Thermal-Hydromechanical Coupling

Thermal-hydromechanical (THM) coupling describes the interaction between temperature, fluid pressure and flow, and deformation. Heating or cooling changes the temperature field and produces thermal strain; deformation can in turn change fracture aperture and fluid storage, while fluid flow transports heat. The relative importance of these feedbacks depends on the material, loading, and time scale.

The high-temperature rock treatment described in the accompanying reference uses a one-way thermo-mechanical approximation: temperature drives thermal stress, while deformation has a negligible effect on heat transfer. The equations below present a broader THM framework and should be read as governing principles; particular constitutive laws, coupling terms, and sign conventions depend on the model setup.

Heat transport

In a fluid-bearing porous medium, heat is conducted through the solid-fluid mixture and may also be carried by moving fluid. A common local energy balance is

\[C_{\mathrm{eff}}\frac{\partial T}{\partial t} + \rho_f c_f\,\mathbf{v}\cdot\nabla T = \nabla\cdot(\mathbf{K}_{\mathrm{eff}}\nabla T) + Q,\]

where \(T\) is temperature, \(C_{\mathrm{eff}}\) is the effective volumetric heat capacity, \(\rho_f\) and \(c_f\) are the fluid density and specific heat, \(\mathbf{v}\) is the Darcy velocity, \(\mathbf{K}_{\mathrm{eff}}\) is the effective thermal-conductivity tensor, and \(Q\) is a volumetric heat source. The advection term may be omitted when fluid transport is absent or negligible. In an impermeable solid, this reduces to transient matrix heat conduction.

Fluid flow and pressure

For a single-phase fluid under Darcy flow, the Darcy velocity can be written as

\[\mathbf{v}=-\frac{\mathbf{k}}{\mu_f} \left(\nabla p-\rho_f\mathbf{g}\right),\]

where \(\mathbf{k}\) is intrinsic permeability, \(\mu_f\) is dynamic viscosity, \(p\) is pore pressure, and \(\mathbf{g}\) is gravitational acceleration. A simplified fluid mass balance is

\[S\frac{\partial p}{\partial t} + \alpha_B\frac{\partial \varepsilon_v}{\partial t} + \nabla\cdot\mathbf{v}=q_f,\]

where \(S\) is fluid-storage capacity, \(\alpha_B\) is the Biot coefficient, \(\varepsilon_v\) is volumetric strain, and \(q_f\) is a fluid source per unit bulk volume. This form illustrates pressure-deformation coupling; compressibility, temperature-dependent properties, and fracture storage can require additional terms.

Thermal and poroelastic deformation

For small strains, an isotropic thermal strain is

\[\boldsymbol{\varepsilon}_{\mathrm{th}} =\alpha_T(T-T_0)\mathbf{I},\]

where \(\alpha_T\) is the linear thermal-expansion coefficient, \(T_0\) is a reference temperature, and \(\mathbf{I}\) is the identity tensor. A representative linear poroelastic stress relation, with tension-positive stress and compression-positive pore pressure, is

\[\boldsymbol{\sigma} =\mathsf{C}: \left(\boldsymbol{\varepsilon}-\boldsymbol{\varepsilon}_{\mathrm{th}}\right) -\alpha_B p\mathbf{I},\]

where \(\mathsf{C}\) is the elastic stiffness tensor and \(\boldsymbol{\varepsilon}\) is total strain. Under idealized complete restraint of an isotropic solid, the thermal stress increment is hydrostatic and has magnitude \(3K\alpha_T\Delta T\), where \(K\) is the bulk modulus. The actual stress state depends on constraints, geometry, and constitutive assumptions; this limiting expression is not a general plane- strain formula.

In fractured media, the coupling may also occur through deformation-dependent fracture aperture and permeability. These feedbacks connect the thermal, hydraulic, and mechanical fields and can be important even when the intact matrix is treated as impermeable.