[TRM] A detailed documentation of ThermoRichardsMechanicsProcess

Documentation/ProjectFile/prj/processes/process/THERMO_RICHARDS_MECHANICS/c_THERMO_RICHARDS_MECHANICS.md

-Non-isothermal Richards flow process coupled with deformation process in
-porous media. It is for coupled thermal hydraulic mechanical processes.
+\copydoc ProcessLib::ThermoRichardsMechanics::ThermoRichardsMechanicsProcess

ProcessLib/ThermoRichardsMechanics/ThermoRichardsMechanicsProcess.h

@@ -18,9 +18,93 @@ namespace ProcessLib
 {
 namespace ThermoRichardsMechanics
 {
-/// Global assembler for the monolithic scheme of the non-isothermal Richards
-/// flow coupled with mechanics.
-///
+/**
+ * \brief Global assembler for the monolithic scheme of the non-isothermal
+ * Richards flow coupled with mechanics.
+ *
+ * <b>Governing equations without vapor diffusion</b>
+ *
+ * The energy balance equation is given by
+ * \f[
+ *  (\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
+ * \f]
+ *  with\f$T\f$ the temperature, \f$(\rho c_p)^{eff}\f$ the  effective
+ * volumetric heat
+ * capacity, \f$\mathbf{k}_T^{eff} \f$
+ *  the effective thermal conductivity, \f$\rho^l\f$ the density of liquid,
+ * \f$c_p^l\f$ the specific heat  capacity of liquid, \f$\mathbf{v}^l\f$ the
+ * liquid velocity, and \f$Q_T\f$ 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
+ * \f[
+ * (\rho c_p)^{eff} = (1-\phi) \rho^s c_p^s + S^l \phi \rho^l c_p^l
+ * \f]
+ * with \f$\phi\f$ the porosity, \f$S^l\f$  the liquid saturation, \f$\rho^s \f$
+ * the solid density, and \f$c_p^s\f$ the specific heat capacity of solid.
+ * Similarly, the effective thermal conductivity is given by
+ * \f[
+ * \mathbf{k}_T^{eff} = (1-\phi) \mathbf{k}_T^s + S^l \phi k_T^l \mathbf I
+ * \f]
+ * where \f$\mathbf{k}_T^s\f$ is the thermal conductivity tensor of solid, \f$
+ *  k_T^l\f$ is the thermal conductivity of liquid, and \f$\mathbf I\f$ is the
+ * identity tensor.
+ *
+ * The mass balance equation is given by
+ * \f{eqnarray*}{
+ * \left(S^l\beta - \phi\frac{\partial S}{\partial p_c}\right) \rho^l\dot p
+ * - H(S-1)\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
+ * \f}
+ * where \f$p\f$ is the pore pressure,  \f$p_c\f$ is the
+ * capillary pressure, which is \f$-p\f$ under the single phase assumption,
+ *  \f$\beta\f$ is a composite coefficient by the liquid compressibility and
+ * solid compressibility, \f$\alpha_B\f$ is the Biot's constant,
+ * \f$\alpha_T^s\f$ is the linear thermal  expansivity of solid, \f$Q_H\f$
+ * is the point source or sink term,  \f$ \mathbf u\f$ is the displacement, and
+ * \f$H(S-1)\f$ is the Heaviside function.
+ * The liquid velocity \f$\mathbf{v}^l\f$ is
+ * described by the Darcy's law as
+ * \f[
+ * \mathbf{v}^l=-\frac{{\mathbf k} k_{ref}}{\mu} (\nabla p - \rho^l \mathbf g)
+ * \f]
+ * with \f${\mathbf k}\f$ the intrinsic permeability, \f$k_{ref}\f$ the relative
+ * permeability, \f$\mathbf g\f$ the gravitational force.
+ *
+ * The momentum balance equation takes the form of
+ * \f[
+ * \nabla (\mathbf{\sigma}-b(S)\alpha_B p^l \mathbf I) +\mathbf f=0
+ * \f]
+ * with \f$\mathbf{\sigma}\f$  the effective stress tensor, \f$b(S)\f$ the
+ * Bishop model, \f$\mathbf f\f$ the body force, and \f$\mathbf I\f$ the
+ * identity. The primary unknowns of the momentum balance equation are the
+ * displacement \f$\mathbf u\f$, which is associated with the stress by the
+ * the generalized Hook's law as
+ * \f[
+ * {\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 )
+ * \f]
+ *  with \f$C\f$ the forth order elastic tensor,
+ *  \f${\dot {\mathbf \epsilon}}\f$ the total strain rate,
+ *  \f${\dot {\mathbf \epsilon}}^e\f$ the elastic strain rate,
+ *  \f${\dot {\mathbf \epsilon}}^T\f$ the thermal strain rate,
+ *  \f${\dot {\mathbf \epsilon}}^p\f$ the plastic strain rate,
+ *  \f${\dot {\mathbf \epsilon}}^{sw}\f$ the swelling strain rate.
+ *
+ *  The strain tensor is given by displacement vector as
+ *  \f[
+ *   \mathbf \epsilon =
+ *   \frac{1}{2} \left((\nabla \mathbf u)^{\text T}+\nabla \mathbf u\right)
+ * \f]
+ * where the superscript \f${\text T}\f$ means transpose,
+ */
 template <int DisplacementDim>
 class ThermoRichardsMechanicsProcess final : public Process
 {