation process is consisting of three layers of iterations. At first, the phase velocities and fractions at different space blocks are calculated based on a drift-flux model. Subsequently, the pressure field is estimated using a predictor-corrector shooting technique. With updating the fluid properties and multiphase flow parame-ters, the temperature distribution in the drilling pipe and annulus are iteratively simulated until a desired convergence tolerance is achieved.

Model
Verification. The proposed model is validated using the measured data of a field well, which comes across a CO 2 kick accident. The pit gain approaches 1.9m 3 at 12:00 when the alarm sounded. Then, the well was shut to measure the standpipe pressure and casing pressure. At 13:05, the gas influx was circulated gradually while a constant casing pressure (2 MPa) is maintained. At 15:30, a large amount of CO 2 blows out from the wellhead. The main parameters of the kicking well are presented in Table 2. Figure 2 shows the simulated and measured pit gains during CO 2 kick. At point A, the gas kick was detected with a pit gain of 1.9 m 3 . The multiphase flow process of gas circulation is simulated from point A to point B, in which the pressure underbalance and gas influx rate at the bottom hole is low. CO 2 dissolves into the drilling fluid gradually, which leads to the nonobvious variations in pit gain. After point B, the  Figure 3: The distribution of fluid temperature and CO 2 density and solubility in the wellbore at 500 s. 5 Geofluids dissolved CO 2 rapidly releases out from the drilling fluid due to the decrease of gas solubility near the surface.
As seen, the simulated results of pit gain agree well with the measured data, and the average error is 3.253%. It indicates that the proposed model can accurately describe the effect of phase transition on gas migration in wellbore.

Case Analysis
The formation fluids can enter the wellbore driven by pressure underbalance, if the bottom hole pressure at open hole section is less than the pore pressure during drilling. The understanding of phase transition and kick migration is important for early kick detection and wellbore control procedure. Using the proposed model, we simulate and analyze the flow behaviors of CO 2 and water-based fluid in the scenario of a drilled CO 2 kick. The basic parameters for kick simulation [19] are shown in Table 3. Figure 3 shows the profiles of fluid temperature and CO 2 density and solubility in the wellbore at 500 s. As seen, the wellbore temperatures increase gradually with the well depth increasing, affected by variations of the formation temperatures. At a given depth, the fluid temperature in the annulus is larger than that in the drill pipe.

Analysis of the Gas Migration Process.
Due to the dynamically changing temperature and pressure fields, the thermophysical properties of CO 2 vary significantly along the wellbore. Furthermore, the variation trend can be abrupt at 500 m to 1000 m because of phase transition.
There results indicate that the free gas and dissolved gas can rapidly expand and separate out, respectively, which will pose a challenge to kick management and well control. Figures 4(a) and 4(b) show the dynamical distributions of the free gas and dissolved gas in the wellbore at different times. As shown in Figure 4, the volume of gas influx in the wellbore increases gradually with time. And the front of gas 6 Geofluids profiles rises nearly linearly with time. In general, the void fraction of free gas increases as the increment of well depth. As shown in Figure 4, the gas dissolves into the drilling fluids gradually as it migrates upward in the wellbore. The void fraction of dissolved gas reaches saturation state with time increasing, which can be a dynamical process. For example, the void fraction of dissolved gas at 3400 m increases rapidly at first and then decreases slightly due to variation of the gas solubility. It can be expected that the gas dissolution process will significantly suppress the migration and expansion of gas kick in the wellbore. After the gas kick initializes 500 s, the void fraction of dissolved gas is 42 percent of that of free gas near the bottom hole.
In this section, we study the migration process of a gas bubble which is released at the bottom hole as the gas kick initializes, as shown in Figure 5.
The gas influx rate at the open hole section is mainly governed by the wellbore pressure distribution, which consists of hydrostatic pressure and friction pressure. As the gas kick enters the wellbore, the flow velocity and friction pressure of fluid mixture will increase abruptly, which can lead to the sudden increment of bottom hole pressure. Subsequently, the decrease of hydrostatic pressure plays a dominant role, and the bubble pressure decreases gradually.
As seen, the bubble radius and mass decrease gradually with time. However, the bubble shrinkage rate decreases gradually due to the decrease of gas/liquid contact area and mass transfer rate. Furthermore, the volume change of gas bubble is also affected by the pressure change. At the early stage, the increment of bubble pressure will increase the bubble shrinkage rate. Then the decrease of bubble pressure will decrease the rate of bubble shrinkage.

Effect of Gas Dissolution.
A comparison of CO 2 and CH 4 kicks is conducted to analyze the effect of gas dissolution and phase transition on kick migration. Figure 6 shows the distributions of free gas along the wellbore during methane and CO 2 kicks. For a given depth, the void fraction of methane is significantly larger than that of CO 2 . Because the gas influx rate for methane kick is larger under the same pressure underbalance, since the viscosity and percolation resistance of methane in reservoir are much smaller than that of CO 2 . Affected by the large gas influx rate, the velocity of the fluid mixture is larger and gas front rises faster during the methane kick. Figure 7 shows the distributions of dissolved gas along the wellbore during methane and CO 2 kicks. Under the same temperature and pressure condition, the solubility of carbon dioxide in water is much larger than that of methane. At 3000 m, we find that the void fraction of dissolved CO 2 is 20 times larger than that of dissolved methane.
As shown in Figure 8, the effect of gas dissolution on kick migration can be negligible in the methane kick, because the solubility of methane in water is low. As for the CO 2 kick, it can be found that gas dissolution has an important influence on the process of kick development. In the figure, the simulation error of pit gain can reach 50% if the effect of gas dissolution is neglected at 500 s.

Conclusions
(1) A transient fully coupled model for the two-phase flow of CO 2 and water-based fluid in wellbore is developed, considering the mass and heat transfer in different flow patterns and the dynamic coupling between wellbore and reservoir. Model validation against the measured data of a kicking well indicates that it can produce an average error of 3.253% (2) Affected by variations of the wellbore temperature and pressure fields, the solubility of CO 2 increases with depth, which shows an abrupt increasing trend at 500 m to 1000 m due to phase transitions (3) A methane kick can develop more rapidly than a CO 2 kick, i.e., the velocity of gas migration and void fraction of gas in a methane kick are much larger (4) The simulation error of pit gain can reach 50% if the effect of gas dissolution is neglected in the CO 2 kick. However, it is negligible in the methane kick