Finite Volume discretization of a two-phase flow pressure equation of the sequential IMPES model. More...

#include <dumux/porousmediumflow/2p/sequential/diffusion/cellcentered/pressure.hh>

Inheritance diagram for Dumux::FVPressure2P< TypeTag >:

Description

template<class TypeTag>
class Dumux::FVPressure2P< TypeTag >

Finite Volume discretization of a two-phase flow pressure equation of the sequential IMPES model.

This model implements two-phase flow of two immiscible fluids \(\alpha \in \{ w, n \}\) using a standard multi-phase Darcy approach as the equation for the conservation of momentum, i.e.

\[ v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \textbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_{\alpha} {\textbf g} \right). \]

By inserting this into the equation for the conservation of the phase mass, one gets

\[ \phi \frac{\partial \varrho_\alpha S_\alpha}{\partial t} - \text{div} \left\{ \varrho_\alpha \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\textbf{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g} \right) \right\} = q_\alpha \;. \]

In the incompressible case the phase densities \( \varrho_\alpha \) can be eliminated from the equations. The two equations are then added up, and because \(S_w + S_n = 1\), the first term cancels out. This leads to the so-called pressure equation. For the wetting ( \(w\)) phase pressure as primary variable this yields

\[ - \text{div}\, \left[\lambda \boldsymbol K \left(\textbf{grad}\, p_w + f_n \textbf{grad}\, p_c - \sum f_\alpha \varrho_\alpha {\textbf g}\right)\right] = q, \]

for the non-wetting ( \( n \)) phase pressure as primary variable it yields

\[ - \text{div}\, \left[\lambda \boldsymbol K \left(\textbf{grad}\, p_n - f_w \textbf{grad}\, p_c - \sum f_\alpha \varrho_\alpha {\textbf g}\right)\right] = q, \]

and for the global pressure as primary variable it leads to

\[ - \text{div}\, \left[\lambda \boldsymbol K \left(\textbf{grad}\, p_{global} - \sum f_\alpha \varrho_\alpha {\textbf g}\right)\right] = q. \]

Here, \( p_\alpha \) is a phase pressure, \( p_ {global} \) the global pressure of a classical fractional flow formulation (see e.g. Binning and Celia (1999) [11]), \( p_c = p_n - p_w \) is the capillary pressure, \( \boldsymbol K \) the absolute permeability tensor, \( \lambda = \lambda_w + \lambda_n \) the total mobility depending on the saturation ( \( \lambda_\alpha = k_{r_\alpha} / \mu_\alpha \)), \( f_\alpha = \lambda_\alpha / \lambda \) the fractional flow function of a phase, \( \varrho_\alpha \) a phase density, \( {\textbf g} \) the gravitational acceleration vector and \( q = q_w + q_n \) the total source term. Depending on the primary variable chosen, one of the pressure equations above is solved sequentially together with the conservation of the phase mass for one phase.

For all cases, \( p = p_D \) on \( \Gamma_{Dirichlet} \), and \( \boldsymbol v_{total} \cdot \boldsymbol n = q_N \) on \( \Gamma_{Neumann} \).

The slightly compressible case is only implemented for phase pressures! In this case for a wetting \((w) \) phase pressure as primary variable the pressure equation is formulated as

\[ \phi \left( \varrho_w \frac{\partial S_w}{\partial t} + \varrho_n \frac{\partial S_n}{\partial t}\right) - \text{div}\, \left[\lambda \boldsymbol{K} \left(\textbf{grad}\, p_w + f_n \, \textbf{grad}\, p_c - \sum f_\alpha \varrho_\alpha {\textbf g}\right)\right] = q, \]

and for a non-wetting ( \( n \)) phase pressure as primary variable as

\[ \phi \left( \varrho_w \frac{\partial S_w}{\partial t} + \varrho_n \frac{\partial S_n}{\partial t}\right) - \text{div}\, \left[\lambda \boldsymbol{K} \left(\textbf{grad}\, p_n - f_w \textbf{grad}\, p_c - \sum f_\alpha \varrho_\alpha {\textbf g}\right)\right] = q. \]

In this slightly compressible case the following definitions are valid: \( \lambda = \varrho_w \lambda_w + \varrho_n \lambda_n \), \( f_\alpha = (\varrho_\alpha \lambda_\alpha) / \lambda \) This model assumes that temporal changes in density are very small and thus terms of temporal derivatives are negligible in the pressure equation. Depending on the formulation the terms including time derivatives of saturations are simplified by inserting \( S_w + S_n = 1 \).

In the IMPES models the default setting is:

formulation: \( p_w-S_w \) (Property: Formulation defined as SequentialTwoPCommonIndices::pwsw)
compressibility: disabled (Property: EnableCompressibility set to false)

Template Parameters

TypeTag The Type Tag

Public Member Functions
void	getSource (EntryType &entry, const Element &element, const CellData &cellData, const bool first)
	Function which calculates the source entry. More...

void	getStorage (EntryType &entry, const Element &element, const CellData &cellData, const bool first)
	Function which calculates the storage entry. More...

void	getFlux (EntryType &entry, const Intersection &intersection, const CellData &cellData, const bool first)
	Function which calculates the flux entry. More...

void	getFluxOnBoundary (EntryType &entry, const Intersection &intersection, const CellData &cellData, const bool first)
	Function which calculates the boundary flux entry. More...

void	updateMaterialLaws ()
	Updates and stores constitutive relations. More...

void	initialize (bool solveTwice=true)
	Initializes the pressure model. More...

void	update ()
	Pressure update. More...

void	updateVelocity ()
	Velocity update. More...

void	storePressureSolution ()
	Globally stores the pressure solution. More...

void	storePressureSolution (const Element &element)
	Stores the pressure solution of a cell. More...

template<class MultiWriter >
void	addOutputVtkFields (MultiWriter &writer)
	Adds pressure output to the output file. More...

	FVPressure2P (Problem &problem)
	Constructs a FVPressure2P object. More...

const Scalar	pressure (const int eIdxGlobal) const
	Public access function for the primary pressure variable. More...

const Matrix &	globalMatrix ()
	Returns the global matrix of the last pressure solution step. More...

const RHSVector &	rightHandSide ()
	Returns the right hand side vector of the last pressure solution step. More...

void	initialize ()
	Initialize pressure model. More...

void	calculateVelocity ()

void	serializeEntity (std::ostream &outstream, const Element &element)
	Function for serialization of the pressure field. More...

void	deserializeEntity (std::istream &instream, const Element &element)
	Function for deserialization of the pressure field. More...

void	setFixPressureAtIndex (Scalar pressure, int eIdxGlobal)
	Set a pressure to be fixed at a certain cell. More...

void	unsetFixPressureAtIndex (int eIdxGlobal)
	Reset the fixed pressure state. More...

void	resetFixPressureAtIndex ()

Protected Types
enum	{ rhs = 1 , matrix = 0 }
	Indices of matrix and rhs entries. More...

enum	{ pressEqIdx = Indices::pressureEqIdx }

using	EntryType = Dune::FieldVector< Scalar, 2 >

Protected Member Functions
void	initializeMatrix ()
	Initialize the global matrix of the system of equations to solve. More...

void	initializeMatrixRowSize ()
	Initialize the global matrix of the system of equations to solve. More...

void	initializeMatrixIndices ()
	Initialize the global matrix of the system of equations to solve. More...

void	assemble (bool first)
	Function which assembles the system of equations to be solved. More...

void	solve ()
	Solves the global system of equations to get the spatial distribution of the pressure. More...

PressureSolution &	pressure ()
	Returns the vector containing the pressure solution. More...

const PressureSolution &	pressure () const
	Returns the vector containing the pressure solution. More...

void	initializePressure ()
	Initialization of the pressure solution vector: Initialization with meaningful values may. More...

Protected Attributes
Matrix	A_
	Global stiffnes matrix (sparse matrix which is build by the `initializeMatrix()` function) More...

RHSVector	f_
	Right hand side vector. More...

Member Typedef Documentation

◆ EntryType

template<class TypeTag >

using Dumux::FVPressure< TypeTag >::EntryType = Dune::FieldVector<Scalar, 2>

protectedinherited

Type of the vector of entries

Contains the return values of the get*()-functions (matrix or right-hand side entry).

Member Enumeration Documentation

◆ anonymous enum

template<class TypeTag >

anonymous enum

protectedinherited

Indices of matrix and rhs entries.

During the assembling of the global system of equations get-functions are called (getSource(), getFlux(), etc.), which return global matrix or right hand side entries in a vector. These can be accessed using following indices:

Enumerator
rhs	index for the right hand side entry
matrix	index for the global matrix entry

◆ anonymous enum

template<class TypeTag >

anonymous enum

protectedinherited

Enumerator
pressEqIdx

Constructor & Destructor Documentation

◆ FVPressure2P()

template<class TypeTag >

Dumux::FVPressure2P< TypeTag >::FVPressure2P ( Problem & problem )

inline

Constructs a FVPressure2P object.

Parameters

problem A problem class object

Member Function Documentation

◆ addOutputVtkFields()

template<class TypeTag >

template<class MultiWriter >

void Dumux::FVPressure2P< TypeTag >::addOutputVtkFields ( MultiWriter & writer )

inline

Adds pressure output to the output file.

Adds the phase pressures or a global pressure (depending on the formulation) as well as the capillary pressure to the output. In the compressible case additionally density and viscosity are added.

Template Parameters

MultiWriter Class defining the output writer

Parameters

writer The output writer (usually a VTKMultiWriter object)

◆ assemble()

template<class TypeTag >

void Dumux::FVPressure< TypeTag >::assemble ( bool first )

protectedinherited

Function which assembles the system of equations to be solved.

This function assembles the Matrix and the right hand side (RHS) vector to solve for a pressure field with a Finite-Volume (FV) discretization. Implementations of this base class have to provide the methods getSource(), getStorage(), getFlux() and getFluxOnBoundary() if the assemble() method is called!

Parameters

first Indicates if function is called at the initialization step or during the simulation (If first is true, no pressure field of previous iterations is required)

◆ calculateVelocity()

template<class TypeTag >

void Dumux::FVPressure< TypeTag >::calculateVelocity ( )

inlineinherited

◆ deserializeEntity()

template<class TypeTag >

void Dumux::FVPressure< TypeTag >::deserializeEntity	(	std::istream &	instream,
		const Element &	element
	)

inlineinherited

Function for deserialization of the pressure field.

Function needed for restart option. Reads the pressure of a grid element from a restart file.

Parameters

instream	Stream from the restart file.
element	Grid element

◆ getFlux()

template<class TypeTag >

void Dumux::FVPressure2P< TypeTag >::getFlux	(	EntryType &	entry,
		const Intersection &	intersection,
		const CellData &	cellData,
		const bool	first
	)

Function which calculates the flux entry.

Function computes the inter-cell flux term and writes it to the corresponding entry of the entry vector

Parameters

entry	Vector containing return values of the function
intersection	Intersection of two grid elements
cellData	Object containing all model relevant cell data
first	Indicates if function is called in the initialization step or during the simulation

◆ getFluxOnBoundary()

template<class TypeTag >

void Dumux::FVPressure2P< TypeTag >::getFluxOnBoundary	(	EntryType &	entry,
		const Intersection &	intersection,
		const CellData &	cellData,
		const bool	first
	)

Function which calculates the boundary flux entry.

Function which calculates the flux entry at a boundary.

Function computes the boundary-flux term and writes it to the corresponding entry of the entry vector

Parameters

entry	Vector containing return values of the function
intersection	Intersection of two grid elements
cellData	Object containing all model relevant cell data
first	Indicates if function is called in the initialization step or during the simulation

Dirichlet boundary condition for pressure depends on the formulation ( \(p_w\) (default), \(p_n\), \(p_{global}\)), Neumann boundary condition are the phase mass fluxes ( \(q_w\) and \(q_n\), [ \(\text{kg}/(\text{m}^2 \text{s}\)])

◆ getSource()

template<class TypeTag >

void Dumux::FVPressure2P< TypeTag >::getSource	(	EntryType &	entry,
		const Element &	element,
		const CellData &	cellData,
		const bool	first
	)

Function which calculates the source entry.

Function computes the source term and writes it to the corresponding entry of the entry vector

Parameters

entry	Vector containing return values of the function
element	Grid element
cellData	Object containing all model relevant cell data
first	Indicates if function is called in the initialization step or during the simulation

Source of each fluid phase has to be added as mass flux ( \(\text{kg}/(\text{m}^3 \text{s}\)).

◆ getStorage()

template<class TypeTag >

void Dumux::FVPressure2P< TypeTag >::getStorage	(	EntryType &	entry,
		const Element &	element,
		const CellData &	cellData,
		const bool	first
	)

Function which calculates the storage entry.

Function computes the storage term and writes it to the corresponding entry of the entry vector

Parameters

entry	Vector containing return values of the function
element	Grid element
cellData	Object containing all model relevant cell data
first	Indicates if function is called in the initialization step or during the simulation

If compressibility is enabled this functions calculates the term

\[ \phi \sum_\alpha \varrho_\alpha \frac{\partial S_\alpha}{\partial t} V \]

In the incompressible case an volume correction term is calculated which corrects for non-physical saturation overshoots/undershoots. These can occur if the estimated time step for the explicit transport was too large. Correction by an artificial source term allows to correct this errors due to wrong time-stepping without losing mass conservation. The error term looks as follows:

\[ q_{error} = \begin{cases} S < 0 & a_{error} \frac{S}{\Delta t} V \\ S > 1 & a_{error} \frac{(S - 1)}{\Delta t} V \\ 0 \le S \le 1 & 0 \end{cases} \]

where \(a_{error}\) is a weighting factor (default: \(a_{error} = 0.5\))

◆ globalMatrix()

template<class TypeTag >

const Matrix & Dumux::FVPressure< TypeTag >::globalMatrix ( )

inlineinherited

Returns the global matrix of the last pressure solution step.

◆ initialize() [1/2]

template<class TypeTag >

void Dumux::FVPressure< TypeTag >::initialize ( )

inlineinherited

Initialize pressure model.

Function initializes the sparse matrix to solve the global system of equations and sets/calculates the initial pressure

◆ initialize() [2/2]

template<class TypeTag >

void Dumux::FVPressure2P< TypeTag >::initialize ( bool solveTwice = true )

inline

Initializes the pressure model.

Function initializes the sparse matrix to solve the global system of equations and sets/calculates the initial pressure

Parameters

solveTwice indicates if more than one iteration is allowed to get an initial pressure solution

◆ initializeMatrix()

template<class TypeTag >

void Dumux::FVPressure< TypeTag >::initializeMatrix

protectedinherited

Initialize the global matrix of the system of equations to solve.

◆ initializeMatrixIndices()

template<class TypeTag >

void Dumux::FVPressure< TypeTag >::initializeMatrixIndices

protectedinherited

Initialize the global matrix of the system of equations to solve.

◆ initializeMatrixRowSize()

template<class TypeTag >

void Dumux::FVPressure< TypeTag >::initializeMatrixRowSize

protectedinherited

Initialize the global matrix of the system of equations to solve.

◆ initializePressure()

template<class TypeTag >

void Dumux::FVPressure< TypeTag >::initializePressure ( )

inlineprotectedinherited

Initialization of the pressure solution vector: Initialization with meaningful values may.

◆ pressure() [1/3]

template<class TypeTag >

PressureSolution & Dumux::FVPressure< TypeTag >::pressure ( )

inlineprotectedinherited

Returns the vector containing the pressure solution.

◆ pressure() [2/3]

template<class TypeTag >

const PressureSolution & Dumux::FVPressure< TypeTag >::pressure ( ) const

inlineprotectedinherited

Returns the vector containing the pressure solution.

◆ pressure() [3/3]

template<class TypeTag >

const Scalar Dumux::FVPressure< TypeTag >::pressure ( const int eIdxGlobal ) const

inlineinherited

Public access function for the primary pressure variable.

Function returns the cell pressure value at index eIdxGlobal

Parameters

eIdxGlobal Global index of a grid cell

◆ resetFixPressureAtIndex()

template<class TypeTag >

void Dumux::FVPressure< TypeTag >::resetFixPressureAtIndex ( )

inlineinherited

◆ rightHandSide()

template<class TypeTag >

const RHSVector & Dumux::FVPressure< TypeTag >::rightHandSide ( )

inlineinherited

Returns the right hand side vector of the last pressure solution step.

◆ serializeEntity()

template<class TypeTag >

void Dumux::FVPressure< TypeTag >::serializeEntity	(	std::ostream &	outstream,
		const Element &	element
	)

inlineinherited

Function for serialization of the pressure field.

Function needed for restart option. Writes the pressure of a grid element to a restart file.

Parameters

outstream	Stream into the restart file.
element	Grid element

◆ setFixPressureAtIndex()

template<class TypeTag >

void Dumux::FVPressure< TypeTag >::setFixPressureAtIndex	(	Scalar	pressure,
		int	eIdxGlobal
	)

inlineinherited

Set a pressure to be fixed at a certain cell.

Allows to fix a pressure somewhere (at one certain cell) in the domain. This can be necessary e.g. if only Neumann boundary conditions are defined. The pressure is fixed until the unsetFixPressureAtIndex() function is called

Parameters

pressure	Pressure value at eIdxGlobal
eIdxGlobal	Global index of a grid cell

◆ solve()

template<class TypeTag >

void Dumux::FVPressure< TypeTag >::solve

protectedinherited

Solves the global system of equations to get the spatial distribution of the pressure.

◆ storePressureSolution() [1/2]

template<class TypeTag >

void Dumux::FVPressure2P< TypeTag >::storePressureSolution ( )

inline

Globally stores the pressure solution.

◆ storePressureSolution() [2/2]

template<class TypeTag >

void Dumux::FVPressure2P< TypeTag >::storePressureSolution ( const Element & element )

inline

Stores the pressure solution of a cell.

Calculates secondary pressure variables and stores pressures.

Parameters

element Grid element

◆ unsetFixPressureAtIndex()

template<class TypeTag >

void Dumux::FVPressure< TypeTag >::unsetFixPressureAtIndex ( int eIdxGlobal )

inlineinherited

Reset the fixed pressure state.

No pressure is fixed inside the domain until setFixPressureAtIndex() function is called again.

Parameters

eIdxGlobal Global index of a grid cell

◆ update()

template<class TypeTag >

void Dumux::FVPressure2P< TypeTag >::update ( )

inline

Pressure update.

Function reassembles the system of equations and solves for a new pressure solution.

◆ updateMaterialLaws()

template<class TypeTag >

void Dumux::FVPressure2P< TypeTag >::updateMaterialLaws

Updates and stores constitutive relations.

Updates constitutive relations and stores them in the variable class.

Stores mobility, fractional flow function and capillary pressure for all grid cells. In the compressible case additionally the densities and viscosities are stored.

Template Parameters

TypeTag The problem TypeTag

◆ updateVelocity()

template<class TypeTag >

void Dumux::FVPressure2P< TypeTag >::updateVelocity ( )

inline

Velocity update.

Resets the velocities in the cellData.

Member Data Documentation

◆ A_

template<class TypeTag >

Matrix Dumux::FVPressure< TypeTag >::A_

protectedinherited

Global stiffnes matrix (sparse matrix which is build by the initializeMatrix() function)

◆ f_

template<class TypeTag >

RHSVector Dumux::FVPressure< TypeTag >::f_

protectedinherited

Right hand side vector.

The documentation for this class was generated from the following file:

2p/sequential/diffusion/cellcentered/pressure.hh

Description

Public Member Functions

Protected Types

Protected Member Functions

Protected Attributes

Member Typedef Documentation

◆ EntryType

Member Enumeration Documentation

◆ anonymous enum

◆ anonymous enum

Constructor & Destructor Documentation

◆ FVPressure2P()

Member Function Documentation

◆ addOutputVtkFields()

◆ assemble()

◆ calculateVelocity()

◆ deserializeEntity()

◆ getFlux()

◆ getFluxOnBoundary()

◆ getSource()

◆ getStorage()

◆ globalMatrix()

◆ initialize() [1/2]

◆ initialize() [2/2]

◆ initializeMatrix()

◆ initializeMatrixIndices()

◆ initializeMatrixRowSize()

◆ initializePressure()

◆ pressure() [1/3]

◆ pressure() [2/3]

◆ pressure() [3/3]

◆ resetFixPressureAtIndex()

◆ rightHandSide()

◆ serializeEntity()

◆ setFixPressureAtIndex()

◆ solve()

◆ storePressureSolution() [1/2]

◆ storePressureSolution() [2/2]

◆ unsetFixPressureAtIndex()

◆ update()

◆ updateMaterialLaws()

◆ updateVelocity()

Member Data Documentation

◆ A_

◆ f_