Hydro-Thermal Process (HT)

Introduction

The Hydro-Thermal (HT) process models coupled groundwater flow and heat transport in porous media. Both sub-processes are described by parabolic partial differential equations that are coupled through temperature-dependent fluid properties (density, viscosity) and the advective heat transport term.

Key features

• Monolithic and staggered coupling schemes.
• Fluid compressibility and optional solid thermal expansion coupling.
• Hydrodynamic thermal dispersion (longitudinal and transversal dispersivities).
• Numerical stabilisation for advection-dominated problems.
• Fracture flow support via aperture size parameter.
• Optional solid thermal expansion with Biot constant.
• Surface flux calculation.

Physical variables

• Primary variables: pressure $p$ and temperature $T$.
• Secondary variable: Darcy velocity $\mathbf{q}$.

Theoretical background

Balance equation framework

Both the flow and heat transport processes are derived from integral conservation laws. Starting with the conservation of mass (or energy), we write:

where $S(u)$ is the volume density of the conserved quantity, $\mathbf{J}$ is the flux, $\mathbf{n}$ is the outward normal, and $Q$ represents sources/sinks.

Applying the divergence theorem (Gauss) to convert the boundary integral to a volume integral:

Since this holds for any domain $\Omega$, the strong form follows:

Substituting constitutive laws for flux $\mathbf{J}$ (such as Darcy’s law for groundwater and Fourier’s law for heat) yields the specific governing equations.

Groundwater flow equation

The Darcy velocity is given by:

The pressure equation including fluid compressibility reads:

where $\Psi = p + \varrho_f(T) g z$ is the piezometric head, $S$ is the storage coefficient, $\phi$ is the porosity, and $Q$ is a source/sink term.

Derivation of the storage term: Applying the balance equation framework with $S(p) = \phi \varrho_f(p,T)$ (fluid mass per unit volume) yields:

Assuming the solid matrix is incompressible ($\partial \phi/\partial t = 0$), the storage term expands as:

Under the Boussinesq approximation, the temperature dependence of density in storage is neglected, and fluid compressibility $\partial \varrho_f/\partial p$ is assumed constant, allowing the definition of an effective storage coefficient $S = \phi \frac{\partial \varrho_f}{\partial p}$. Temperature dependence is retained only in the buoyancy term through $\varrho_f(T)$ in the piezometric head.

Heat transport equation

where:

Coupling

The fluid density $\varrho_f(T, p)$ and viscosity $\mu(T)$ couple the hydraulic and thermal equations. Fluid compressibility $\partial\varrho_f/\partial p$ enters the storage term, and temperature-dependent density drives buoyancy.

Note: The classical Boussinesq (Oberbeck–Boussinesq) approximation, where density variations appear only in the buoyancy term, can be recovered by choosing a density model with no pressure dependence (i.e. $\partial\varrho_f/\partial p = 0$, e.g. a constant or temperature-only dependent density) and omitting the solid_thermal_expansion configuration.

Finite element discretization

Both equations are discretized into the standard form $\mathbf{M} \dot{\mathbf{u}} + \mathbf{K} \mathbf{u} = \mathbf{f}$.

Pressure equation

Weak formulation

The weak form is obtained by multiplying the strong form by a test function $v \in H_0^1(\Omega)$ and integrating over the domain:

Integration by parts applied to the diffusion term:

Applying the Neumann boundary condition and setting $v=0$ on Dirichlet boundaries yields the weak form:

Substituting the Galerkin discretization $p \approx \sum_j N_j a_j$ and choosing $v = N_i$ recovers the matrix form $\mathbf{M}^p \dot{\mathbf{a}} + \mathbf{K}^p \mathbf{a} = \mathbf{f}^p$ presented above.

When solid_thermal_expansion is configured, an additional thermal expansion source term is added to the load vector:

where $\alpha_B$ is the Biot constant, $\alpha_s$ is the solid thermal expansion coefficient, and $\dot{T}$ is approximated by backward finite differences.

Temperature equation

Definition in the project file

The HT process is declared in the <processes> block of the project file.

Monolithic scheme (default)

<process>
    <name>HydroThermal</name>
    <type>HT</type>
    <integration_order>2</integration_order>
    <process_variables>
        <temperature>T</temperature>
        <pressure>p</pressure>
    </process_variables>
    <specific_body_force>0 -9.81</specific_body_force>
    <secondary_variables>
        <secondary_variable name="darcy_velocity"/>
    </secondary_variables>
</process>

Staggered scheme

To use the staggered coupling scheme, add:

<coupling_scheme>staggered</coupling_scheme>

In the staggered scheme, the heat transport equation (process ID 0) and the hydraulic equation (process ID 1) are solved sequentially.

Process variables

The HT process requires two process variables: temperature and pressure. Each should have 1 component. For more details, see Process variables.

Media properties

The HT process requires properties for the porous medium, the liquid phase, and the solid phase.

Medium properties

Liquid phase properties

$\large^{\star}$ Functional dependencies (e.g. on temperature) can be specified. See the OGS User Guide.

Solid phase properties

Features

Specific body force

The gravity vector is specified as:

<specific_body_force>0 -9.81</specific_body_force>

Solid thermal expansion

Optional coupling of thermal expansion effects on the pore pressure:

<solid_thermal_expansion>
    <thermal_expansion>alpha_s</thermal_expansion>
    <biot_constant>biot</biot_constant>
</solid_thermal_expansion>

where alpha_s and biot are parameter names defined in the <parameters> block.

Aperture size

For lower-dimensional fracture elements, an aperture size parameter can be specified:

<aperture_size>
    <parameter>fracture_aperture</parameter>
</aperture_size>

Surface flux calculation

Surface flux output can be configured:

<calculatesurfaceflux>
    ...
</calculatesurfaceflux>

Numerical stabilisation

The HT process supports numerical stabilisation methods for advection-dominated transport.

Benchmarks

See the OGS benchmark gallery for Hydro-Thermal examples.