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

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

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, \( \mathbf u\) 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

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

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

The momentum balance equation takes the form of

\[ \nabla (\mathbf{\sigma}-b(S)\alpha_B p^l \mathbf I) +\mathbf f=0 \]

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 the generalized Hook's law as

\[ {\dot {\mathbf {\sigma}}} = C {\dot {\mathbf \epsilon}}^e = C ( {\dot {\mathbf \epsilon}} - {\dot {\mathbf \epsilon}}^T -{\dot {\mathbf \epsilon}}^p - {\dot {\mathbf \epsilon}}^{sw}-\cdots ) \]

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

\[ \mathbf \epsilon = \frac{1}{2} \left((\nabla \mathbf u)^{\text T}+\nabla \mathbf u\right) \]

where the superscript \({\text T}\) means transpose,

Child parameters, attributes and cases

Additional info

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Used in the following test data files