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. 2025 Mar 7;10(10):10302–10307. doi: 10.1021/acsomega.4c09926

Joule–Thomson Effect on Bottom Hole Temperature in Ultra-High-Temperature and High-Pressure Gas Wells

Boyuan Yang †, Hui Zhang †,*, Baokang Wu †, Kunhong Lv †, Yuting Zhou †, Xingyu Li †, Ze Yang †, Rui Yuan ‡
PMCID: PMC11923656  PMID: 40124027

Abstract

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Accurate calculation of temperature and pressure in the borehole of ultrahigh-temperature and high-pressure wells is essential for casing stress analysis. Typically, formation temperature is assumed to be equal to bottom-hole temperature, but this assumption deviates from actual conditions, affecting the accuracy of casing stress calculations. In ultrahigh-temperature and high-pressure environments, the Joule–Thomson effect can cause the bottom-hole temperature to exceed the formation temperature, further impacting temperature calculations throughout the wellbore. This study employs the BWRS equation of state to model the Joule–Thomson effect in ultrahigh-temperature and high-pressure gas wells. A predictive model for bottom-hole temperature in such wells is developed. The influences of production rate, wellbore diameter, and natural gas relative density on bottom-hole temperature are also analyzed. Results show that in these conditions, the bottom-hole temperature decreases as the perforation hole diameter, production rate, and relative density of natural gas increase, due to the Joule–Thomson effect.

1. Introduction

In response to the growing energy demands and technological advancements resulting from rapid societal development, natural gas exploration and development have progressively advanced into the deeper regions of the Earth. Deep and ultradeep wells are subjected to extreme downhole conditions characterized by high temperatures and high pressures. These conditions significantly influence the thermodynamic properties of natural gases. Currently, theoretical models for calculating temperature and pressure in gas wellbores typically assume that the formation temperature equals the bottom-hole temperature. However, this assumption deviates from the actual bottom-hole temperature. As a consequence, making accurate temperature and pressure calculations essential for assessing the mechanical behavior of tubing strings in ultrahigh temperature and high-pressure gas wells.

Current theoretical models for calculating temperature and pressure in gas wellbores typically assume that the formation temperature is equal to the bottom-hole temperature. However, this assumption does not accurately reflect the actual conditions. In high-temperature and high-pressure environments, the formation, perforation channels, and wellbore act as a throttle valve, causing a significant Joule–Thomson effect that can alter the bottom-hole fluid temperature. Specifically, under high-pressure conditions, the Joule–Thomson effect causes the bottom-hole fluid temperature to exceed the formation temperature, complicating the calculation of accurate bottom-hole temperatures. The Joule–Thomson effect refers to the phenomenon where a high-pressure fluid undergoes a temperature change due to pressure variation after passing through a throttling expansion. The effect can either be positive, where the temperature decreases as the pressure decreases, or negative, where the temperature increases under the same conditions. While the positive Joule–Thomson effect has been extensively studied, the negative Joule–Thomson effect of natural gas under ultrahigh temperature and high-pressure conditions remains relatively underexplored. Kortekaas1 studied the behavior of hydrocarbons under high-pressure conditions and found that the Joule–Thomson effect may cause heating during the expansion of high-temperature, high-pressure gas condensates. Baker2 documented a high-pressure natural gas drill pipe test conducted in the North Sea, where a temperature increase of nearly 15 °C was observed when the pressure dropped to 48.2 MPa. X. Wang3 derived a calculation model for the Joule–Thomson coefficient based on Maxwell’s equations, indicating that a positive Joule–Thomson effect occurs in a single gas phase at pressures below 41 MPa, while a negative Joule–Thomson effect is observed when the pressure rises to 55 MPa. Alireza Hosseini4 and Wei-Min Yuan5 used the PR and SRK equations of state to calculate the Joule–Thomson coefficient and inversion curves of natural gas at different temperatures, providing expressions for both single fluids and mixtures. I.D. Pinzon6 employed a DTS fiber optic temperature sensing system to measure bottomhole temperature variations caused by the Joule–Thomson effect, observing fluid heating due to pressure drop near the wellbore under high-pressure conditions. Yu-Guo Wu,7 Zheng-Yuan Dong,8 and Yang Cheng9 derived the BWRS model, verified its accuracy, and calculated the Joule–Thomson coefficient at various temperatures and pressures. N. Tarom,10 Li-Na Hong,11 Fan Yang,12 and Xiang-Ming Feng13 utilized the PR equation of state to calculate the Joule–Thomson coefficient of gases under different temperature and pressure conditions, comparing their results with the PR model and HYSYS. Feng Liu14 and Li-Hui Zheng15 designed an experimental apparatus for measuring the Joule–Thomson coefficient and conducted theoretical simulation studies using HYSYS, identifying various influencing parameters and factors for the Joule–Thomson coefficient. Gang Ai,16,17 Liupeng Huo,18 and Lou Cheng19 conducted experimental and theoretical studies on the Joule–Thomson effect of high-pressure gases during depressurization under various conditions, focusing on temperature–pressure relationships. Chengxiang Deng20 calculated the corresponding Joule–Thomson coefficients and compared them with experimental data from the literature to select the optimal equation of state for generating JT coefficient curves and Joule–Thomson inversion curves for different gases. Datong Zhang21 and Huan Chen22 investigated the mechanism of the throttling effect of CO2 from a molecular perspective, studying the Joule–Thomson coefficient associated with phase changes of CO2. Zhongxi Zhu23 analyzed the occurrence of the Joule–Thomson effect at the drill bit nozzle during the gas well drilling process.

This study seeks to fill this research gap by systematically investigating the negative Joule–Thomson effect in ultrahigh temperature and high-pressure natural gas wells. By employing the BWRS equation of state, we model the thermodynamic behavior of natural gas and develop a predictive model for bottom-hole temperature that incorporates the Joule–Thomson effect. The proposed model’s accuracy is validated through comparison with field-measured data. The results provide significant insights into the influence of the Joule–Thomson effect on temperature variations in ultrahigh temperature and high-pressure pressure gas wells, contributing a theoretical foundation for improved bottom-hole temperature prediction. Additionally, this work offers practical implications for tubing string optimization and safe operational practices in high-temperature, high-pressure gas wells.

2. Bottom-Hole Temperature Calculation Model

2.1. Calculation of the Joule–Thomson Coefficient

The Joule–Thomson coefficient is a parameter that quantifies the temperature change of a gas per unit pressure variation during an isenthalpic process.

2.1. 1

In the formula: μJT is the Joule–Thomson coefficient, K/Pa; T is the gas temperature, K; P is the gas pressure, Pa; H is the gas enthalpy, J/mol.

2.1. 2

In the formula: Cp is the specific heat capacity of the gas at constant pressure, J/(mol·K); V is the gas volume, m3.

2.1. 3

In the formula: ρ is the molar density of the gas, mol/m3.

As shown in eq 3, obtaining the Joule–Thomson coefficient, the relationship between gas pressure, temperature, and density must be understood. The partial derivatives involved can be calculated using the real gas equation of state. Upon review, it was found that the Benedict–Webb–Rubin–Starling (BWRS) equation of state provides the highest accuracy for calculating gas properties under high-pressure conditions. Therefore, this study employs the BWRS equation of state to determine the Joule–Thomson coefficient.

The BWRS equation of state is a multiparameter equation, and its fundamental form is expressed as follows:

2.1. 4

In the formula: R is the universal gas constant, with a value of 8.314 J/(mol·K). The formulas for calculating the 11 parameters (A0, B0, C0, D0, E0, a, b, c, d, α, γ) in the equation can be obtained from ref (24), and the critical parameters for natural gas are shown in Table 1.

Table 1. Critical Physical Parameters of Natural Gas.

substance critical temperature, K critical pressure, MPa critical density, kmol/m3 acentric factor molecular weight
methane 190.69 4.604 10.05 0.013 16.04
ethane 305.38 4.88 6.75 0.101 30.06
propane 369.89 4.25 4.99 0.157 44.09
isobutane 408.13 3.648 3.80 0.183 58.12
n-butane 425.18 3.797 3.92 0.197 58.12
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After computing the critical parameters, a total of 11 variables were derived. The original BWRS equation was then subjected to rigorous partial differentiation, resulting in a refined partial differential expression for the accurate calculation of the Joule–Thomson coefficient.

2.1. 5
2.1. 6

By simultaneously solving eqs 3, 5, and 6, the Joule–Thomson coefficient based on the BWRS equation of state can be derived:

2.1. 7

Where:

2.1. 8

2.2. Bottom-Hole Temperature Calculation Model Considering Perforation Parameters

If the process of high-pressure gas entering the wellbore through the perforation channels is regarded as a throttling process, due to the short length of the channels and the high gas flow velocity at the bottom of the well, the gas flow within the perforation channels can be considered adiabatic. The throttling mechanism is similar to that of surface throttling, except that this process is transferred from the surface to the interior of the wellbore, as shown in Figure 1.

Figure 1.

Figure 1

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Schematic diagram of gas flow through perforation channels.

The mass flow rate of gas at the perforation channel outlet is given by

2.2. 9

In the formula: A1 is the cross-sectional area at the perforation channel inlet, m2; υ1 is the velocity at the inlet end, m/s; ρ1 is the gas density in the formation, kg/m3; A2 is the cross-sectional area at the perforation channel outlet, m2; υ2 is the velocity at the outlet end, m/s; ρ2 is the gas density at the wellbore, kg/m3.

When the gas reaches a steady state, due to the adiabatic process during its flow through the perforation channel, ensures no heat exchange with the surroundings, and the heat generated by friction is negligible. At this point, only the effect of the change in the gas’s kinetic energy needs to be considered. The energy equation can be expressed in its integral form as follows:

2.2. 10
2.2. 11

In the formula: K is the adiabatic index of natural gas.

By combining eqs 10 and 11, the velocity of the gas in the formation can be neglected as it is significantly lower than velocity of the fluid within the wellbore. Consequently, the equation can be simplified as follows:

2.2. 12

By simultaneously solving eqs 9 and 12, and assuming a flow coefficient of 0.865, the result is obtained as follows:

2.2. 13

In the formula: Qsc is the volumetric flow rate through the perforation channel under standard conditions, m3/d; dv is the perforation channel diameter, mm; γg is the relative density of natural gas; T1 is the temperature at the inlet of the perforation channel, °C; Z1 is the gas compressibility factor at the inlet conditions.

Based on eq 13, the bottomhole pressure can be determined. Furthermore, by applying the definition of the Joule–Thomson effect, the bottomhole temperature variation due to the Joule–Thomson effect can be derived as follows:

2.2. 14

In the formula: T2 is the temperature at the outlet of the perforation channel, °C.

3. Model Validation

3.1. Reliability Analysis of the Model

To validate the accuracy of the BWRS equation of state in calculating the Joule–Thomson coefficient, this study conducted calculations for methane gas under various temperature and pressure conditions. The results of the calculations are presented in Figure 2.

Figure 2.

Figure 2

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Joule–Thomson coefficient of methane at various temperature and pressure conditions.

A comparative analysis was conducted between the model-calculated Joule–Thomson coefficients of methane at different temperatures and pressures and the experimental measurements. The comparison results are shown in Table 2.

Table 2. Comparison between Calculated and Measured Joule–Thomson Coefficients.

pressure/MPa method temperature/°C
    0 25 50 75 100
0.098 measurement results 4.89 4.18 3.57 3.06 2.65
model results 4.99 4.39 3.89 3.47 3.11
0.51 measurement results 4.79 4.08 3.47 3.06 2.65
model results 4.5 3.95 3.49 3.11 2.78
2.53 measurement results 4.38 3.67 3.16 2.65 2.35
model results 4.13 3.57 3.12 2.75 2.44
5.05 measurement results 3.87 3.37 2.86 2.45 2.14
model results 3.93 3.35 2.9 2.53 2.22
10.1 measurement results 3.26 2.75 2.55 2.14 1.94
model results 3.09 2.74 2.39 2.08 1.82
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The data comparison reveals that the average relative error between the experimental measurements and the calculated values is 4.47%, demonstrating the high accuracy of the model.

3.2. Field Case Calculation

Using the high-temperature, high-pressure X-1 well in the Sichuan Basin as a case study, with a vertical depth of 7271 m,25 the formation temperature and pressure are recorded as 155 °C and 98 MPa, respectively. The bottomhole pressure, measured directly, is 68 MPa, while the observed bottomhole temperature is 164.8 °C. The model’s prediction for the bottomhole temperature is 162.5 °C, yielding a relative error of 1.39%. The comparative analysis is shown in Figure 3.

Figure 3.

Figure 3

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Impact of the Joule–Thomson effect on bottomhole temperature in well X-1.

4. Results and Discussion

4.1. Effect of Perforation Diameter on Bottomhole Temperature

Figure 4 illustrates the relationship between bottomhole temperature and perforation diameter in a high-temperature, high-pressure gas well, under conditions of a production rate of 600,000 m3/day and a gas relative density of 0.7, and a formation temperature of 198 °C. As shown in the figure, the bottomhole temperature gradually decreases with the increase in perforation diameter. When the diameter is 7 mm, the bottomhole temperature is approximately 205.1 °C, while an increase in diameter to 9 mm results in a temperature drop to 203.5 °C. This indicates that the increase in perforation diameter leads to significant reduction in bottomhole temperature. The reason for this is that, as the perforation diameter increases, the gas flow channel widens, reducing flow friction. Consequently, due to the Joule–Thomson effect, the bottomhole temperature decreases.

Figure 4.

Figure 4

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Influence of perforation diameter on bottomhole temperature.

4.2. Effect of Production Rate on Bottomhole Temperature

Figure 5 illustrates the relationship between bottomhole temperature and production rate in a high-temperature, high-pressure gas well, under conditions of a perforation diameter of 7 mm, gas relative density of 0.7, and a formation temperature of 198 °C. The figure shows that the bottomhole temperature gradually decreases as the production rate increases. When the production rate is 600,000 m3/day, the bottomhole temperature is approximately 205.1 °C, whereas an increase in the production rate to 800,000 m3/day results in a temperature drop to 204.4 °C. This indicates that a higher production rate significantly reduces the bottomhole temperature. The underlying mechanism is that, as the production rate increases, both gas velocity and pressure drop increase, which, due to the Joule–Thomson effect, leads to a decrease in bottomhole temperature.

Figure 5.

Figure 5

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Impact of production rate on bottomhole temperature.

4.3. Effect of Gas Relative Density on Bottomhole Temperature

Figure 6 illustrates the relationship between bottomhole temperature and gas relative density in a high-temperature, high-pressure gas well, under conditions of a perforation diameter of 7 mm, a production rate of 600,000 m3/day, and a formation temperature of 198 °C. The figure shows that the bottomhole temperature decreases gradually as the gas relative density increases. When the gas relative density is 0.74, the bottomhole temperature is approximately 204.7 °C, whereas an increase in gas relative density to 0.82 results in a temperature drop to 204.2 °C. This demonstrates that an increase in gas relative density significantly reduces the bottomhole temperature. The reason is that as the gas relative density increases, the heat capacity of the gas also increases, and due to the Joule–Thomson effect, this leads to a reduction in bottomhole temperature.

Figure 6.

Figure 6

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Effect of gas relative density on bottomhole temperature.

5. Conclusions

  • 1.

    Based on the BWRS equation of state, a bottom-hole temperature prediction model for ultrahigh temperature and high-pressure gas wells, incorporating the Joule–Thomson effect, was established. This model provides a more accurate estimation of the bottom-hole temperature, aligning closely with field observations. The Joule–Thomson effect, by altering the temperature profile due to pressure variations during throttling, plays a significant role in adjusting the predicted temperature from the traditional assumption of equal formation and bottom-hole temperatures.

  • 2.

    The effects of perforation diameter, production rate, and gas relative density on bottom-hole temperature were analyzed. The results show that, due to the Joule–Thomson effect, an increase in the perforation diameter, production rate, or gas relative density leads to a decrease in the bottom-hole temperature. This is attributed to the cooling effect caused by the expansion of high-pressure gas through the perforations, which is amplified under ultrahigh temperature and high-pressure conditions.

  • 3.

    For ultrahigh temperature and high pressure gas wells, considering the Joule–Thomson effect can accurately predict the bottom hole temperature, thereby more accurately predicting the temperature and pressure distribution of the entire wellbore. This provides a theoretical basis for optimizing the tubing design strategy and ensuring the integrity and safety of the wellbore under extreme downhole conditions.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant No. 52174010, Grant No. 52227804).

The authors declare no competing financial interest.

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