Detailed Description

RichardsComponentTransport process

Governing Equations

Richards Flow

The flow process is described by

\[ \phi \frac{\partial \rho_w}{\partial p} \frac{\partial p}{\partial t} S - \phi \rho_w \frac{\partial S}{\partial p_c} \frac{\partial p_c}{\partial t} - \nabla \cdot \left[\rho_w \frac{k_{\mathrm{rel}} \kappa}{\mu} \nabla \left( p + \rho_w g z \right)\right] - Q_p = 0, \]

where

\(\phi\) is the porosity,
\(S\) is the saturation,
\(p\) is the pressure,
\(k_{\mathrm{rel}}\) is the relative permeability (depending on \(S\)),
\(\kappa\) is the specific permeability,
\(\mu\) is viscosity of the fluid,
\(\rho_w\) is the mass density of the fluid, and
\(g\) is the gravitational acceleration.

Here it is assumed, that

the porosity is constant and
the pressure of the gas phase is zero.

The capillary pressure is given by

\[ p_c = \frac{\rho_w g}{\alpha} \left[S_{\mathrm{eff}}^{-\frac{1}{m}} - 1\right]^{\frac{1}{n}} \]

and the effective saturation by

\[ S_{\mathrm{eff}} = \frac{S-S_r}{S_{\max} - S_r} \]

Mass Transport

The mass transport process is described by

\[ \phi R \frac{\partial C}{\partial t} + \nabla \cdot \left(\vec{q} C - D \nabla C \right) + \phi R \vartheta C - Q_C = 0 \]

where

\(R\) is the retardation factor,
\(C\) is the concentration,
\(\vec{q} = \frac{k_{\mathrm{rel}} \kappa}{\mu(C)} \nabla \left( p + \rho_w g z \right)\) is the Darcy velocity,
\(D\) is the hydrodynamic dispersion tensor,
\(\vartheta\) is the decay rate.

For the hydrodynamic dispersion tensor the relation

\[ D = (\phi D_d + \beta_T \|\vec{q}\|) I + (\beta_L - \beta_T) \frac{\vec{q} \vec{q}^T}{\|\vec{q}\|} \]

is implemented, where \(D_d\) is the molecular diffusion coefficient, \(\beta_L\) the longitudinal dispersivity of chemical species, and \(\beta_T\) the transverse dispersivity of chemical species.

The implementation uses a monolithic approach, i.e., both processes are assembled within one global system of equations.

Process Coupling

The advective term of the concentration equation is given by the Richards flow process, i.e., the concentration distribution depends on darcy velocity of the Richards flow process. On the other hand the concentration dependencies of the viscosity and density in the groundwater flow couples the unsaturated H process to the C process.

Note: At the moment there is not any coupling by source or sink terms, i.e., the coupling is implemented only by density changes due to concentration changes in the buoyance term in the groundwater flow. This coupling schema is referred to as the Boussinesq approximation.

Definition at line 102 of file RichardsComponentTransportProcess.h.

#include <RichardsComponentTransportProcess.h>

Inheritance diagram for ProcessLib::RichardsComponentTransport::RichardsComponentTransportProcess:

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Collaboration diagram for ProcessLib::RichardsComponentTransport::RichardsComponentTransportProcess:

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Public Member Functions
	RichardsComponentTransportProcess (std::string name, MeshLib::Mesh &mesh, std::unique_ptr< ProcessLib::AbstractJacobianAssembler > &&jacobian_assembler, std::vector< std::unique_ptr< ParameterLib::ParameterBase > > const &parameters, unsigned const integration_order, std::vector< std::vector< std::reference_wrapper< ProcessVariable > > > &&process_variables, RichardsComponentTransportProcessData &&process_data, SecondaryVariableCollection &&secondary_variables, bool const use_monolithic_scheme)

ODESystem interface
bool	isLinear () const override

Public Member Functions inherited from ProcessLib::Process
	Process (std::string name_, MeshLib::Mesh &mesh, std::unique_ptr< AbstractJacobianAssembler > &&jacobian_assembler, std::vector< std::unique_ptr< ParameterLib::ParameterBase > > const &parameters, unsigned const integration_order, std::vector< std::vector< std::reference_wrapper< ProcessVariable > > > &&process_variables, SecondaryVariableCollection &&secondary_variables, const bool use_monolithic_scheme=true)

void	preTimestep (std::vector< GlobalVector * > const &x, const double t, const double delta_t, const int process_id)
	Preprocessing before starting assembly for new timestep.

void	postTimestep (std::vector< GlobalVector * > const &x, std::vector< GlobalVector * > const &x_prev, const double t, const double delta_t, int const process_id)
	Postprocessing after a complete timestep.

void	postNonLinearSolver (std::vector< GlobalVector * > const &x, std::vector< GlobalVector * > const &x_prev, const double t, double const dt, int const process_id)

void	preIteration (const unsigned iter, GlobalVector const &x) final

void	computeSecondaryVariable (double const t, double const dt, std::vector< GlobalVector * > const &x, GlobalVector const &x_prev, int const process_id)
	compute secondary variables for the coupled equations or for output.

NumLib::IterationResult	postIteration (GlobalVector const &x) final

void	initialize (std::map< int, std::shared_ptr< MaterialPropertyLib::Medium > > const &media)

void	setInitialConditions (std::vector< GlobalVector * > &process_solutions, std::vector< GlobalVector * > const &process_solutions_prev, double const t, int const process_id)

MathLib::MatrixSpecifications	getMatrixSpecifications (const int process_id) const override

void	updateDeactivatedSubdomains (double const time, const int process_id)

virtual bool	isMonolithicSchemeUsed () const

virtual void	extrapolateIntegrationPointValuesToNodes (const double, std::vector< GlobalVector * > const &, std::vector< GlobalVector * > &)

void	preAssemble (const double t, double const dt, GlobalVector const &x) final

void	assemble (const double t, double const dt, std::vector< GlobalVector * > const &x, std::vector< GlobalVector * > const &x_prev, int const process_id, GlobalMatrix &M, GlobalMatrix &K, GlobalVector &b) final

void	assembleWithJacobian (const double t, double const dt, std::vector< GlobalVector * > const &x, std::vector< GlobalVector * > const &x_prev, int const process_id, GlobalVector &b, GlobalMatrix &Jac) final

void	preOutput (const double t, double const dt, std::vector< GlobalVector * > const &x, std::vector< GlobalVector * > const &x_prev, int const process_id)

std::vector< NumLib::IndexValueVector< GlobalIndexType > > const *	getKnownSolutions (double const t, GlobalVector const &x, int const process_id) const final

virtual NumLib::LocalToGlobalIndexMap const &	getDOFTable (const int) const

MeshLib::Mesh &	getMesh () const

std::vector< std::vector< std::reference_wrapper< ProcessVariable > > > const &	getProcessVariables () const

std::vector< std::reference_wrapper< ProcessVariable > > const &	getProcessVariables (const int process_id) const

std::vector< std::size_t > const &	getActiveElementIDs () const

SecondaryVariableCollection const &	getSecondaryVariables () const

std::vector< std::unique_ptr< MeshLib::IntegrationPointWriter > > const &	getIntegrationPointWriters () const

virtual Eigen::Vector3d	getFlux (std::size_t, MathLib::Point3d const &, double const, std::vector< GlobalVector * > const &) const

virtual void	solveReactionEquation (std::vector< GlobalVector * > &, std::vector< GlobalVector * > const &, double const, double const, NumLib::EquationSystem &, int const)

bool	requiresNormalization () const override

Public Member Functions inherited from ProcessLib::SubmeshAssemblySupport
virtual std::vector< std::vector< std::string > >	initializeAssemblyOnSubmeshes (std::vector< std::reference_wrapper< MeshLib::Mesh > > const &meshes)

virtual	~SubmeshAssemblySupport ()=default

Private Member Functions
void	initializeConcreteProcess (NumLib::LocalToGlobalIndexMap const &dof_table, MeshLib::Mesh const &mesh, unsigned const integration_order) override
	Process specific initialization called by initialize().

void	assembleConcreteProcess (const double t, double const dt, std::vector< GlobalVector * > const &x, std::vector< GlobalVector * > const &x_prev, int const process_id, GlobalMatrix &M, GlobalMatrix &K, GlobalVector &b) override

void	assembleWithJacobianConcreteProcess (const double t, double const dt, std::vector< GlobalVector * > const &x, std::vector< GlobalVector * > const &x_prev, int const process_id, GlobalVector &b, GlobalMatrix &Jac) override

Private Attributes
RichardsComponentTransportProcessData	_process_data

std::vector< std::unique_ptr< RichardsComponentTransportLocalAssemblerInterface > >	_local_assemblers

Additional Inherited Members
Public Attributes inherited from ProcessLib::Process
std::string const	name

Static Public Attributes inherited from ProcessLib::Process
static PROCESSLIB_EXPORT const std::string	constant_one_parameter_name = "constant_one"

Protected Member Functions inherited from ProcessLib::Process
std::vector< NumLib::LocalToGlobalIndexMap const * >	getDOFTables (int const number_of_processes) const

NumLib::Extrapolator &	getExtrapolator () const

NumLib::LocalToGlobalIndexMap const &	getSingleComponentDOFTable () const

void	initializeProcessBoundaryConditionsAndSourceTerms (const NumLib::LocalToGlobalIndexMap &dof_table, const int process_id, std::map< int, std::shared_ptr< MaterialPropertyLib::Medium > > const &media)

virtual void	constructDofTable ()

void	constructMonolithicProcessDofTable ()

void	constructDofTableOfSpecifiedProcessStaggeredScheme (const int specified_process_id)

virtual std::tuple< NumLib::LocalToGlobalIndexMap *, bool >	getDOFTableForExtrapolatorData () const

std::vector< GlobalIndexType >	getIndicesOfResiduumWithoutInitialCompensation () const override

Protected Attributes inherited from ProcessLib::Process
MeshLib::Mesh &	_mesh

std::unique_ptr< MeshLib::MeshSubset const >	_mesh_subset_all_nodes

std::unique_ptr< NumLib::LocalToGlobalIndexMap >	_local_to_global_index_map

SecondaryVariableCollection	_secondary_variables

CellAverageData	cell_average_data_

std::unique_ptr< ProcessLib::AbstractJacobianAssembler >	_jacobian_assembler

VectorMatrixAssembler	_global_assembler

const bool	_use_monolithic_scheme

unsigned const	_integration_order

std::vector< std::unique_ptr< MeshLib::IntegrationPointWriter > >	_integration_point_writer

GlobalSparsityPattern	_sparsity_pattern

std::vector< std::vector< std::reference_wrapper< ProcessVariable > > >	_process_variables

std::vector< BoundaryConditionCollection >	_boundary_conditions

Constructor & Destructor Documentation