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Generalized multi-phase, multi-component Darcy flow. More...

Description

A fully implicit model for MpNc flow using vertex centered finite volumes.

This model implements a \(M\)-phase flow of a fluid mixture composed of \(N\) chemical species. The phases are denoted by lower index \(\alpha \in \{ 1, \dots, M \}\). All fluid phases are mixtures of \(N \geq M - 1\) chemical species which are denoted by the upper index \(\kappa \in \{ 1, \dots, N \} \).

The momentum approximation can be selected via "BaseFluxVariables": Darcy (ImplicitDarcyFluxVariables) and Forchheimer (ImplicitForchheimerFluxVariables) relations are available for all Box models. For details on Darcy's law see dumux/flux/darcyslaw.hh.

By inserting this into the equations for the conservation of the mass of each component, one gets one mass-continuity equation for each component \(\kappa\),

\[ \sum_{\kappa} \left( \frac{\partial \left(\phi \varrho_\alpha x_\alpha^\kappa S_\alpha\right)}{\partial t} + \mathrm{div}\; \left\{ v_\alpha \frac{\varrho_\alpha}{\overline M_\alpha} x_\alpha^\kappa \right\} \right) = q^\kappa \]

with \(\overline M_\alpha\) being the average molar mass of phase \(\alpha\):

\[ \overline M_\alpha = \sum_\kappa M^\kappa \; x_\alpha^\kappa \]

Additionally:

For the missing \(M\) model assumptions, the model assumes that if a fluid phase is not present, the sum of the mole fractions of this fluid phase is smaller than \(1\), i.e.

\[ \forall \alpha: S_\alpha = 0 \Rightarrow \sum_\kappa x_\alpha^\kappa \leq 1 \]

Also, if a fluid phase may be present at a given spatial location its saturation must be positive:

\[ \forall \alpha: \sum_\kappa x_\alpha^\kappa = 1 \Rightarrow S_\alpha \geq 0 \]

Since at any given spatial location, a phase is always either present or not present, one of the strict equalities on the right hand side is always true, i.e.

\[ \forall \alpha: S_\alpha \left( \sum_\kappa x_\alpha^\kappa - 1 \right) = 0 \]

always holds.

These three equations constitute a non-linear complementarity problem, which can be solved using so-called non-linear complementarity functions \(\Phi(a, b)\) which have the property

\[\Phi(a,b) = 0 \iff a \geq0 \land b \geq0 \land a \cdot b = 0 \]

Several non-linear complementarity functions have been suggested, e.g. the Fischer-Burmeister function

\[ \Phi(a,b) = a + b - \sqrt{a^2 + b^2} \;. \]

This model uses

\[ \Phi(a,b) = \min \{a, b \}\;, \]

because of its piecewise linearity.

The model assumes local thermodynamic equilibrium and uses the following primary variables:

Files

file  indices.hh
 The primary variable and equation indices for the MpNc model.
 
file  initialconditionhelper.hh
 A helper function to get the correct initial conditions by updating the fluidstate and defining the primary variables needed for equilibrium mpnc models for the MPNC model.
 
file  iofields.hh
 Adds I/O fields specific to the mpnc model.
 
file  localresidual.hh
 MpNc specific details needed to approximately calculate the local defect in the fully implicit scheme.
 
file  model.hh
 A fully implicit model for MpNc flow using vertex centered finite volumes.
 
file  pressureformulation.hh
 Enumeration of the formulations accepted by the MpNc model.
 
file  volumevariables.hh
 Contains the secondary variables (Quantities which are constant within a finite volume) of the MpNc model.
 

Classes

struct  Dumux::MPNCIndices< numPhases, numEqBalance >
 The primary variable and equation indices for the MpNc model. More...
 
class  Dumux::MPNCIOFields
 Adds I/O fields specific to the mpnc model. More...
 
class  Dumux::MPNCLocalResidual< TypeTag >
 MpNc specific details needed to approximately calculate the local defect in the fully implicit scheme. More...
 
struct  Dumux::MPNCModelTraits< nPhases, nComp, formulation, useM, repCompEqIdx >
 Specifies a number properties of the m-phase n-component model. More...
 
struct  Dumux::MPNCNonequilibriumModelTraits< NonEquilTraits >
 Specifies a number properties of the m-phase n-component model in conjunction with non-equilibrium. This is necessary because the mpnc indices are affected by the non-equilibrium which can thus not be plugged on top of it that easily. More...
 
struct  Dumux::MPNCVolumeVariablesTraits< PV, FSY, FST, SSY, SST, PT, MT, DT, EDM >
 Traits class for the mpnc volume variables. More...
 

Typedefs

template<class Traits >
using Dumux::MPNCVolumeVariables = MPNCVolumeVariablesImplementation<Traits, Traits::ModelTraits::enableChemicalNonEquilibrium()>
 Contains the quantities which are constant within a finite volume in the MpNc model.
 

Enumerations

enum class  Dumux::MpNcPressureFormulation { Dumux::MpNcPressureFormulation::mostWettingFirst , Dumux::MpNcPressureFormulation::leastWettingFirst }
 Enumerates the formulations which the MpNc model accepts. More...
 

Typedef Documentation

◆ MPNCVolumeVariables

template<class Traits >
using Dumux::MPNCVolumeVariables = MPNCVolumeVariablesImplementation<Traits, Traits::ModelTraits::enableChemicalNonEquilibrium()>
Template Parameters
TraitsClass encapsulating types to be used by the vol vars

Enumeration Type Documentation

◆ MpNcPressureFormulation

enum class Dumux::MpNcPressureFormulation
strong
Enumerator
mostWettingFirst 
leastWettingFirst