# A New Model of Bubble Migration Velocity in Deep Water Wellbore Considering Hydrate Phase Transition

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## Abstract

**:**

## 1. Introduction

## 2. Model Description

^{3}. ${\tau}_{y}$ is yield stress, Pa. $g$ is the acceleration of gravity, m/s

^{2}. ${V}_{b}$ is bubble volume, m

^{3}. ${R}_{\mathrm{max}}$ is the maximum width of the bubble in the horizontal direction, m.

^{3}. ${\rho}_{h}$ is the density of hydrate shell, kg/m

^{3}. $\delta $ represents the thickness of hydrate shell, m. ${F}_{v}$ is the buoyancy force of the bubble, N.

## 3. Determination of Key Model Parameters

#### 3.1. Dynamic Growth Thickness of Hydrate Shell

- (1)
- Hydrate shell is a kind of plastic-like material, which has a certain ability to resist damage [26]. Therefore, the mechanical failure of hydrate shell during bubble migration is not considered;
- (2)
- The hydrate shell has a porous medium-like structure, consisting of hydrate crystals and microscopic pore throats.

^{3}. ${f}_{w}$ is the permeation rate of water through hydrate shell, m

^{3}/s.

^{3}, which depends on the number and size of micro-cracks in the hydrate shell:

^{2}. $s$ is the tortuosity of the micro-cracks, dimensionless.

^{3}/s. ${f}_{d}$ is the dissolution rate of gas in the liquid phase, m

^{3}/s.

^{3}/m

^{3}. ${d}_{h}$ is the gas diffusion coefficient inside the hydrate shell, m

^{2}/s. The diffusion coefficient ${d}_{h}$ inside the hydrate shell reflects the diffusivity of the gas along the concentration gradient. Ogasawara et al. [28] obtained that the gas diffusion coefficient inside the hydrate is about 10

^{−11}~10

^{−12}m

^{2}/s on the basis of experimental research.

^{3}/m

^{3}. ${C}_{o}$ is the gas concentration outside the hydrate shell, m

^{3}/m

^{3}.

^{3}/m

^{3}. ${C}_{l}$ is the gas concentration in the liquid phase, m

^{3}/m

^{3}. ${k}_{hl}$ is the mass transfer coefficient between hydrate shell and the liquid phase, m/s. The mass transfer coefficient determines the dissolution rate of gas in the liquid phase. Rehder et al. [30] showed that the existence of hydrate shells would increase the mass transfer resistance of gas dissolution and reduce the mass transfer coefficient. The formation of the hydrate shell leads to the transition from the original mass transfer across the gas–liquid interface to the mass transfer across the solid hydrate shell, and the mass transfer coefficient of gas in the solid phase is significantly lower than that in the liquid and gas phases. In order to accurately characterize the effect of the dissolution rate of gas mass transfer across the hydrate shell on the thickness of the hydrate shell, error analysis and optimization of the typical mass transfer model of bubble dissolution in Table 1 should be carried out by using experimental data. Rehder et al. [30] experimentally studied the variation law of the equivalent radius of methane bubbles released in the water depth range from 400 m to 1500 m with the bubble migration time. The experimental data were collected and recorded by ROV (Remote Operated Vehicle) observation. The data are of high practical value and have a strong reference significance for the establishment and verification of the bubble migration model. Therefore, in this study, error analysis and optimization of the typical mass transfer model of bubble dissolution were carried out based on the experimental data of bubble equivalent radius variation with migration time in the hydrate shell formation segment obtained by Rehder et al., which released methane bubbles at a depth of 1098.0 m.

#### 3.2. Bubble Dynamic Equivalent Radius

^{3}. ${m}_{l}$ is the gas mass concentration in the drilling fluid, kg/m

^{3}.

^{2}/s.

#### 3.3. Drag Coefficient during Bubble Migration

#### 3.4. Model Verification

## 4. Discussion

#### 4.1. Influence of Initial Bubble Size

^{3}.

#### 4.2. Influence of Annular Fluid Viscosity

^{3}.

#### 4.3. Influence of Annulus Fluid Density

^{3}, 1250 kg/m

^{3}, and 1300 kg/m

^{3}. The influence of annulus fluid density on the migration velocity of hydrated bubbles and clean bubbles is consistent. With the increase of the bottom hole fluid density, the migration velocity of hydrated bubbles and clean bubbles increases slightly. When the fluid density is 1200 kg/m

^{3}, the migration velocity of hydrated bubbles and clean bubbles to the wellhead is 0.036 m/s and 0.052 m/s, respectively. While when the fluid density is 1300 kg/m

^{3}, the migration velocity of hydrated bubbles and clean bubbles to the wellhead is 0.038 m/s and 0.055 m/s, respectively. This indicates that the effect of fluid density on bubble migration velocity is not significant. This is consistent with the variation of the bubble migration cycle in fluid with different densities in Figure 14. When the bottom hole fluid density is 1200 kg/m

^{3}and 1300 kg/m

^{3}, the migration cycle of hydrated bubbles is about 2.0 h and 1.9 h longer than that of clean bubbles, respectively. There is little difference in safe shut-in cycle under different annulus fluid densities. The above analysis shows that the migration velocity of hydrated bubbles and clean bubbles is not sensitive to the change of annular fluid density.

## 5. Conclusions

- (1)
- The migration velocity of hydrated bubbles is divided into a gradually decreasing stage and a slowly increasing stage. The gas consumption and the thickening of hydrate shell in the gradually decreasing stage play a dominant role, and the increase of bubble volume caused by the decrease of pressure in the slowly increasing stage is the most important factor;
- (2)
- The formation of a hydrated bubble can significantly reduce the migration velocity of bubble and effectively prolong the safe shut-in period. The migration cycle of the hydrated bubble can be significantly increased by decreasing bubble size and increasing annular fluid viscosity;
- (3)
- The initial size of the bubble and the viscosity of annulus fluid are the main factors affecting the migration velocity of the bubble, while the density of annulus fluid has little effect on the migration velocity of hydrated bubbles and clean bubbles.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Error analysis of new mass transfer coefficient model for bubble dissolution based on Rehder et al. [30] experimental data.

**Figure 6.**Comparison between the calculated results of hydrated bubble migration drag coefficient model and experimental data.

**Figure 11.**Variation of bubble migration velocity with well depth under different annular fluid viscosity.

**Figure 13.**Bubble migration velocity changes with well depth under different fluid densities in deepwater wellbore annulus.

Researchers | Mass Transfer Coefficient Model |
---|---|

Clift [31] | ${k}_{hl}=\frac{E}{2{r}_{g}}+\frac{E}{2{r}_{g}}{\left(1+\frac{2{r}_{g}{v}_{g}}{E}\right)}^{1/3}$ |

Levich [32] | ${k}_{hl}=0.624{v}_{g}{}^{1/3}{\left(E/{r}_{g}\right)}^{2/3}$ |

Oellrich [33] | ${k}_{hl}=\frac{E}{{r}_{g}}+\frac{0.651{\left(2{r}_{g}{v}_{g}/E\right)}^{1.72}}{1+{\left(2{r}_{g}{v}_{g}/E\right)}^{1.22}}\frac{E}{2{r}_{g}}$ |

Leclair [34] | ${k}_{hl}=\left(0.65+0.06{\mathrm{Re}}^{1/2}\right){\left(\frac{E{v}_{g}}{2{r}_{g}}\right)}^{1/2}$ |

Johnson [35] | ${k}_{hl}=1.13\sqrt{\frac{E{v}_{g}}{0.45+40{r}_{g}}}$ |

Winnikow [36] | ${k}_{hl}=\frac{2}{\sqrt{\pi}}\left(1-\frac{2.89}{{\mathrm{Re}}^{1/2}}\right){\left(\frac{E{v}_{g}}{2{r}_{g}}\right)}^{1/2}$ |

Acrivos [37] | ${k}_{hl}=0.624{v}_{g}{}^{1/3}{\left(E/{r}_{g}\right)}^{2/3}+0.46E/{r}_{g}$ |

Researchers | Drag Coefficient Model |
---|---|

Mei [40] | ${C}_{D}=\frac{16}{\mathrm{Re}}\left\{1+{\left[\frac{8}{\mathrm{Re}}+\frac{1}{2}\left(1+3.315{\mathrm{Re}}^{-0.5}\right)\right]}^{-1}\right\}$ |

Bigalke [13] | ${C}_{D}=f{\left(\frac{a}{R}\right)}^{2}$ |

Peebles [9] | ${C}_{D}=\mathrm{max}\left\{\mathrm{max}\left[\frac{24}{\mathrm{Re}},\frac{18}{{\mathrm{Re}}^{0.68}}\right],\mathrm{min}\left[0.0275Eo\cdot W{e}^{2},0.82E{o}^{0.25}\cdot W{e}^{0.5}\right]\right\}$ |

Turton [41] | ${C}_{D}=\frac{24}{\mathrm{Re}}\left(1+10.173{\mathrm{Re}}^{0.657}\right)+\frac{0.413}{1+16300{\mathrm{Re}}^{-1.09}}$ |

Tomiyama [42] | ${C}_{D}=\mathrm{max}\left\{\frac{24}{\mathrm{Re}}\left(1+0.15{\mathrm{Re}}^{0.687}\right),\frac{8}{3}\frac{Eo}{Eo+4}\right\}$ |

Wallis [43] | ${C}_{D}=\mathrm{max}\left\{\mathrm{min}\left[\mathrm{max}\left(\frac{16}{\mathrm{Re}},\frac{13.6}{{\mathrm{Re}}^{0.8}}\right),\frac{48}{\mathrm{Re}}\right],\mathrm{min}\left[\frac{Eo}{3},0.47E{o}^{0.25}W{e}^{0.5},0.8\right]\right\}$ |

Bozzano [44] | ${C}_{D}=F{\left(\frac{A}{Ro}\right)}^{2}$ |

Ishii [45] | ${C}_{D}=\mathrm{max}\left\{\frac{24}{\mathrm{Re}}\left(1+0.1{\mathrm{Re}}^{0.75}\right),\mathrm{min}\left[\frac{8}{3},\frac{2}{3}\sqrt{Eo}\right]\right\}$ |

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

Depth of reservoir $H$ | 4700 m | Bottom hole liquid density ${\rho}_{l}$ | 1.2~1.3 g/cm^{3} |

Depth of water ${H}_{w}$ | 1500 m | Initial bubble diameter ${d}_{e}$ [14] | 2~6 mm |

Design well depth ${H}_{d}$ | 4850 m | Bottom hole liquid viscosity $\mu $ | 10~30 mPa·s |

Hydrate density ${\rho}_{h}$ | 910 kg/m^{3} | Gas-liquid interfacial tension $\sigma $ [15] | 0.0194 N/m |

Contact angle $\beta $ [15] | 0° | Sea surface temperature ${T}_{s}$ | 28 °C |

Molar mass of water ${M}_{w}$ | 18 g/mol | Diffusion coefficient ${d}_{h}$ [29] | 10^{−11} m^{2}/s |

Mudline temperature ${T}_{m}$ | 3.4 °C | Bassett force coefficient ${K}_{b}$ [21] | 6.0 |

Geothermal gradient ${T}_{D}$ | 0.03 °C/m | Coefficient of undercooling $b$ | $3.1205\times {10}^{-5}$ m·K |

Micropore radius ${r}_{c}$ [15] | 0.05 μm | Molar mass of methane ${M}_{g}$ | 16 g/mol |

Hydration number $n$ | 6.0 | Dissolved methane concentration ${C}_{m}$ | 0.1 nmol/L |

Micropore tortuosity $s$ [15] | 2.0 | Number of pores per unit area ${n}_{d}$ [14] | $1\times {10}^{12}$1/m^{2} |

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**MDPI and ACS Style**

Zhao, X.; Yin, F.; Yao, H.; Qi, Y.; Cao, X.
A New Model of Bubble Migration Velocity in Deep Water Wellbore Considering Hydrate Phase Transition. *J. Mar. Sci. Eng.* **2023**, *11*, 2206.
https://doi.org/10.3390/jmse11112206

**AMA Style**

Zhao X, Yin F, Yao H, Qi Y, Cao X.
A New Model of Bubble Migration Velocity in Deep Water Wellbore Considering Hydrate Phase Transition. *Journal of Marine Science and Engineering*. 2023; 11(11):2206.
https://doi.org/10.3390/jmse11112206

**Chicago/Turabian Style**

Zhao, Xinxin, Faling Yin, Haiyuan Yao, Yaqiang Qi, and Xin Cao.
2023. "A New Model of Bubble Migration Velocity in Deep Water Wellbore Considering Hydrate Phase Transition" *Journal of Marine Science and Engineering* 11, no. 11: 2206.
https://doi.org/10.3390/jmse11112206