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3.4
DUNE for Multi-{Phase, Component, Scale, Physics, ...} flow and transport in porous media
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3.4
DUNE for Multi-{Phase, Component, Scale, Physics, ...} flow and transport in porous media
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The finite volume model for the solution of the compositional pressure equation. More...
#include <dumux/porousmediumflow/2p2c/sequential/fvpressure.hh>
The finite volume model for the solution of the compositional pressure equation.
Provides a Finite Volume implementation for the pressure equation of a compressible system with two components. An IMPES-like method is used for the sequential solution of the problem. Diffusion is neglected, capillarity can be regarded. Isothermal conditions and local thermodynamic equilibrium are assumed. Gravity is included.
\[ c_{total}\frac{\partial p}{\partial t} + \sum_{\kappa} \frac{\partial v_{total}}{\partial C^{\kappa}} \nabla \cdot \left( \sum_{\alpha} X^{\kappa}_{\alpha} \varrho_{\alpha} \bf{v}_{\alpha}\right) = \sum_{\kappa} \frac{\partial v_{total}}{\partial C^{\kappa}} q^{\kappa}, \]
where \(\bf{v}_{\alpha} = - \lambda_{\alpha} \bf{K} \left(\nabla p_{\alpha} + \rho_{\alpha} \bf{g} \right) \). \( c_{total} \) represents the total compressibility, for constant porosity this yields \( - \frac{\partial V_{total}}{\partial p_{\alpha}} \), \(p_{\alpha} \) denotes the phase pressure, \( \bf{K} \) the absolute permeability, \( \lambda_{\alpha} \) the phase mobility, \( \rho_{\alpha} \) the phase density and \( \bf{g} \) the gravity constant and \( C^{\kappa} \) the total Component concentration. See paper SPE 99619 or "Analysis of a Compositional Model for Fluid Flow in Porous Media" by Chen, Qin and Ewing for derivation.
The pressure base class FVPressure assembles the matrix and right-hand-side vector and solves for the pressure vector, whereas this class provides the actual entries for the matrix and RHS vector. The partial derivatives of the actual fluid volume \( v_{total} \) are gained by using a secant method.
Public Member Functions | |
| void | getSource (EntryType &sourceEntry, const Element &elementI, const CellData &cellDataI, const bool first) |
| Assembles the source term. More... | |
| void | getStorage (EntryType &storageEntry, const Element &elementI, const CellData &cellDataI, const bool first) |
| Assembles the storage term. More... | |
| void | getFlux (EntryType &entries, const Intersection &intersection, const CellData &cellDataI, const bool first) |
| Get flux at an interface between two cells. More... | |
| void | getFluxOnBoundary (EntryType &entries, const Intersection &intersection, const CellData &cellDataI, const bool first) |
| Get flux on Boundary. More... | |
| void | updateMaterialLawsInElement (const Element &elementI, bool postTimeStep) |
| Updates secondary variables of one cell. More... | |
| FVPressure2P2C (Problem &problem) | |
| Constructs a FVPressure2P2C object. More... | |
| void | initialMaterialLaws (bool compositional) |
| initializes the fluid distribution and hereby the variables container More... | |
| void | initialize (bool solveTwice=false) |
| Initializes the simulation run. More... | |
| void | initialize () |
| Initialize pressure model. More... | |
| void | update () |
| Compositional pressure solution routine: update estimate for secants, assemble, solve. More... | |
| void | updateMaterialLaws (bool postTimeStep=false) |
| Updates secondary variables. More... | |
| void | volumeDerivatives (const GlobalPosition &, const Element &ep) |
| Partial derivatives of the volumes w.r.t. changes in total concentration and pressure. 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 | calculateVelocity () |
| void | updateVelocity () |
| 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 | |
| using | EntryType = Dune::FieldVector< Scalar, 2 > |
| enum | { rhs = 1 , matrix = 0 } |
| Indices of matrix and rhs entries. More... | |
| enum | { pressEqIdx = Indices::pressureEqIdx } |
Protected Member Functions | |
| Problem & | problem () |
| const Problem & | problem () const |
| 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 | |
| Problem & | problem_ |
| bool | enableVolumeIntegral |
| Enables the volume integral of the pressure equation. More... | |
| bool | regulateBoundaryPermeability |
| Enables regulation of permeability in the direction of a Dirichlet Boundary Condition. More... | |
| Scalar | minimalBoundaryPermeability |
| Minimal limit for the boundary permeability. More... | |
| Scalar | ErrorTermFactor_ |
| Handling of error term: relaxation factor. More... | |
| Scalar | ErrorTermLowerBound_ |
| Handling of error term: lower bound for error dampening. More... | |
| Scalar | ErrorTermUpperBound_ |
| Matrix | A_ |
Global stiffnes matrix (sparse matrix which is build by the initializeMatrix() function) More... | |
| RHSVector | f_ |
| Right hand side vector. More... | |
Static Protected Attributes | |
| static constexpr int | pressureType = getPropValue<TypeTag, Properties::PressureFormulation>() |
| gives kind of pressure used ( \( 0 = p_w \), \( 1 = p_n \), \( 2 = p_{global} \)) More... | |
general methods for output | |
| TransportSolutionType | updateEstimate_ |
| Update estimate for changes in volume for the pressure equation. More... | |
| VtkMultiWriter< GridView > | initializationOutputWriter_ |
| output for the initialization procedure More... | |
| Scalar | maxError_ |
| Maximum volume error of all cells. More... | |
| Scalar | incp_ |
| Increment for the volume derivative w.r.t pressure. More... | |
| template<class MultiWriter > | |
| void | addOutputVtkFields (MultiWriter &writer) |
| Write data files. More... | |
| void | initializationOutput (double pseudoTS=0.) |
| Write additional debug info in a special writer. More... | |
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protected |
Type of the vector of entries
Contains the return values of the get*()-functions (matrix or right-hand side entry).
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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 |
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inline |
Constructs a FVPressure2P2C object.
| problem | a problem class object |
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inlineinherited |
Write data files.
Adds pressure-related quantities, including numerical things such as the volume Error entering the pressure equation. Verobosity of the output can be triggered by the property / parameter VtkOutputLevel, with 0 putting out only primary variables and 4 being very verbose.
| MultiWriter | Class defining the output writer |
| writer | The output writer (usually a VTKMultiWriter object) |
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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!
| 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) |
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inlineinherited |
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inlineinherited |
Function for deserialization of the pressure field.
Function needed for restart option. Reads the pressure of a grid element from a restart file.
| instream | Stream from the restart file. |
| element | Grid element |
| void Dumux::FVPressure2P2C< TypeTag >::getFlux | ( | EntryType & | entries, |
| const Intersection & | intersection, | ||
| const CellData & | cellDataI, | ||
| const bool | first | ||
| ) |
Get flux at an interface between two cells.
for first == true, the flux is calculated in traditional fractional-flow forn as in FVPressure2P. for first == false, the flux thorugh \( \gamma \) is calculated via a volume balance formulation
\[ - A_{\gamma} \mathbf{n}^T_{\gamma} \mathbf{K} \sum_{\alpha} \varrho_{\alpha} \lambda_{\alpha} \mathbf{d}_{ij} \left( \frac{p_{\alpha,j}^t - p^{t}_{\alpha,i}}{\Delta x} + \varrho_{\alpha} \mathbf{g}^T \mathbf{d}_{ij} \right) \sum_{\kappa} X^{\kappa}_{\alpha} \frac{\partial v_{t}}{\partial C^{\kappa}} + V_i \frac{A_{\gamma}}{U_i} \mathbf{d}^T \mathbf{K} \sum_{\alpha} \varrho_{\alpha} \lambda_{\alpha} \mathbf{d}_{ij} \left( \frac{p_{\alpha,j}^t - p^{t}_{\alpha,i}}{\Delta x} + \varrho_{\alpha} \mathbf{g}^T \mathbf{d}_{ij} \right) \sum_{\kappa} X^{\kappa}_{\alpha} \frac{\frac{\partial v_{t,j}}{\partial C^{\kappa}_j}-\frac{\partial v_{t,i}}{\partial C^{\kappa}_i}}{\Delta x} \]
This includes a boundary integral and a volume integral, because \( \frac{\partial v_{t,i}}{\partial C^{\kappa}_i} \) is not constant. Here, \( \mathbf{d}_{ij} \) is the normalized vector connecting the cell centers, and \( \mathbf{n}_{\gamma} \) represents the normal of the face \( \gamma \).
| entries | The Matrix and RHS entries |
| intersection | Intersection between cell I and J |
| cellDataI | Data of cell I |
| first | Flag if pressure field is unknown |
| void Dumux::FVPressure2P2C< TypeTag >::getFluxOnBoundary | ( | EntryType & | entries, |
| const Intersection & | intersection, | ||
| const CellData & | cellDataI, | ||
| const bool | first | ||
| ) |
Get flux on Boundary.
for first == true, the flux is calculated in traditional fractional-flow forn as in FVPressure2P. for first == false, the flux thorugh \( \gamma \) is calculated via a volume balance formulation
\[ - A_{\gamma} \mathbf{n}^T_{\gamma} \mathbf{K} \sum_{\alpha} \varrho_{\alpha} \lambda_{\alpha} \mathbf{d}_{ij} \left( \frac{p_{\alpha,j}^t - p^{t}_{\alpha,i}}{\Delta x} + \varrho_{\alpha} \mathbf{g}^T \mathbf{d}_{ij} \right) \sum_{\kappa} \frac{\partial v_{t}}{\partial C^{\kappa}} X^{\kappa}_{\alpha} \;, \]
where we skip the volume integral assuming \( \frac{\partial v_{t,i}}{\partial C^{\kappa}_i} \) to be constant at the boundary. Here, \( \mathbf{d}_{ij} \) is the normalized vector connecting the cell centers, and \( \mathbf{n}_{\gamma} \) represents the normal of the face \( \gamma \).
If a Neumann BC is set, the given (mass-)flux is directly multiplied by the volume derivative and inserted.
| entries | The Matrix and RHS entries |
| intersection | Intersection between cell I and J |
| cellDataI | Data of cell I |
| first | Flag if pressure field is unknown |
| void Dumux::FVPressure2P2C< TypeTag >::getSource | ( | EntryType & | sourceEntry, |
| const Element & | elementI, | ||
| const CellData & | cellDataI, | ||
| const bool | first | ||
| ) |
Assembles the source term.
for first == true, a source is implemented as in FVPressure2P. for first == false, the source is translated into a volumentric source term:
\[ V_i \sum_{\kappa} \frac{\partial v_{t}}{\partial C^{\kappa}} q^{\kappa}_i \]
.
| sourceEntry | The Matrix and RHS entries |
| elementI | The element I |
| cellDataI | Data of cell I |
| first | Flag if pressure field is unknown |
| void Dumux::FVPressure2P2C< TypeTag >::getStorage | ( | EntryType & | storageEntry, |
| const Element & | elementI, | ||
| const CellData & | cellDataI, | ||
| const bool | first | ||
| ) |
Assembles the storage term.
for first == true, there is no storage contribution. for first == false, the storage term comprises the compressibility (due to a change in pressure from last timestep):
\[ V_i c_{t,i} \frac{p^t_i - p^{t-\Delta t}_i}{\Delta t} \]
and the damped error introduced by the incorrect transport of the last timestep:
\[ V_i \alpha_r \frac{v_{t} - \phi}{\Delta t} \]
. The latter is damped according to Fritz 2011.
| storageEntry | The Matrix and RHS entries |
| elementI | The element I |
| cellDataI | Data of cell I |
| first | Flag if pressure field is unknown |
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inlineinherited |
Returns the global matrix of the last pressure solution step.
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inlineinherited |
Write additional debug info in a special writer.
To visualize the different steps through the initialization procedure, we use very small pseudo time steps only for the writer! This is only for debugging of the initialization procedure.
| pseudoTS | Time steps that only appear in the writer, not real. |
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inlineinherited |
Initialize pressure model.
Function initializes the sparse matrix to solve the global system of equations and sets/calculates the initial pressure
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inherited |
Initializes the simulation run.
Initializes the simulation to gain the initial pressure field. Output throughout initialization procedure is only done in debug mode.
| solveTwice | flag to determine possible iterations of the initialization process |
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protectedinherited |
Initialize the global matrix of the system of equations to solve.
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protectedinherited |
Initialize the global matrix of the system of equations to solve.
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protectedinherited |
Initialize the global matrix of the system of equations to solve.
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inlineprotectedinherited |
Initialization of the pressure solution vector: Initialization with meaningful values may.
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inherited |
initializes the fluid distribution and hereby the variables container
It differs from updateMaterialLaws() because there are two possible initial conditions: saturations and concentration.
| compositional | flag that determines if compositional effects are regarded, i.e. a reasonable pressure field is known with which compositions can be calculated. |
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inlineprotectedinherited |
Returns the vector containing the pressure solution.
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inlineprotectedinherited |
Returns the vector containing the pressure solution.
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inlineinherited |
Public access function for the primary pressure variable.
Function returns the cell pressure value at index eIdxGlobal
| eIdxGlobal | Global index of a grid cell |
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inlineprotected |
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inlineprotected |
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inlineinherited |
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inlineinherited |
Returns the right hand side vector of the last pressure solution step.
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inlineinherited |
Function for serialization of the pressure field.
Function needed for restart option. Writes the pressure of a grid element to a restart file.
| outstream | Stream into the restart file. |
| element | Grid element |
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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
| pressure | Pressure value at eIdxGlobal |
| eIdxGlobal | Global index of a grid cell |
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protectedinherited |
Solves the global system of equations to get the spatial distribution of the pressure.
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inlineinherited |
Reset the fixed pressure state.
No pressure is fixed inside the domain until setFixPressureAtIndex() function is called again.
| eIdxGlobal | Global index of a grid cell |
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inlineinherited |
Compositional pressure solution routine: update estimate for secants, assemble, solve.
An update estime (transport step acoording to old pressure field) determines changes in mass, composition, which is used to calculate volume derivatives entering the pressure equation, as well as an approximate guess for time step size for the storage terms in the p.e. Afterwards, the system is assembled and solved for pressure.
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inherited |
Updates secondary variables.
A loop through all elements updates the secondary variables stored in the variableclass by using the updated primary variables.
| postTimeStep | Flag indicating method is called from Problem::postTimeStep() |
| void Dumux::FVPressure2P2C< TypeTag >::updateMaterialLawsInElement | ( | const Element & | element, |
| bool | postTimeStep | ||
| ) |
Updates secondary variables of one cell.
For each element, the secondary variables are updated according to the primary variables. In case the method is called after the Transport, i.e. at the end / post time step, CellData2p2c.reset() resets the volume derivatives for the next time step.
| element | The element |
| postTimeStep | Flag indicating if we have just completed a time step |
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inlineinherited |
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inherited |
Partial derivatives of the volumes w.r.t. changes in total concentration and pressure.
This method calculates the volume derivatives via a secant method, where the secants are gained in a pre-computational step via the transport equation and the last TS size. The partial derivatives w.r.t. mass are defined as \( \frac{\partial v}{\partial C^{\kappa}} = \frac{\partial V}{\partial m^{\kappa}}\)
| globalPos | The global position of the current element |
| element | The current element |
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protectedinherited |
Global stiffnes matrix (sparse matrix which is build by the initializeMatrix() function)
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protected |
Enables the volume integral of the pressure equation.
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protected |
Handling of error term: relaxation factor.
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protected |
Handling of error term: lower bound for error dampening.
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protected |
Handling of error term: upper bound for error dampening
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protectedinherited |
Right hand side vector.
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protectedinherited |
Increment for the volume derivative w.r.t pressure.
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protectedinherited |
output for the initialization procedure
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protectedinherited |
Maximum volume error of all cells.
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protected |
Minimal limit for the boundary permeability.
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staticconstexprprotected |
gives kind of pressure used ( \( 0 = p_w \), \( 1 = p_n \), \( 2 = p_{global} \))
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protected |
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protected |
Enables regulation of permeability in the direction of a Dirichlet Boundary Condition.
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protectedinherited |
Update estimate for changes in volume for the pressure equation.
1.9.3