Thermal Transfer#
Matrix Thermal#
Matrix thermal transfer describes heat storage and conduction within the intact solid material. Temperature gradients drive heat from warmer regions toward cooler regions, while heat sources, boundary conditions, and material properties determine the transient temperature field.
Fourier’s law#
The conductive heat-flux density is given by Fourier’s law:
where \(\mathbf{q}\) is heat flux per unit area, \(T\) is temperature, and \(\mathbf{K}\) is the thermal-conductivity tensor. The negative sign indicates that heat flows down the temperature gradient. For a two-dimensional domain, \(\nabla T=(\partial T/\partial x,\,\partial T/\partial y)\). For an isotropic material, \(\mathbf{K}=k\mathbf{I}\), where \(k\) is the scalar thermal conductivity.
Energy balance#
For a stationary solid with constant density and specific heat capacity, local energy conservation is
where \(\rho\) is mass density, \(C_p\) is specific heat capacity, and \(Q\) is the volumetric heat-generation rate. Combining this balance with Fourier’s law gives
For homogeneous isotropic material with constant \(k\), this becomes
The thermal diffusivity \(a_T=k/(\rho C_p)\) measures how quickly temperature disturbances spread through the matrix. If properties vary with temperature or position, the divergence form should be retained rather than replacing it with \(k\nabla^2T\).
Initial and boundary conditions#
Transient conduction requires an initial temperature field \(T(\mathbf{x},0)=T_0(\mathbf{x})\). Common boundary conditions include a prescribed temperature, a prescribed normal heat flux \(-\mathbf{n}\cdot\mathbf{K}\nabla T\), or an exchange condition with the surrounding environment. These conditions define how the matrix receives or loses heat and should be consistent with the physical heating or cooling process being represented.
The equations above assume a continuum matrix and do not by themselves represent heat exchange across a discontinuity. Heat transfer between separated blocks or across a fracture is described by the contact and fracture thermal-resistance models, respectively.
Fracture Heat Resistance#
A fracture interrupts direct heat conduction through the intact matrix. Heat may still cross the discontinuity through gas, liquid, or solid material in the fracture, as well as through microscopic asperity contacts between its faces. These mechanisms can be represented by an effective thermal resistance.
Series-resistance model#
Consider a fracture of aperture \(b\) filled with a material of thermal conductivity \(k_f\). For one-dimensional steady conduction normal to the fracture, the resistance per unit area of the filling layer is
If the two fracture faces also have interfacial resistances \(R''_{c,1}\) and \(R''_{c,2}\), the total resistance per unit area is
The corresponding effective conductance is \(h_f=1/R''_{\mathrm{tot}}\), and the heat flux from side 1 to side 2 is
Here \(T_1\) and \(T_2\) are the temperatures on the opposing sides of the fracture. For a fracture with negligible filling-layer resistance, \(b/k_f\) may be omitted; for a dry, open fracture, the effective conductance can instead be calibrated to account for gas conduction and radiation if those mechanisms are included in the intended model.
Interpretation and assumptions#
The series model assumes heat flows approximately normal to a locally planar fracture and that the layer and interface resistances act in series. The resistance has units of \(\mathrm{m^2\,K/W}\), while conductance has units of \(\mathrm{W/(m^2\,K)}\). For a contact area \(A\), the total heat transfer rate is \(\dot Q=Aq''_{1\rightarrow2}\). A larger aperture or lower filling conductivity increases the layer resistance and reduces heat transfer, all else being equal.
This effective fracture resistance represents heat transmission across the fracture itself. It is distinct from the contact-thermal exchange law used when two discrete elements or blocks are in thermal contact. In a numerical model, the effective conductance should reflect the chosen representation so that fracture resistance and contact resistance are not inadvertently counted twice.
Contact Thermal#
Contact thermal transfer exchanges heat between the surfaces of two discrete elements or blocks in contact. It provides a thermal connection across a mechanical contact and is commonly modeled with a conductance proportional to the temperature difference across the interface.
Interface law#
Let \(\Gamma_c\) denote the active contact interface, and let \(T_t\) and \(T_c\) be the temperatures on the target and contactor sides. The normal heat flux from target to contactor is
where \(h_c\) is the thermal contact conductance. The heat-transfer rate over the interface is
For a uniform conductance and temperature difference over an interface of area \(A_c\), this reduces to \(\dot Q_{t\rightarrow c}=h_cA_c(T_t-T_c)\). In a two-dimensional model, the measure \(\mathrm{d}\Gamma\) is a boundary length and the out-of-plane thickness must be included if a three-dimensional heat rate is required.
Energy conservation and interpretation#
The equal-and-opposite heat rate is applied to the other body:
Thus, contact exchange redistributes energy between the bodies but does not create or remove energy from the pair. If \(T_t>T_c\) and \(h_c>0\), heat flows from the target to the contactor. A larger conductance approaches the perfect-contact limit, while a small conductance represents a more thermally resistive interface. The conductance may be prescribed or calibrated as an effective parameter; its dependence on contact pressure, roughness, aperture, and interface material is model-specific.
Contact thermal transfer applies only where a contact interface is defined. Heat transfer across an open fracture or a gap requires a fracture-resistance or other gap-transfer model. Care is needed to avoid applying both laws to the same interface unless their resistances have been intentionally combined.
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
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
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
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
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
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.