CLADDING TECHNOLOGY SHANXI CO., LTDTemperature-Pressure Field Coupling Calculation Method for Supercritical CO2 Dry Fracturing
Literature Overview and Methodological Approach
This paper presents a coupled temperature-pressure field calculation method for supercritical CO2 dry fracturing operations, addressing the complex thermodynamic interactions that occur during the fracturing process. The mathematical model couples the energy equation with the momentum equation and equation of state for supercritical CO2, providing a comprehensive framework for predicting temperature and pressure distributions within the formation and equipment. For pressure vessel engineers, this coupled analysis is essential for thermal stress evaluation and fatigue assessment of high-pressure equipment.
Coupled Mathematical Model
The governing equations for the coupled temperature-pressure field include the mass conservation equation, momentum equation (Darcy's law modified for fracture flow), energy equation with heat transfer terms, and the equation of state for supercritical CO2. The key challenge in this coupled system is the strong non-linearity arising from the temperature-dependent properties of supercritical CO2, particularly its density, viscosity, and thermal conductivity, which vary dramatically near the critical point.
| Thermodynamic Property | Near Critical Point | Far from Critical Point | Variation Rate |
|---|---|---|---|
| Density | 467.6 kg/m³ | 1.8-1000 kg/m³ | Extremely sensitive |
| Viscosity | 0.064 mPa·s | 0.01-0.1 mPa·s | Moderate sensitivity |
| Thermal conductivity | 0.074 W/(m·K) | 0.014-0.08 W/(m·K) | Moderate sensitivity |
| Specific heat | 2.3 kJ/(kg·K) | 0.85-1.5 kJ/(kg·K) | Highly variable |
| Speed of sound | 220 m/s | 230-400 m/s | Highly variable |
The coupling effect between temperature and pressure fields manifests primarily through the equation of state and the temperature-dependent transport properties. Small changes in temperature near the critical point can cause large changes in density, which in turn affects the pressure field and flow patterns. This non-linearity requires iterative numerical solutions and careful convergence criteria to ensure accurate results.
Thermal Stress Implications for Pressure Equipment
The temperature-pressure coupling analysis has direct implications for the thermal stress assessment of pressure vessels and piping systems used in supercritical CO2 fracturing. The temperature gradients predicted by the coupled model can be used to calculate thermal stresses in vessel walls, nozzle penetrations, and pipe bends.
For a typical vessel wall thickness of 50-100 mm designed for 35 MPa operating pressure, the thermal gradient across the wall during a single fracturing cycle can reach 20-40°C. This thermal gradient produces thermal stresses that, when combined with mechanical stresses from internal pressure, may approach or exceed the allowable stress limits specified in design codes.
The cyclic nature of fracturing operations means that these thermal stresses are repeated thousands of times over the service life of the equipment. Fatigue analysis per ASME VIII Div.2 Part 5 or equivalent national standards must be performed to ensure adequate fatigue life. The number of allowable cycles depends on the stress range, mean stress, and material properties at the operating temperature.
Numerical Solution Considerations
The numerical solution of the coupled temperature-pressure field requires careful attention to several technical aspects. The mesh must be sufficiently refined near the wellbore and fracture tips where gradients are largest. The time step must be small enough to capture the rapid pressure transients during the gas explosion phase while remaining computationally efficient for the long simulation periods required for permeability evolution studies.
Convergence criteria for the iterative solution should be set at residual values below 10⁻⁶ for the pressure equation and 10⁻⁵ for the energy equation. The equation of state for supercritical CO2 should use a high-accuracy model such as the Span-Wagner equation rather than simplified cubic equations of state, as the latter introduce significant errors near the critical point.
Equipment Design Parameters Derived from Coupled Analysis
The coupled temperature-pressure field analysis provides critical input for pressure equipment design. The maximum and minimum temperatures experienced during operation determine the material selection and impact testing requirements. The maximum pressure transients determine the design pressure and pressure relief system sizing. The thermal gradients determine the required thickness and material grade for stress relief welds.
Based on the coupled analysis results, the following design parameters are recommended for supercritical CO2 fracturing pressure vessels:
- Design pressure: 1.5 times maximum transient pressure
- Design temperature: Maximum operating temperature plus 20°C margin
- Minimum design temperature: Minimum expected temperature minus 10°C
- Material impact testing temperature: Minimum design temperature
- Fatigue analysis required when pressure cycles exceed 5,000
Study Insights and Engineering Applications
This coupled calculation method provides a rigorous analytical framework for predicting the thermal and pressure conditions experienced by fracturing equipment during operation. Engineers should incorporate these coupled analysis results into the design process rather than relying on simplified single-variable approaches. The strong coupling effects near the critical point mean that temperature and pressure cannot be treated independently, and the interaction between these variables must be fully accounted for in equipment design, material selection, and inspection planning.