 OGS
[case] THERMO_RICHARDS_MECHANICS

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,