# A Predictive Model for Wellbore Temperature in High-Sulfur Gas Wells Incorporating Sulfur Deposition

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. PPP Models

#### 2.1.1. Deviation Coefficient

_{2}and H

_{2}S components in acidic gas influences the critical temperature and pressure of natural gas, resulting in an elevation of the gas deviation coefficient Z value and subsequent calculation inaccuracies. As such, it is imperative to adjust the critical parameter properties for acidic natural gas. This study employs the Wichert–Aziz correction method [35] for this purpose.

#### 2.1.2. Viscosity

_{2}S. Therefore, it is essential to adjust for non-hydrocarbon effects when predicting the viscosity of HSG. The empirical method is used [37]:

#### 2.1.3. Thermal Conductivity Coefficient

#### 2.2. WTD Model

#### 2.2.1. Equation of Heat Conduction

#### 2.2.2. Transient Temperature Model

- (1)
- The transmission of heat from high-temperature gas to the sulfur layer occurs through thermal convection.
- (2)
- Heat transfer in the sulfur layer, tubing, casing, and cement takes place through heat conduction.
- (3)
- The heat transfer in the annulus is characterized by convection and radiation.

#### 2.3. Wellbore Pressure Model

#### 2.4. Sulfur Solubility Model

#### 2.5. Initial Conditions and Boundary Conditions

#### 2.5.1. Initial Conditions

#### 2.5.2. Boundary Conditions

#### 2.6. Numerical Methods

#### 2.7. The Calculation Process

- (1)
- Initiate the program and input the initial parameters, which encompass well structure data, production performance data, physical property data, and the heat transfer coefficient.
- (2)
- Spatial and temporal discretization of the simulation domain is determined based on the depth and radial orientation of the well.
- (3)
- Calculate the initial temperature, pressure, and PPPs according to Section 2.1, Section 2.4 and Section 2.5.
- (4)
- At one time step, the pressure field distribution is first calculated. The pressure is calculated in stages from the wellhead to the bottom:
- Based on the findings outlined in Section 2.4, it is imperative to ascertain the attainment of critical solubility and the subsequent precipitation of sulfur. Should sulfur precipitation occur, the utilization of a multi-phase pressure calculation model is warranted; conversely, if sulfur does not precipitate, the adoption of a single-phase pressure calculation model is recommended.
- Determine the pressure values P
_{i}and P_{i}_{+1}from Section 2.3. Calculate the PPPs and pressure value P’_{i}_{+1}as outlined in Section 2.1, and compare P_{i}_{+1}with P’_{i}_{+1}to assess compliance with the accuracy requirements. If met, proceed to the subsequent step; if not, substitute the value of P_{i}_{+1}with P’_{i}_{+1}and iterate the aforementioned process until the accuracy requirements are satisfied. - When the pressure of the i grid reaches the calculated accuracy, the grid moves down, and the pressure in the next grid is calculated until that of the bottom of the hole is calculated.

- (5)
- The temperature is calculated from the bottom to the wellhead:
- Determine the temperature of the wellbore in grid i using the methodology outlined in Section 2.3. The precision of this calculation is similar to that of the pressure calculation. Following the completion of the calculation, assess the temperature distribution radially from the tubing to the reservoir.
- As the pressure in the lower part reaches the predetermined level of accuracy, advance the grid upwards and compute the temperature in the subsequent grid until reaching the wellhead.

- (6)
- The time step calculation method utilized in steps (4) through (5) should be iteratively applied until all time steps have been computed.
- (7)
- Save the data and output the calculation results.

## 3. Validation

^{3}. The operators performed well test operations to analyze the wellbore pressure, temperature distribution, reservoir seepage, and productivity characteristics, thereby establishing a technical foundation for the scientific and rational development of the well. The testing procedure is automated using an electronic pressure gauge and is configured to collect data. The experimental test yielded temperature and pressure data, which are presented in Table 3 and Figure 5. The pressure exhibits a linear increase, while the temperature demonstrates rapid changes in the upper region and slower changes in the lower region.

^{3}under the prevailing wellhead pressure and temperature conditions. The declining wellhead pressure in well X prompted the disassembly of the well valve to investigate the potential presence of sulfur deposition. Subsequent experimental analysis revealed that 84.5% of the precipitated substances consisted of sulfur deposits (Table 4). A significant quantity of solid sediment adheres to the inner wall of the import and export pipe, measuring approximately 3–5 mm in thickness and possessing a hardened consistency (Figure 8). Sulfur is consistently adsorbed and accumulated on the pipe wall, transitioning from dispersion to concentration, and subsequently forms a consolidated and thickened layer in a sequential manner from the interior to the exterior. This overall process can be categorized into two distinct stages: the initial phase involving the aggregation and adsorption of individual particles, followed by the subsequent stage of deposition, consolidation, and thickening.

## 4. Discussion

#### 4.1. Gas Productivity

^{4}m

^{3}/d, 90 × 10

^{4}m

^{3}/d, and 120 × 10

^{4}m

^{3}/d. The results of these calculations are presented in Figure 10 and Figure 11. The highest recorded wellhead temperature was 49.6 °C, while the lowest temperature reached 41.3 °C. As the production levels doubled, there was a corresponding 20.09% increase in the wellhead temperature. The rise in gas production leads to a notable elevation in the wellhead gas temperature. The proximity of gas to the wellhead results in a more pronounced temperature differential among various gas production levels. This phenomenon can be attributed to the heightened heat loss of gas within the tubing, caused by the substantial temperature variance between the gas within the tubing and the reservoir. Additionally, as gas production increases, the fluctuation in the gas density within the tubing remains minimal, while the difference of pressure is not obvious.

#### 4.2. Sulfur Thickness

#### 4.3. HSC

_{2}S content and 14.0 MPa for a 40% H

_{2}S content. Higher HSCs were found to result in increased natural gas density. Consequently, higher HSCs led to more rapid pressure drops, a greater dissolution capacity, and reduced sulfur precipitation. The influence of specific heat capacity on temperature during the precipitation of sulfur particles is comparatively lower, as compared to that of single-phase gas, owing to the reduced solid content.

#### 4.4. Reservoir Pressure

#### 4.5. Reservoir Temperature

## 5. Conclusions

- (1)
- A temperature field prediction model incorporating sulfur deposition was developed, emphasizing variations in the PPPs of hydrogen-sulfide-containing gas and the corresponding modified model. The pressure model employed a GSTP flow approach, while the sulfur solubility prediction model selected a theoretical model tailored to the specific target block.
- (2)
- The accuracy of the calculation model is validated through comparison with field-measured data, revealing an error of 2.53% in the temperature calculations and 4.80% in the pressure calculations. The results demonstrate a high level of agreement, indicating that the model is suitable for predicting wellbore temperatures in HSG wells.
- (3)
- The production of the HSG well has a notable impact on temperature. Specifically, a 20.09% increase in the wellhead temperature is observed when comparing a gas production rate of 120 × 10
^{4}m^{3}/d to that of 60 × 10^{4}m^{3}/d. The thickness of the sulfur scale within the wellbore influences the flow rate, subsequently leading to a decrease in wellhead pressure. - (4)
- An elevation in the hydrogen sulfide concentration correlates with an increase in the density of natural gas, thereby causing a more rapid decline in pressure. This reduction in reservoir pressure and temperature will subsequently lower the wellbore pressure and temperature, impeding the removal of sulfur deposits at the well bottom and accelerating a decrease in the productivity of HSG wells.
- (5)
- This paper primarily examines a prediction model for temperature fields, without delving extensively into the specific locations and mechanisms of sulfur deposition under varying temperature conditions. Future research may explore these aspects further, enabling decision-makers to implement tailored strategies for sulfur deposition removal.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

HSG | high-sulfur gas |

GSTP | gas–solid two-phase |

WTD | wellbore temperature distribution |

PPP | physical property parameter |

HSC | hydrogen sulfide content |

Z | Deviation coefficient |

${P}_{r}$ | Pseudo pressure |

${T}_{r}$ | Pseudo pressure |

$\epsilon $ | Correction coefficient |

M | The sum of the mole fractions of H_{2}S and CO_{2} in the system |

N | The mole fraction of H_{2}S in the system |

${T}_{ci}$ | Critical temperature, K |

${p}_{ci}$ | Critical pressure, Pa |

${T}_{ci}^{\prime}$ | Corrected critical temperature, K |

${p}_{ci}^{\prime}$ | Corrected critical pressure, Pa |

${\gamma}_{g}$ | Relative density of natural gas |

${\mu}_{{H}_{2}S}$,${\mu}_{C{O}_{2}}$${\mu}_{{N}_{2}}$ | Viscosity correction values for H_{2}S, CO_{2}, and N_{2}, respectively, mPa·s |

H_{2}S, CO_{2}, N_{2} | Molar content in the gas mixture, % |

$\lambda $ | Thermal conductivity coefficient, W/(m·K) |

T | Temperature, K |

r | Distance from the center of the tubing, m |

$\rho \mathrm{c}$ | Heat capacity per unit volume, J/(m^{3}·K) |

t | Time, s |

$\lambda $ | Heat transfer coefficient, w/(m·K) |

${h}_{c}$ | Heat transfer coefficients of heat convection, w/(m·K) |

${h}_{r}$ | Heat transfer coefficients of heat radiation, w/(m·K) |

$\alpha $ | Content of sulfur particles |

${\rho}_{\mathrm{s}}$ | Densities of sulfur particles, kg/m^{3} |

${\rho}_{\mathrm{g}}$ | Densities of gas, kg/m^{3} |

${v}_{\mathrm{s}}$ | Velocity of the solid sulfur particle, m/s |

${\mathrm{d}}_{\mathrm{s}}$ | Diameter of the sulfur particle, m |

${v}_{g}$ | Speed of the gas, m/s |

d | Inner diameter of the tubing, m |

${\mu}_{g}$ | Viscosity of the gas, mPa·s |

${T}_{\mathrm{i}}$ | Initial reservoir temperature at depth $z$, K |

${T}_{0}$ | Wellhead temperature, K |

$a$ | Temperature gradient, K/m |

$z$ | Depth, m |

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A_{1} | 0.31506237 | A_{5} | −0.61232032 |

A_{2} | −1.0467099 | A_{6} | −0.10488813 |

A_{3} | −0.57832729 | A_{7} | 0.68157001 |

A_{4} | 0.53530771 | A_{8} | 0.68446549 |

B_{0} | −2.4621182 | B_{6} | 0.36037302 | B_{12} | 0.0839387178 |

B_{1} | 2.97054714 | B_{7} | −0.0104432413 | B_{13} | −0.186408846 |

B_{2} | −0.286264054 | B_{8} | −0.793385684 | B_{14} | 0.0203367881 |

B_{3} | 0.00805420522 | B_{9} | 1.39643306 | B_{15} | −0.000609579263 |

B_{4} | 2.80860949 | B_{10} | −0.149144925 | ||

B_{5} | −3.49803305 | B_{11} | 0.00441015512 |

Depth (m) | True Vertical Depth (m) | Pressure (MPa) | Pressure Gradient (MPa/100 m) | Temperature (°C) | Temperature Gradient (°C/100 m) |
---|---|---|---|---|---|

0.00 | 0.00 | 16.620 | / | 40.78 | / |

1000.00 | 999.99 | 18.584 | 0.196 | 68.06 | 2.73 |

2000.00 | 1999.26 | 20.497 | 0.191 | 78.72 | 1.07 |

3000.00 | 2998.94 | 22.394 | 0.190 | 87.73 | 0.90 |

3100.00 | 3097.32 | 22.580 | 0.189 | 88.37 | 0.65 |

3250.00 | 3229.07 | 21.953 | 0.257 | 89.37 | 0.76 |

Component | Na_{2}O | MgO | Al_{2}O_{3} | SiO_{2} | P_{2}O_{5} | SO_{3} | Cl | K_{2}O |

Mass ratio | 0.0905 | 0.1089 | 0.8055 | 1.6912 | 0.0174 | 3.6514 | 0.0305 | 0.0828 |

Component | CaO | Fe_{2}O_{3} | NiO | CuO | ZnO | SrO | BaO | S |

Mass ratio | 0.1676 | 0.4706 | 0.0087 | 0.0154 | 0.1239 | 0.0202 | 8.0479 | 84.668 |

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## Share and Cite

**MDPI and ACS Style**

Fang, Q.; He, J.; Wang, Y.; Pan, H.; Ren, H.; Liu, H.
A Predictive Model for Wellbore Temperature in High-Sulfur Gas Wells Incorporating Sulfur Deposition. *Processes* **2024**, *12*, 1073.
https://doi.org/10.3390/pr12061073

**AMA Style**

Fang Q, He J, Wang Y, Pan H, Ren H, Liu H.
A Predictive Model for Wellbore Temperature in High-Sulfur Gas Wells Incorporating Sulfur Deposition. *Processes*. 2024; 12(6):1073.
https://doi.org/10.3390/pr12061073

**Chicago/Turabian Style**

Fang, Qiang, Jinghong He, Yang Wang, Hong Pan, Hongming Ren, and Hao Liu.
2024. "A Predictive Model for Wellbore Temperature in High-Sulfur Gas Wells Incorporating Sulfur Deposition" *Processes* 12, no. 6: 1073.
https://doi.org/10.3390/pr12061073