# A Hydraulic Model for Multiphase Flow Based on the Drift Flux Model in Managed Pressure Drilling

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Governing Equations

_{G}represents the gravity term:

_{f}is the friction term:

_{m}can be calculated by [26]:

_{m}is the viscosity.

#### 2.2. Closure Equations

_{0}is the distribution parameter and υ

_{t}is the gas drift velocity.

#### 2.3. Shi Slip Relation

_{1}= 0.2 and a

_{2}= 0.4.

_{u}is the critical Kutateladze number:

_{w}denote the fitted parameters, which are set to 142 and 0.008, respectively.

_{B}is the Bond number:

_{gl}is the gas–liquid interfacial tension.

_{c}is the characteristic velocity:

_{0}can be determined by:

_{max}represents the value of the gas distribution parameter, which is usually set to 1.2.

_{υ}is a multiplier of the flooding velocity fraction that defaults to 1.0.

_{0}= 1.28, n

_{1}= 0.5, n

_{2}= 1.7 (θ < 88°). When 88°≤ θ ≤ 90°, the relation has a large deviation for horizontal wells.

#### 2.4. Primitive Variables

_{1}and u

_{2}:

_{l}and υ

_{g}:

#### 2.5. Boundary Conditions

_{c}is the valve discharge coefficient, z

_{c}is the choke valve opening index, and γ is the gas expansion factor. P

_{choke}and P

_{atm}represent the pressure of the choke and the atmosphere, respectively.

_{r}and P

_{wf}are the reservoir pressure and bottom-hole pressure, respectively. q is the inflow rate. A and B are regression coefficients, which are mainly determined by the reservoir.

## 3. Numerical Scheme

#### 3.1. Advection Upwind Splitting Method Scheme

#### 3.2. Second-Order Accuracy

#### 3.3. Solution Method

#### 3.4. Workflow

## 4. Experimental Validation

#### 4.1. Laboratory Test

^{3}/s and then injected with gas at 25 s, of which the flow rate was 0.01 kg/s. Figure 8 shows the annulus flow dynamics. The flow state is mainly divided into three stages: the liquid phase flow stage (0–25 s), the transient gas–liquid two-phase flow stage (25–32 s), and the stable flow stage (32–60 s). Generally, the pressure in the wellbore increases first and then decreases, and finally, tends to be stable.

#### 4.2. Full-Scale Experiment

^{3}/s prior to the air being injected into the wellbore. The change in gas flow is presented in Figure 11. Over the course of gas injection, there was a significant change in the gas volume. The total time taken to conduct this experiment amounted to 1500 s, and the injection depth was relatively deep, which was used to exhibit the characteristics of the gas invasion process in practice.

## 5. Sensitivity Analysis

^{3}. A commonly-seen practice for controlling the downhole pressure is to change the density of the drilling fluid. However, as it takes time for the drilling fluid to reach the downhole from the wellhead, the downhole pressure is incapable of being adjusted promptly, which means that the possibility of various dangerous incidents occurring remains possible. Therefore, a combined approach taken while drilling with backpressure and displacement would be more effective and reasonable to prevent the occurrence of incidents.

## 6. Conclusions

- (1)
- In laboratory experiments, the flow state is mainly divided into three stages: the liquid phase flow stage (0–25 s), the transient gas–liquid two-phase flow stage (25–32 s), and the stable flow stage (32–60 s). The simulated data are in good agreement with the experimental results, and the error range is within ±10%.
- (2)
- The pressure shows a gradual decline, which is followed by a sharp fall, before an incremental increase again. The drop in pressure is due to the original fluid being dispelled with the ingress of gas. As the gas moves upwards, the pressure it is subjected to decreases, which causes the gas to expand further. As a result, more fluid is dispelled, and the drop in pressure occurs faster. Up to the point when the gas reaches the wellhead, the downhole pressure is at its minimum.
- (3)
- The adjustment of wellhead back pressure is mainly realized by throttle valve. The higher the wellhead back pressure is, the smaller the downhole pressure will be. When the gas–liquid two-phase flow in the wellbore reaches an equilibrium state, the downhole pressure will decrease less with the increase of drilling fluid displacement, and the time of gas reaching the wellhead will be earlier.
- (4)
- The downhole pressure can be controlled by changing the density of drilling fluid. However, the adjustment of drilling fluid density has a serious lag. Considering the variation of gas invasion caused by reservoir pressure difference, there will be no stable gas–liquid two-phase flow equilibrium.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Relation between the models [11]. PDE: partial differential equation; DFM: drift flow model; and ODE: ordinary differential equation.

**Figure 2.**The typical hydraulic cycle of managed pressure drilling (MPD). BHA: bottom hole assembly.

**Figure 11.**Gas mass flow rate vs. time [37].

**Figure 17.**Variation in the bottom hole pressure under different back pressures at different: (

**a**) times; and (

**b**) depths.

**Figure 18.**Variation in the bottom hole pressure at different liquid densities at different: (

**a**) times; and (

**b**) depths.

**Figure 19.**Variation in the bottom hole pressure at different liquid flow rates at different: (

**a**) times; and (

**b**) depths.

**Figure 20.**Variation in the bottom hole pressure at different pressure differences at different: (

**a**) times; and (

**b**) depths.

Well Depth (m) | Outer Tubing Diameter (m) | Casing Diameter (m) |
---|---|---|

0–500 | 0.089 | 0.2523 |

500–2800 | 0.089 | 0.15708 |

2800–3600 | 0.127 | 0.1469 |

Parameter | Unit | Value | Parameter | Unit | Value |
---|---|---|---|---|---|

Liquid density | kg/m^{3} | 1100 | Surface temperature | K | 293.15 |

Back pressure | Pa | 101325 | Temperature gradient | K/(100 m) | 2.2 |

Liquid viscosity | Pa·s | 0.02 | Surface tension | N/m | 0.0761 |

Gas viscosity | Pa·s | 0.0000015 | Sound velocity in liquid | m/s | 1200 |

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**MDPI and ACS Style**

Fang, Q.; Meng, Y.; Wei, N.; Xu, C.; Li, G.
A Hydraulic Model for Multiphase Flow Based on the Drift Flux Model in Managed Pressure Drilling. *Energies* **2019**, *12*, 3930.
https://doi.org/10.3390/en12203930

**AMA Style**

Fang Q, Meng Y, Wei N, Xu C, Li G.
A Hydraulic Model for Multiphase Flow Based on the Drift Flux Model in Managed Pressure Drilling. *Energies*. 2019; 12(20):3930.
https://doi.org/10.3390/en12203930

**Chicago/Turabian Style**

Fang, Qiang, Yingfeng Meng, Na Wei, Chaoyang Xu, and Gao Li.
2019. "A Hydraulic Model for Multiphase Flow Based on the Drift Flux Model in Managed Pressure Drilling" *Energies* 12, no. 20: 3930.
https://doi.org/10.3390/en12203930