Coupled Temperature–Pressure Field Calculation Method for Supercritical CO₂ Dry Fracturing
1. Definition and Fundamental Principles
Supercritical carbon dioxide (sCO₂) dry fracturing is an advanced reservoir stimulation technique in which carbon dioxide, maintained above its critical point (T_c = 304.13 K, P_c = 7.377 MPa), is injected into subsurface formations to generate and propagate hydraulic fractures without the use of water. Unlike conventional water-based hydraulic fracturing, dry fracturing eliminates the need for large volumes of proppant-laden fluid, thereby reducing formation damage, water consumption, and flowback contamination. The coupled temperature–pressure field calculation method is a multiphysics computational framework that simultaneously solves the governing equations for thermal transport, fluid flow, and fracture mechanics under supercritical conditions, capturing the strong nonlinearity introduced by the rapid phase transitions, density variations, and Joule–Thomson cooling effects inherent to sCO₂ systems.
The method is grounded in three coupled physical domains:
- Thermodynamic domain: Real-gas equations of state (e.g., Peng–Robinson, Span–Wagner) govern the density, enthalpy, and internal energy of supercritical CO₂ as a function of local temperature and pressure. The Joule–Thomson coefficient of sCO₂ is strongly pressure-dependent, leading to significant cooling (up to 30–50 K) as pressure drops across the fracture network.
- Hydraulic domain: The in-situ fluid flow within fractures is modeled using the lubrication approximation (lubrication theory) for variable-viscosity, compressible fluids, coupled with the surrounding porous medium via the modified Biot theory of poroelasticity.
- Fracture mechanics domain: Fracture propagation is governed by the Griffith energy criterion or cohesive zone models, where the stress intensity factor at the fracture tip depends on the coupled thermal stress (from temperature gradients) and mechanical stress (from fluid pressure and in-situ stress).
The governing coupled system can be expressed as:
- Momentum (fracture flow): ∂p/∂x = (12μ/h³) · q, where μ is the sCO₂ viscosity, h is the fracture aperture, and q is the volumetric flux.
- Mass conservation: ∂(ρh)/∂t + ∂q/∂x = -2k_f(ρ/μ)(p - p_f)/h, accounting for fluid leak-off into the formation.
- Energy conservation: ρc_p(∂T/∂t + v·∇T) = k_eff∇²T + φ·(∂p/∂t) + viscous dissipation terms, where the φ·(∂p/∂t) term captures the thermoelastic coupling.
- Thermoelastic stress: σ_ij = C_ijkl·(ε_kl - α·ΔT·δ_kl), linking temperature gradients to residual stress fields that influence fracture path and width.
2. Category and Business Positioning
Within the operational scope of Cladding Technology Shanxi Co., Ltd, this computational methodology occupies a cross-disciplinary technology support position. While the company's core deliverables—bimetallic clad plates, clad pipes, weld-overlay components, and explosively bonded products—are material and fabrication-focused, the supercritical CO₂ dry fracturing domain represents an emerging high-value application market where advanced material solutions are critically needed.
The business positioning can be articulated along three axes:
- Upstream technology intelligence: Mastery of the coupled calculation method provides the company with deep understanding of the extreme thermomechanical environments (T: 100–250°C, P: 30–120 MPa) that sCO₂ fracturing equipment must withstand. This knowledge directly informs material selection, cladding thickness design, and weld-overlay specification for downstream manufacturing.
- Customer qualification enabler: Energy companies developing sCO₂ fracturing programs require suppliers who demonstrate technical competence in understanding the operating envelope. Proficiency in coupled field calculations positions the company as a qualified engineering partner, not merely a component manufacturer.
- Product development driver: The coupled calculation outputs—specifically, the transient thermal cycling profiles and peak pressure/temperature combinations at equipment interfaces—define the exact performance requirements for clad and overlay products, enabling the company to develop purpose-engineered solutions.
3. Technical Purpose and Value
3.1 Primary Technical Purpose
The coupled temperature–pressure field calculation serves the following primary purposes:
- Predicting fracture geometry: Determining fracture length, width, and complexity under realistic thermal-hydraulic conditions to optimize reservoir contact area.
- Quantifying thermal effects: Evaluating the magnitude and spatial distribution of Joule–Thomson cooling and its impact on formation stress, fluid properties, and fracture propagation direction.
- Defining equipment design envelopes: Establishing the maximum and minimum temperature/pressure excursions that surface and downhole equipment (tubing, valves, heat exchangers, pumps, and storage vessels) must withstand.
- Optimizing injection parameters: Identifying the optimal injection rate, temperature, and proppant schedule to maximize fracture conductivity while minimizing thermal shock to the wellbore and equipment.
3.2 Value to Cladding Technology Shanxi Co., Ltd
The value of this capability to the company is multi-dimensional:
- Engineering credibility: Demonstrates to energy-sector clients that the company possesses the technical depth to specify appropriate material solutions for the most demanding sCO₂ applications.
- WPS and qualification support: The calculated thermal cycling profiles directly inform the thermal cycling conditions for weld procedure qualification (WPS) under ASME Section IX or ISO 15614, ensuring that weld-overlay and clad products are qualified for the actual service conditions.
- Failure analysis and risk mitigation: Understanding the coupled field enables proactive identification of potential failure modes (thermal fatigue cracking, pressure-induced cladding delamination, stress corrosion cracking of overlay layers) and the design of appropriate countermeasures.
- IP and standardization contribution: Methodological expertise supports participation in emerging industry standards for sCO₂ equipment (e.g., ISO/TC 6/SC 14, API RP 1005) and contributes to the company's intellectual property portfolio.
4. Key Process and Implementation Points
4.1 Computational Framework Architecture
The coupled calculation is typically implemented using a sequential or fully coupled numerical approach. The following table summarizes the key components of the computational framework:
| Module | Governing Equation / Model | Key Parameters | Typical Discretization |
|---|---|---|---|
| Thermodynamic Property | Peng–Robinson or Span–Wagner EOS | T_c = 304.13 K, P_c = 7.377 MPa, acentric factor ω = 0.224 | Property tables interpolated at each node |
| Fluid Flow in Fracture | Lubrication theory with variable μ(T, P) | μ: 0.03–0.08 mPa·s, h: 1–10 mm | 1D/2D finite element mesh along fracture |
| Formation Flow | Darcy's law with Biot poroelasticity | Permeability k: 0.1–100 mD, porosity φ: 0.05–0.25 | 3D finite element or finite difference grid |
| Heat Transfer | Energy equation with Joule–Thomson source term | k_eff: 2–10 W/m·K, c_p: 1.0–2.5 kJ/kg·K | Implicit time stepping, Δt = 1–10 s |
| Fracture Propagation | Griffith criterion or cohesive zone model | K_IC: 0.5–2.0 MPa·m^0.5, fracture toughness | Extended finite element method (XFEM) or boundary element |
| Thermoelastic Stress | Linear elastic constitutive law with thermal expansion | α: 2.5–12×10⁻⁶ /K, E: 30–200 GPa | Coupled with fracture mechanics module |
4.2 Typical Operating Envelope for sCO₂ Dry Fracturing
The following table defines the parameter ranges that the coupled calculation must capture, which directly inform the material and manufacturing requirements for associated equipment:
| Parameter | Range | Material Implication |
|---|---|---|
| Injection Pressure | 30–120 MPa | High-pressure containment components require thick-wall clad pipes and pressure-rated weld overlays |
| Injection Temperature | 100–250°C (subsurface); 40–80°C (surface) | Thermal cycling between surface and subsurface conditions demands overlay materials with good thermal fatigue resistance |
| Joule–Thomson Temperature Drop | 20–50 K across fracture network | Thermal shock resistance of clad interfaces is critical; interface bonding quality must withstand rapid cooling |
| Pressure Cycling Rate | 0.5–5 MPa/min during injection/flowback | Clad plate and overlay welds must resist fatigue under cyclic pressure loading |
| Fluid Composition | CO₂ with up to 5% H₂S, CH₄, N₂, and trace moisture | Corrosion-resistant overlay alloys (e.g., 310, 625, C-276) required for wet CO₂ environments |
| Flow Velocity (in fractures) | 1–20 m/s | Erosion-corrosion resistance of overlay surfaces is essential for high-velocity sCO₂ flow |
4.3 Implementation Workflow
- Reservoir characterization: Input geological data including formation permeability, porosity, in-situ stress field, geothermal gradient, and lithological properties.
- Thermodynamic property setup: Implement the selected equation of state and generate property lookup tables for sCO₂ across the expected T-P domain (40–250°C, 5–150 MPa).
- Boundary condition definition: Specify injection rate, injection temperature, wellbore geometry, and formation boundary conditions (constant pressure, constant temperature, or no-flow boundaries).
- Mesh generation and coupling: Construct a discretized domain that resolves the fracture, near-wellbore region, and far-field formation. Implement the sequential coupling scheme (pressure → temperature → stress → fracture update) with appropriate convergence criteria (residual tolerance < 10⁻⁶).
- Time-stepping and solution: Execute the transient simulation over the fracturing timeline (typically 2–24 hours), with adaptive time stepping to capture rapid pressure transients during injection and flowback.
- Post-processing and equipment envelope extraction: Extract the maximum, minimum, and transient T-P trajectories at critical equipment locations (wellhead, downhole tools, surface heat exchangers) to define the material design envelope.
5. Applicable Standards and Acceptance Criteria
5.1 Computational Method Validation Standards
- ISO 15683-1: Petroleum and natural gas industries—Geothermal reservoirs—Numerical simulation of reservoirs (provides general framework for reservoir simulation validation).
- ISO 10426-1: Petroleum and natural gas industries—Petroleum—Numerical methods for reservoir simulation (methodology for validating numerical schemes).
- API RP 1005: Evaluation and Analysis of Production Data (relevant for field validation of simulation results against actual production performance).
- NACE SP0775: Considerations for Materials Selection for Carbon Dioxide Service (materials qualification criteria that the simulation outputs must support).
5.2 Equipment Design and Acceptance Standards
The coupled calculation results feed directly into equipment design and qualification, governed by the following standards:
- ASME BPV Code Section I / VIII: Boiler and Pressure Vessel Code for pressure-containing equipment exposed to sCO₂ conditions. The calculated maximum design pressure and temperature define the design conditions for cladding integrity assessment.
- ASME Section IX: Welding, Brazing, and Fusing Qualifications—governs WPS/PQR qualification for weld overlay layers and clad interfaces. Thermal cycling profiles from the simulation inform the number and severity of cycles in qualification testing.
- ASTM A377 / A167: Standard specifications for clad plate and pipe, respectively. The calculated pressure and temperature envelopes define the minimum base metal and cladding material requirements.
- ASTM E165 / E1099: Nondestructive examination standards for ultrasonic testing of clad interfaces and weld overlays. Acceptance criteria for bonding quality are defined per these standards.
- API 5CT: Specification for casing and tubing—applies to clad tubing used in sCO₂ injection wells. The coupled calculation defines the minimum yield strength and corrosion resistance requirements.
- NACE MR0175 / ISO 15156: Materials for use in H₂S-containing environments—relevant when CO₂ contains H₂S impurities. Defines maximum hardness limits and material restrictions for overlay alloys.
- GB/T 19078: Chinese national standard for clad steel plates (relevant for domestic market compliance).
- NB/T 47014: Chinese standard for qualification of welding procedures for pressure vessels—applies to WPS qualification for clad and overlay components used in sCO₂ systems.
5.3 Acceptance Criteria for Clad/Overlay Products in sCO₂ Service
| Acceptance Item | Standard Reference | Typical Criterion |
|---|---|---|
| Clad interface bonding quality | ASTM E165 / GB/T 19078 | 100% bond area, no defects ≥ 1 mm at interface |
| Weld overlay hardness | NACE MR0175 | ≤ 22 HRC (for H₂S-containing environments) |
| Overlay layer thickness uniformity | ASME Section IX / WPS | ± 10% of specified thickness, minimum 1.5 mm |
| Thermal fatigue resistance | ASTM E466 / custom qualification | No cracking after specified cycles (derived from coupled calculation) |
| Corrosion resistance | NACE TM0177 / ASTM G150 | Corrosion rate ≤ 0.025 mm/year in wet CO₂ at service conditions |
| UT scanning coverage | ASTM E165 / NB/T 47013 | 100% UT scanning of clad interface and overlay welds |
6. Common Risks and Controls
6.1 Computational Risks
- Risk: Inaccurate thermodynamic property predictions. The Peng–Robinson EOS can deviate by 5–15% in density predictions near the critical point. Control: Use the Span–Wagner reference EOS for CO₂ and validate against NIST REFPROP data. Implement uncertainty quantification on property inputs.
- Risk: Inadequate resolution of thermal gradients. The Joule–Thomson cooling zone near the fracture tip can have steep temperature gradients (10–50 K/mm). Control: Use adaptive mesh refinement (AMR) near the fracture tip and validate against analytical solutions for simple geometries.
- Risk: Coupling convergence failure. The strong nonlinearity of the coupled system can lead to non-convergence of the iterative solver. Control: Implement sub-iterations within each time step, use a semi-implicit coupling scheme, and limit the pressure/temperature change per iteration.
- Risk: Over-reliance on simplified fracture models. Linear elastic fracture mechanics (LEFM) may not capture the thermal stress effects on fracture toughness. Control: Use cohesive zone models with temperature-dependent fracture energy and validate against laboratory thermal fracture experiments.
6.2 Material and Manufacturing Risks (Directly Relevant to Company Operations)
- Risk: Thermal fatigue cracking of clad interfaces. Repeated temperature cycling between surface (40–80°C) and subsurface (100–250°C) conditions can initiate cracks at the clad interface, particularly where thermal expansion mismatch exists between base and cladding materials. Control: Select cladding materials with thermal expansion coefficients closely matched to the base metal (Δα ≤ 1.0×10⁻⁶/K). Use the coupled calculation to define the number of thermal cycles and qualify the clad product under equivalent cycling conditions per ASTM E466.
- Risk: Cladding delamination under cyclic pressure. Pressure cycling at 0.5–5 MPa/min can cause fatigue-driven delamination of the clad layer, especially at weld seams or geometric discontinuities. Control: Ensure 100% UT bonding quality per ASTM E165. Design cladding thickness with a minimum 30% margin above the calculated maximum pressure requirement. Use explosion welding or hydraulic explosive bonding to achieve metallurgical bonding with no interfacial defects.
- Risk: Stress corrosion cracking (SCC) of overlay welds in wet CO₂. When CO₂ contains moisture, carbonic acid corrosion and SCC can affect austenitic overlay alloys, particularly at sensitized grain boundaries. Control: Specify overlay alloys with low carbon content (≤ 0.03% C) such as 316L, 310, or 625. Maintain hardness below 22 HRC per NACE MR0175. Perform post-weld heat treatment to relieve residual stresses and reduce sensitization.
- Risk: Erosion-corrosion of overlay surfaces at high flow velocities. sCO₂ flowing at 1–20 m/s through fracture networks or tubing can cause erosive wear of the overlay surface, exposing the base metal to corrosive attack. Control: Use overlay alloys with high hardness (35–50 HRC for non-H₂S environments) such as 5Cr-0.5Mo or Stellite 6. Ensure overlay thickness ≥ 3 mm for high-velocity applications. Validate erosion resistance through accelerated testing per ASTM G98.
7. Application Scenarios Across the Company's Three Technology Routes
7.1 TIG/MIG Weld Overlay Route
The coupled calculation method directly informs the TIG/MIG weld overlay specifications for sCO₂ fracturing equipment:
- Thermal cycling qualification: The transient temperature profiles extracted from the simulation (e.g., 40°C surface → 250°C subsurface → 40°C return, repeated over 500–5000 cycles) define the thermal cycling conditions for weld procedure qualification. The WPS must demonstrate that the overlay welds survive these cycles without cracking, per ASME Section IX and NB/T 47014.
- Overlay material selection: The coupled calculation identifies the maximum temperature and the presence of wet CO₂, guiding the selection of overlay alloys. For dry CO₂ at temperatures below 150°C, 309L or 316L overlay is sufficient. For wet CO₂ or temperatures above 150°C, 310, 625, or C-276 overlay is required. The hardness limit of 22 HRC (NACE MR0175) constrains the choice when H₂S is present.
- Multi-pass overlay design: The calculated pressure envelope determines the required overlay thickness. For 120 MPa service pressure, a minimum overlay thickness of 3–5 mm is typically required, achieved through 3–5 TIG passes with interpass temperature control below 150°C to prevent sensitization.
- Residual stress management: The coupled calculation reveals the magnitude of thermal stresses (up to 300–500 MPa) that can develop during operation. The WPS must include post-weld heat treatment (PWHT) at 600–650°C for 2 hours to relieve residual stresses to below 50 MPa, ensuring that the overlay welds do not become initiation sites for fatigue cracking.
7.2 Hydraulic Explosive Bonding Route
Hydraulic explosive bonding produces clad plates and pipes with metallurgical bonding suitable for sCO₂ applications. The coupled calculation contributes as follows:
- Interface integrity under thermal cycling: The hydraulic explosive bonding process creates a wave-like metallurgical interface. The coupled calculation defines the thermal cycling conditions that this interface must withstand. Qualification testing must demonstrate that the wave amplitude and frequency of the bonded interface are sufficient to resist fatigue-driven delamination under the calculated cycling conditions.
- Clad thickness optimization: The pressure envelope from the calculation (30–120 MPa) determines the minimum clad thickness. Hydraulic explosive bonding can produce clad layers from 1.0 mm to 10.0 mm with high bonding quality. The calculation enables the company to specify the optimal clad thickness that balances pressure containment with material cost.
- Material combination selection: The coupled calculation identifies the corrosion environment (dry CO₂, wet CO₂, H₂S-containing CO₂) and the temperature range. This guides the selection of clad material combinations, such as:
- Carbon steel (Q345R/16MnR) base + 316L clad for dry CO₂ below 150°C
- Low-alloy steel (15CrMo) base + 310 clad for wet CO₂ above 150°C
- Stainless steel (304L) base + C-276 clad for highly corrosive wet CO₂ with H₂S
- Product qualification for energy sector: The coupled calculation results provide the technical justification for the clad product specifications, enabling the company to submit qualified products to energy companies developing sCO₂ fracturing programs. The product datasheet can reference the specific T-P envelope and demonstrate compliance with the applicable standards (ASME, API, NACE).
7.3 Explosion Welding Route
Explosion welding is the most demanding of the company's three routes, producing clad products with the highest bonding quality and the most complex material combinations. The coupled calculation method supports this route in the following ways:
- Extreme condition qualification: Explosion-welded clad plates are often used for the most demanding sCO₂ applications—high pressure (≥ 80 MPa), high temperature (≥ 200°C), and wet CO₂ with H₂S. The coupled calculation provides the definitive service envelope against which the explosion-welded product must be qualified. The company can demonstrate that the explosion-welded interface, with its high bond strength (typically > 95% of the weaker parent material), is capable of withstanding the calculated pressure and thermal cycling conditions.
- Multi-layer clad design: For the most aggressive sCO₂ environments, multi-layer clad structures may be required (e.g., carbon steel base + 316L intermediate layer + C-276 outer layer). The coupled calculation identifies the specific corrosion mechanism (uniform corrosion, SCC, erosion-corrosion) that drives the need for multi-layer designs. The explosion welding process is uniquely capable of producing multi-layer clad plates with full metallurgical bonding at each interface.
- Thermal stress analysis of clad interfaces: The coupled calculation provides the thermal stress distribution that develops at the clad interface during operation. For explosion-welded products, the wave-like interface geometry provides inherent resistance to crack propagation. The company can use the calculated stress fields to perform finite element analysis (FEA) of the explosion-welded interface, demonstrating that the wave geometry arrests any potential fatigue cracks and prevents delamination.
- NDT protocol development: The coupled calculation identifies the critical locations where failure is most likely (e.g., near geometric discontinuities, at weld seams, at the transition between clad and unclad regions). This information guides the development of targeted NDT protocols—UT scanning per ASTM E165 for the clad interface, MT/PT per ASTM E709/E165 for surface defects, and RT per ASTM E94 for weld volume defects—ensuring that the most critical areas receive the most thorough inspection.
8. Contribution to Qualification Building, Product Delivery, and Customer Value
8.1 Qualification Building
The coupled temperature–pressure field calculation method is a cornerstone of the company's qualification infrastructure for the sCO₂ fracturing market. Specifically:
- WPS qualification: The calculated thermal cycling profiles provide the basis for thermal cycling qualification tests of weld overlay procedures (ASME Section IX, NB/T 47014). Without this calculation, WPS qualification would be based on generic assumptions rather than actual service conditions, potentially leading to non-qualified products or over-conservative (costly) specifications.
- Product qualification: The calculated pressure and temperature envelopes define the design conditions for clad plate and pipe products (ASME Section II, ASTM A377, ASTM A167). The company can develop product-specific qualification documents that directly reference the coupled calculation results, demonstrating to customers that the product is engineered for the specific application.
- System qualification: For integrated solutions (e.g., clad heat exchangers, clad storage vessels, clad injection tubing), the coupled calculation provides the system-level performance data required for ASME Section VIII or NB/T 47003 qualification.
8.2 Product Delivery
- Specification accuracy: The coupled calculation ensures that the product specifications (material grade, clad thickness, overlay thickness, hardness, NDT requirements) are precisely matched to the service conditions, avoiding under-specification (risk of failure) and over-specification (unnecessary cost).
- Quality assurance: The calculated failure modes (thermal fatigue, delamination, SCC, erosion-corrosion) define the specific quality attributes that must be verified during production. This enables the development of a targeted quality assurance plan that focuses inspection resources on the most critical aspects.
- Traceability: Each product delivered for sCO₂ applications can be traced back to the specific coupled calculation run that defined its design envelope, providing full traceability from computational analysis to manufactured product.
8.3 Customer Value
- Risk reduction: By providing products that are qualified against actual calculated service conditions, the company reduces the customer's risk of equipment failure, wellbore integrity issues, and production downtime in the sCO₂ fracturing operation.
- Cost optimization: The coupled calculation enables the company to specify the minimum viable material and manufacturing solution, avoiding unnecessary use of premium alloys and reducing the customer's capital expenditure on equipment.
- Technical partnership: The company's ability to perform coupled field calculations positions it as a technical partner to the customer's engineering team, not merely a supplier. This fosters long-term relationships and enables collaborative development of next-generation sCO₂ equipment.
- Regulatory compliance: The coupled calculation provides the technical basis for regulatory submissions (e.g., to national energy authorities, environmental agencies) that demonstrate the safety and environmental soundness of the sCO₂ fracturing operation, including the integrity of the equipment that contains and transports the supercritical fluid.
9. Summary and Strategic Outlook
The coupled temperature–pressure field calculation method for supercritical CO₂ dry fracturing represents a high-value technical capability that bridges the gap between reservoir engineering and materials manufacturing. For Cladding Technology Shanxi Co., Ltd, this capability is not merely an academic exercise—it is a direct enabler of the company's expansion into the rapidly growing sCO₂ energy sector. By mastering this computational method, the company can:
- Deliver precisely engineered clad and overlay products for the most demanding sCO₂ applications.
- Qualify its products against actual service conditions rather than generic assumptions.
- Establish itself as a technical authority and preferred supplier in the sCO₂ fracturing supply chain.
- Contribute to the development of industry standards and best practices for sCO₂ equipment materials.
As the global energy transition accelerates and supercritical CO₂ technology moves from pilot projects to commercial deployment, the demand for high-integrity, corrosion-resistant, and thermally stable clad and overlay products will grow exponentially. The company's investment in this computational methodology positions it at the forefront of this market, ensuring that its products meet the exacting requirements of next-generation energy systems.