Thermal Analysis

The thermal option of FLAC3D incorporates both conduction and advection models. The conduction models allow simulation of transient heat conduction in materials, and the development of thermally induced displacements and stresses. The advection model takes the transport of heat by convection into account; it can simulate temperature-dependent fluid density and thermal advection in the fluid. This thermal option has several specific features:

1. Four thermal material models are available: isotropic conduction, anisotropic conduction, isotropic conduction/advection, and the null thermal model.
2. As in the standard version of FLAC3D, different zones may have different models and properties.
3. Any of the mechanical models may be used with the thermal model.
4. Temperature, flux, convective and adiabatic boundary conditions may be prescribed.
5. Heat sources may be inserted into the material as either point sources or volume sources. These sources may decay exponentially with time.
6. Both explicit- and implicit-solution algorithms are available.
7. The thermal option provides for one-way coupling to the mechanical stress and pore-pressure calculations through the thermal expansion coefficients.
8. Temperatures can be accessed via FISH for users to define temperature-dependent properties.

This chapter contains a description of the thermal formulation and the numerical implementation. Recommendations for solving thermal problems are also provided. The FLAC3D input commands for thermal analysis, and the system of unit for thermal analysis, are given. Finally, several verification problems are described. Refer to these examples as a guide for creating FLAC3D models for thermal analysis and coupled thermal-stress or thermal-groundwater flow analysis.

• Mathematical Model Description
  • Conventions and Definitions
  • Conduction
    • Energy-Balance Equation
    • Transport Law
    • Boundary and Initial Conditions
    • Mechanical Coupling — Thermal Strains
    • Fluid Coupling — Thermally Induced Pore Pressures
  • Advection
    • Energy Balance for Convective-Diffusive Heat Transport
    • Fluid Mass Balance (Slightly Compressible Fluid)
    • Transport Laws
    • Thermal-Mechanical-Pore Pressure Coupling
    • Initial and Boundary Conditions
      • Advection-Conduction Properties
      • Anisotropic Heat Conduction Properties
      • Isotropic Heat Conduction Properties
• Numerical Formulation
  • Conduction
    • Finite-Difference Approximation to Space Derivatives
    • Nodal Formulation of the Energy-Balance Equation
    • Explicit Finite-Difference Formulation
    • Implicit Finite-Difference Formulation
    • Thermal-Stress Coupling
    • Thermal Pore-Pressure Coupling
  • Advection
    • Stability and Accuracy
• Solving Thermal-Only and Coupled-Thermal Problems
  • Thermal Conduction-Only Analysis
  • Thermal Advection-Only Analysis
    • Steady-State Conduction Solution
    • Forced Advection and Free Advection Simulations
    • Synchronization of Fluid and Thermal Times
  • Thermal-Mechanical Analysis
  • Thermal Coupling to Pore Pressure
• Input Instructions for Thermal Analysis
  • Thermal Commands
  • Thermal FISH Variables
  • Zone-Based Temperature
• Systems of Units for Thermal Analysis
• Verification and Example Problems
  • Conduction in a Plane Sheet
  • Conduction in a Hollow Cylinder
  • Infinite Line Heat Source in an Infinite Medium
  • Thermal Pore Pressure Coupled Response
  • One-Dimensional Solution of Thermal Transport by Forced Convection and Conduction
  • Steady-State Convection in a Saturated Porous Medium Heated from Above
  • Steady-State Convection in a Saturated Porous Medium Heated from Below
• References