The transient flow testing of ultra-deepwater gas wells is greatly impacted by the low temperatures of seawater encountered over extended distances. This leads to a redistribution of temperature within the wellbore, which in turn influences the flow behavior. To accurately predict such a temperature distribution, in this study a comprehensive model of the flowing temperature and pressure fields is developed. This model is based on principles of fluid mechanics, heat transfer, mass conservation, and energy conservation and relies on the Runge-Kutta method for accurate integration in time of the resulting equations. The analysis includes the examination of the influence of various factors, such as gas flow production rate, thermal diffusivity of the formation, and thermal diffusivity of seawater, on the temperature and pressure profiles of the wellbore. The key findings can be summarized as follows: 1. Higher production rates during testing lead to increased flowing temperatures and decreased pressures within the wellbore. However, in the presence of a seawater thermocline, a crossover in flowing temperature is observed. 2. An increase in wellbore pressure is associated with larger pipe diameters. 3. Greater thermal diffusivity of the formation results in more rapid heat transfer from the wellbore to the formation, which causes lower flowing temperatures within the wellbore. 4. In an isothermal layer, higher thermal diffusivity of seawater leads to increased wellbore flowing temperatures. Conversely, in thermocline and mixed layer segments, lower temperatures are noted. 5. Production test data from a representative deep-water gas well in the South China Sea, used to calculate the bottom-seafloor-wellhead temperature and pressure fields across three operating modes, indicate that the average error in temperature prediction is 2.18%, while the average error in pressure prediction is 5.26%, thereby confirming the reliability of the theoretical model.
Symbol | Symbol Description | Unit |
---|---|---|
Gas density | kg/m^{3} | |
Gas flow velocity inside the wellbore | m/s | |
Enthalpy entering the infinitesimal fluid element per unit time, including internal energy and pressure energy | J/s | |
Enthalpy leaving the infinitesimal fluid element per unit time, including internal energy and pressure energy | J/s | |
Fluid mass flow rate | kg/s | |
Heat transferred into the wellbore from the formation per unit time | J | |
Well deviation angle | ° | |
z | Wellbore depth | m |
R | Gas constant | |
Molar mass of gas | g/mol | |
Fluid temperature inside the wellbore | ° | |
Pressure inside the wellbore | MPa | |
Thermal conductivity of the formation | W/(m∙K) | |
Transient heat transfer time function | ||
T_{wb} | Temperature at the well-formation interface, in degrees Celsius | °C |
Formation temperature | °C | |
Outer radius of the tubing | m | |
Total heat transfer coefficient from the formation to the inner wall of the tubing | J/s∙m^{2}∙K | |
Temperature at the wellbore-wall interface | °C | |
Convective heat transfer coefficient of the fluid inside the tubing | W/(m∙K) | |
Natural convection heat transfer coefficient | W/(m∙K) | |
Radiative heat transfer coefficient | W/(m∙K) | |
Outer radius of the tubing | m | |
Inner radius of the tubing | m | |
Outer radius of the insulating layer | m | |
Outer radius of the casing | m | |
Inner radius of the casing | m | |
Well diameter | m | |
Thermal conductivity of the tubing | W/(m∙K) | |
Thermal conductivity of the insulating layer | W/(m∙K) | |
Thermal conductivity of the casing | W/(m∙K) | |
Thermal conductivity of the mud | W/(m∙K) | |
Specific heat at constant pressure | J/kg∙K | |
Joule-thomson coefficient of the gas | ||
The heat transfer coefficient of the seawater segment | J/s∙m^{2}∙K | |
The thermal conductivity of seawater | W/(m∙K) | |
The radius of the seawater segment wellbore | mm | |
The convective heat transfer coefficient inside the test pipe column | mm | |
The inner diameter of the test pipe casing | mm | |
The outer diameter of the test pipe casing | mm | |
The outer diameter of the annular mud | mm | |
The inner diameter of the annular mud | mm | |
The diameter of the seawater segment wellbore | mm | |
The thermal conductivity of the annular mud | W/(m∙K) | |
The outer diameter of the casing pipe | mm | |
The inner diameter of the casing pipe | mm | |
The thermal conductivity of the casing pipe | W/(m∙K) | |
Represents the coordinate at the mud line wellhead position | ||
Represents the temperature at the mud line wellhead | °C | |
The density of the mixed fluid | kg/m^{3} | |
The density of the water phase | kg/m^{3} | |
The density of the gas phase | kg/m^{3} | |
The acceleration due to gravity | 9.8m/s^{2} | |
Hold up ratio | ||
The frictional stress in Newtons | N | |
The dimensionless friction factor | ||
The mixed flow velocity | m/s | |
The inner diameter of the casing | m | |
The water phase velocity | m/s | |
The gas phase velocity | m/s | |
The water phase mass flow rate | kg/(m^{2}·s) | |
The gas phase mass flow rate | kg/(m^{2}·s) | |
Represents the mass flow rate of the mixture | kg/s | |
Represents the cross-sectional area of the wellbore | m^{2} |
Currently, offshore oil and gas exploration and development is developing from deep water to ultra-deep water (water depth exceeding 1500 m) [
In general, there are three main approaches or methods for calculating wellbore pressure-temperature distributions of the fluid flowing: (1) Separate the pressure appropriately and calculate the flowing temperature distribution of wellbore fluids approximately based on the steady-state heat transfer principle. (2) Calculate the bottom hole pressure by averaging the wellbore temperature either overall or discretely in sections. (3) The calculation involves the coupling of pressure and temperature. Unlike the production flow in onshore oil and gas wells, the temperature gradient in the seawater section of offshore gas wells is opposite to that of the formation. This leads to significant differences in the characteristics of wellbore flowing temperature and pressure profiles. Taking into account the temperature characteristics of the seawater section. Song et al. [
In summary, although numerous scholars have conducted research on prediction models and variations of wellbore flowing temperature fields in onshore gas wells and deepwater gas wells, there is limited research on the prediction and patterns of wellbore flowing temperature fields in ultra-deepwater test wells. This lack of research results in an unclear understanding of the characteristics and patterns of wellbore flowing temperature-pressure field variations during ultra-deepwater gas well production flowing testing processes. Therefore, considering the temperature characteristics of ultra-deepwater environments, a coupled model of wellbore flowing temperature-pressure in ultra-deepwater gas well tests was established based on fluid mechanics theory, heat transfer theory, mass conservation, and energy conservation laws. The model accounts for non-steady-state heat transfer and nonlinear changes in gas high-pressure properties. The fourth-order Runge-Kutta method was used to calculate the temperature and pressure distributions to reveal the influence patterns of relevant parameters on the flowing temperature and pressure profiles. The research findings have significant theoretical implications for the safe and efficient production flowing testing of ultra-deepwater gas fields as well as for ensuring wellbore flow assurance.
Based on the characteristics of temperature variations during ultra-deepwater gas well testing, the entire ultra-deepwater testing system can be divided into two parts: the ultra-deepwater segment and the formation segment. The deepwater segment primarily involves heat transfer issues among seawater, casing, fluid inside the casing, test string, and fluid inside the string; while the second part is the wellbore section below the mudline, namely the formation segment. This mainly deals with heat transfer among the formation, cement sheath, casing, annular fluid, test string, and fluid inside the string. As shown in
Model assumptions:
1). The gas flow inside the test string is assumed to be steady one-way flow.
2). Heat transfer in the wellbore of ultra-deepwater wells is considered to be steady heat transfer.
3). Heat transfer in the formation is assumed to be unsteady heat transfer, following the dimensionless time function recommended by Remay.
4). Oil and casing (casing string) are assumed to be concentric.
Taking the mud line wellhead as the coordinate origin, with the coordinate
The gas flow inside the test string satisfies the conservation of momentum and energy, as shown in
The equation of state for a gas:
By combining
For the energy conservation
The temperature at the well-formation interface and the energy in the wellbore satisfy:
Overall heat transfer coefficient
By combining
The energy
In the fluid energy
Since the gas flows within the test column and the pipe diameter variation is generally small, the Joule-Thomson coefficient is very small and can be neglected. Therefore, we can consider:
Therefore, we have:
The temperature gradient in the wellbore is [
Similarly, based on the formation temperature model, the mathematical model for the seawater segment is established from an energy perspective as follows:
The overall heat transfer coefficient of the seawater segment [
Boundary conditions:
According to the literature, the temperature distribution of the seawater segment can be expressed as [
In
The physical interpretation expression of the pressure gradient in the fluid flow process in the wellbore [
The gravitational potential energy in
In
In
The mixed flow velocity
The expression for the acceleration pressure drop in
For the liquid phase, the compressibility is much smaller compared to the compressibility of the gas phase, which can be neglected, leading to:
Substituting
The
The general form of the fourth-order Runge-Kutta method can be written as
By substituting
If the expected depth is not reached, the calculated value of the node is used as the starting value for the next step, and the above steps are repeated so that the continuous forward calculation is until the expected depth. The solution steps for ultra-deepwater wellbore temperature and pressure field testing are as follows:
(1) Firstly, the temperature and pressure field of the formation section is solved. Taking the mud line wellhead as the starting point, the step size is selected, and the four-order Runge-Kutta method is used to solve the problem point by point to the bottom of the well.
(2) Substituting the boundary conditions at the bottom of the hole and reversely calculating to the wellhead of the mud line, the temperature and pressure field distribution of the formation section is obtained.
(3) The temperature and pressure field of the seawater segment is then solved. Taking the platform wellhead as the starting point, the step size is selected and the four-order Runge-Kutta method is applied to solve the problem point by point to the mud line wellhead.
(4) Substitute the boundary conditions of the mud line wellhead obtained in Step (2), inversely calculate the platform wellhead, and find the distribution of temperature and pressure field in the seawater section;
(5) Draw the temperature and pressure field distribution of the deepwater wellbore according to the results obtained from Steps (2) and (4).
The detailed flow chart is shown below (
Based on the temperature field prediction model and solution method [
Parameter | Value | Parameter | Value |
---|---|---|---|
Tubing inner diameter (mm) | 76 | Seawater depth (m) | 1816.6 |
Tubing outer diameter (mm) | 114 | Seawater specific heat (J/g·°C) | 4.182 |
Well depth (m) | 2882.4 | Casing inner diameter (mm) | 244 |
Specific heat capacity of natural gas |
2650 | Formation thermal conductivity (W/m·°C) | 0.96 |
Well radius (r_{w}, m) | 0.108 | Seawater thermal conductivity (W/m·°C) | 1.6 |
Formation temperature (°C) | 81.9 | Cement sheath thermal conductivity (W/m·°C) | 10 |
Casing thermal conductivity (W/m·°C) | 2 | Formation-annulus fluid convective heat transfer coefficient (W/m·°C) | 0.96 |
Seawater segment annulus fluid convective heat transfer coefficient (W/m·°C) | 30.09 | Formation-annulus fluid radiative heat transfer coefficient (W/m·°C) | 40.28 |
Seawater segment annulus fluid radiative heat transfer coefficient (W/m·°C) | 60.28 | Gas production rate (10^{4}m^{3}/d) | 26 |
Similarly, the following relevant parameters (e.g., formation thermal diffusivity, annular test fluid thermal conductivity, annular fluid heat radiation coefficient) also nearly have no impact on the wellbore pressure. To avoid redundancy, they will not be elaborated in the following text.
X1 well is a typical ultra-deepwater gas well in the South China Sea, with a seawater depth of 1816.6 m. The well underwent a DST productivity test in the interval from 2828.80 to 2936.0 m. Pressure and temperature measurement devices were installed at the well bottom, seabed mudline, and wellhead, providing temperature and pressure data under different operating conditions during the test (
Test section (m) | Oil nozzle (mm) | Pressure (MPa) | Temperature (°C) | Flow rate (m^{3}/d) | |||||
---|---|---|---|---|---|---|---|---|---|
Well bottom | Well head | Seabed | Well bottom | Well head | Seabed | Gas | Water | ||
2828.80–2936.0 | 7.94 (2 h 15) | 29.46 | 23.73 | 27.95 | 81.9 | 19.6 | 33 | 261, 505 | 0 |
10.32 (1 h 55) | 29.46 | 23.44 | 27.76 | 83.8 | 17.9 | 44.75 | 432, 002 | 0 | |
12.70 (1 h 55) | 29.44 | 22.64 | 27.30 | 84.7 | 19.3 | 54 | 653, 814 | 0 |
Utilizing the established wellbore temperature-pressure field model, a fitting of the temperature-pressure profile from the well bottom to the seabed to the wellhead was performed. Specific results can be found in
The basic parameters obtained through fitting for different operating systems are shown in
Production volume (10^{4}m^{3}/d) | Parameter name | Value | Parameter name | Value |
---|---|---|---|---|
26.15 | Formation thermal diffusivity m^{2}/s | 1 × 10^{−9} | Seawater thermal diffusivity (Isothermal layer) m^{2}/s | 1.21 × 10^{−5} |
Seawater thermal diffusivity (Mixed layer and thermocline) m^{2}/s | 1.21 × 10^{−5} | Annular fluid radiative heat transfer coefficient W/m^{2}·K.s | 40.28 | |
43.20 | Formation thermal diffusivity m^{2}/s | 1 × 10^{−9} | Seawater thermal diffusivity (Isothermal layer) m^{2}/s | 1.21 × 10^{−6} |
Seawater thermal diffusivity (Mixed layer and thermocline) m^{2}/s | 1.21 × 10^{−6} | Annular fluid radiative heat transfer coefficient W/m^{2}·K.s | 60.28 | |
65.38 | Formation thermal diffusivity m^{2}/s | 1 × 10^{−9} | Seawater thermal diffusivity (Isothermal layer) m^{2}/s | 5.21 × 10^{−7} |
Seawater thermal diffusivity (Mixed layer and thermocline) m^{2}/s | 1.21 × 10^{−8} | Annular fluid radiative heat transfer coefficient W/m^{2}·K.s | 70.28 |
(1) By considering non-steady-state heat transfer in the formation and steady-state heat transfer in the seawater section, and combining energy conservation and fluid mechanics theory, a coupled flowing model of wellbore temperature-pressure field during ultra-deepwater gas well testing was established. The model was solved using the Runge-Kutta method to obtain the flowing temperature and pressure distribution in the wellbore.
(2) The analysis of key parameters of wellbore profile shows that: ➀ The higher the gas well production, the higher the wellbore temperature, but the wellbore temperature will cross in the thermocline interval; ➁ The higher the thermal diffusion coefficient, the faster the heat transfer from the wellbore to the formation, the lower the wellbore temperature; ➂ The higher the thermal diffusion coefficient of seawater in the constant temperature zone, the higher the wellbore temperature, while the opposite is true in the thermocline and mixed zone.
(3) Based on the production flow test data of X1 well in the South China Sea, the flow temperature and pressure fields of X1 under three working conditions are fitted. The fitting errors of temperature and pressure are 2.18% and 5.26%, which verify the reliability of the theoretical model. It is confirmed that with the increase of gas well flow test production, the thermal diffusivity of the seawater section decreases and the radiative heat transfer coefficient of annulus fluid increases.
The authors would like to thank the reviewers and editors for their useful suggestions for improving the quality of our manuscript.
The authors received no specific funding for this study.
The authors confirm contribution to the paper as follows: Study conception and design: Xingbin Zhao and Neng Yang; data collection: Dongling Qiu and Neng Yang; analysis and interpretation of results: Benteng Ma, Dongling Qiu and Mingqiang Wei; draft manuscript preparation: Hao Liang and Xingbin Zhao. All authors reviewed the results and approved the final version of the manuscript.
Data will be made available on request.
Not applicable.
The authors declare that they have no conflicts of interest to report regarding the present study.