Thermo-Richards-Mechanics Process

This page describes Thermo-Richards-Mechanics Process (TRM)

Introduction

Theoretical background

Global assembler for the monolithic scheme of the non-isothermal Richards flow coupled with mechanics.

Governing equations without vapor diffusion

The energy balance equation is given by:

with $T$ the temperature, $(\rho c_p)^{eff}$ the effective volumetric heat capacity, $\mathbf{k}_T^{eff}$ the effective thermal conductivity, $\rho^l$ the density of liquid, $c_p^l$ the specific heat capacity of liquid, $\mathbf{v}^l$ the liquid velocity, and $Q_T$ the point heat source.

The effective volumetric heat can be considered as a composite of the contributions of solid phase and the liquid phase as

with $\phi$ the porosity, $S^l$ the liquid saturation, $\rho^s$ the solid density, and $c_p^s$ the specific heat capacity of solid. Similarly, the effective thermal conductivity is given by:

where $\mathbf{k}_T^s$ is the thermal conductivity tensor of solid, $k_T^l$ is the thermal conductivity of liquid, and $\mathbf I$ is the identity tensor.

The mass balance equation is given by:

where $p$ is the pore pressure, $p_c$ is the capillary pressure, which is $-p$ under the single phase assumption, $\beta$ is a composite coefficient by the liquid compressibility and solid compressibility, $\alpha_B$ is the Biot’s constant, $\alpha_T^s$ is the linear thermal expansivity of solid, $Q_H$ is the point source or sink term, $\mathbf u$ is the displacement, and $H(S-1)$ is the Heaviside function. The liquid velocity $\mathbf{v}^l$ is described by the Darcy’s law as

with ${\mathbf k}$ the intrinsic permeability, $k_{ref}$ the relative permeability, $\mathbf g$ the gravitational force.

The momentum balance equation takes the form of

with $\mathbf{\sigma}$ the effective stress tensor, $b(S)$ the Bishop model, $\mathbf f$ the body force, and $\mathbf I$ the identity. The primary unknowns of the momentum balance equation are the displacement $\mathbf u$, which is associated with the stress by the generalized Hook’s law as

with $C$ the forth order elastic tensor, ${\dot {\mathbf \epsilon}}$ the total strain rate, ${\dot {\mathbf \epsilon}}^e$ the elastic strain rate, ${\dot {\mathbf \epsilon}}^T$ the thermal strain rate, ${\dot {\mathbf \epsilon}}^p$ the plastic strain rate, ${\dot {\mathbf \epsilon}}^{sw}$ the swelling strain rate.

The strain tensor is given by displacement vector as

where the superscript ${\text T}$ means transpose.

Implementation

Supported phases

• Aqueous liquid
• Solid

Input variables and parameters

List of medium properties required by TRM process.

Medium phase properties

Those properties are defined on the phase level for each medium. See phase properties for more details on defining them.

Medium properties

Those properties are defined on medium level. See medium properties for more details on defining them.

Thermo-osmosis can be parametrised in two, mutually exclusive, ways:

• thermal_osmosis_coefficient sets the thermo-osmotic coefficient $k_T$ ($[k_T] = m^2/(K \cdot s)$) directly. Use this when a tabulated $k_T$ is available.
• thermal_osmosis_permeability sets the scalar thermo-osmotic permeability $\epsilon_T$ ($[\epsilon_T] = Pa/K$), from which $k_T = \epsilon_T k / \mu$ is formed using the medium’s intrinsic permeability $k$ and the AqueousLiquid phase’s dynamic viscosity $\mu$. Use this when $\epsilon_T$ is the measured quantity and $k_T$ should track a spatially varying $k$ and a temperature-dependent $\mu$.

Note that thermal_osmosis_permeability is not a permeability and does not have a permeability’s dimension; the name was kept for continuity with the existing property naming.

Older project files defining thermal_osmosis_coefficient on the solid phase are moved to the medium level by the scripts/dev/move_thermal_osmosis.py helper script, which with --convert also rewrites the property as thermal_osmosis_permeability. A thermal_osmosis_coefficient already given on the medium level is a valid parametrisation and is left unchanged by the script.

Input parameters in the project file

TRM process has to be declared in the project file in the processes block. For example in following way:

<processes>
    <process>
        <type>THERMO_RICHARDS_MECHANICS</type>
    </process>
</processes>

Process variables

Following process variables are available in TRM process:

• temperature
• pressure
• displacement

For more details, see Process variables .

Example of full section defining TRM process

For more detailed description of tags used in this snippet, please see Processes .

  <processes>
    <process>
      <name>BodyForceTest</name>
      <type>THERMO_RICHARDS_MECHANICS</type>
      <mass_lumping>false</mass_lumping>
      <integration_order>3</integration_order>
      <constitutive_relation id="0">
            ...
      </constitutive_relation>
      <process_variables>
        <temperature>temperature</temperature>
        <pressure>pressure</pressure>
        <displacement>displacement</displacement>
      </process_variables>
      <secondary_variables>
            ...
      </secondary_variables>
      <specific_body_force>0 -9.81</specific_body_force>
      <apply_body_force_for_deformation>false</apply_body_force_for_deformation>
      <initial_stress>Initial_stress</initial_stress>
    </process>
  </processes>

For more information on tags <apply_body_force_to_deformation> and <mass_lumping> see section Features at this page.

Features

Specific body force

Mass lumping

The diagonal lumping of the mass matrix of the Richards equation can be using by adding following tag to the <process> </process> block:

<mass_lumping>true</mass_lumping>

Applying body force on deformation

Thermal porosity mixing

TRM can automatically obtain thermal conductivity for the medium based on thermal conductivities of phases and porosity.

See Thermal conductivity: effective porosity mixing for more information.

Examples

Available benchmarks

To gain more insight into TRM process, you can investigate TRM benchmarks .

References