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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,

Child parameters, attributes and cases

This process is commonly used together with the following media properties

Note: This list has been automatically extracted from OGS's benchmark tests (ctests). Therefore it might not be exhaustive, but it should give users a good overview about which properties they can/have to use with this process. Probably most of the properties occurring in this list are mandatory.

The list might contain different property <type>s for some property <name> to illustrate different possibilities the users have.

Additional info

No additional info.

Used in the following test data files