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

Governing equations without vapor diffusion

The energy balance equation is given by

\[ (\rho c_p)^{eff}\dot T - \nabla (\mathbf{k}_T^{eff} \nabla T)+\rho^l c_p^l \nabla T \cdot \mathbf{v}^l = Q_T \]

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

\[ (\rho c_p)^{eff} = (1-\phi) \rho^s c_p^s + S^l \phi \rho^l c_p^l \]

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

\[ \mathbf{k}_T^{eff} = (1-\phi) \mathbf{k}_T^s + S^l \phi k_T^l \mathbf I \]

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

\begin{eqnarray*} \left(S^l\beta - \phi\frac{\partial S}{\partial p_c}\right) \rho^l\dot p - S \left( \frac{\partial \rho^l}{\partial T} +\rho^l(\alpha_B -S) \alpha_T^s \right)\dot T\\ +\nabla (\rho^l \mathbf{v}^l) + S \alpha_B \rho^l \nabla \cdot \dot {\mathbf u}= Q_H \end{eqnarray*}

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, \(H(S-1)\) is the Heaviside function, and \( \mathbf u\) is the displacement. While this process does not contain a fully mechanical coupling, simplfied expressions can be given to approximate the latter term under certain stress conditions. The liquid velocity \(\mathbf{v}^l\) is described by the Darcy's law as

\[ \mathbf{v}^l=-\frac{{\mathbf k} k_{ref}}{\mu} (\nabla p - \rho^l \mathbf g) \]

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

Child parameters, attributes and cases

Additional info

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Used in the following test data files