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DUNE for Multi-{Phase, Component, Scale, Physics, ...} flow and transport in porous media
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porousmediumflow/mpnc/model.hh File Reference

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

#include <dumux/common/properties.hh>
#include <dumux/material/fluidstates/nonequilibrium.hh>
#include <dumux/material/fluidstates/compositional.hh>
#include <dumux/material/fluidmatrixinteractions/diffusivitymillingtonquirk.hh>
#include <dumux/material/fluidmatrixinteractions/2p/thermalconductivity/simplefluidlumping.hh>
#include <dumux/material/fluidmatrixinteractions/2p/thermalconductivity/somerton.hh>
#include <dumux/porousmediumflow/properties.hh>
#include <dumux/porousmediumflow/compositional/localresidual.hh>
#include <dumux/porousmediumflow/nonisothermal/model.hh>
#include <dumux/porousmediumflow/nonisothermal/indices.hh>
#include <dumux/porousmediumflow/nonisothermal/iofields.hh>
#include <dumux/porousmediumflow/nonequilibrium/model.hh>
#include <dumux/porousmediumflow/nonequilibrium/volumevariables.hh>
#include "indices.hh"
#include "volumevariables.hh"
#include "iofields.hh"
#include "localresidual.hh"
#include "pressureformulation.hh"

Go to the source code of this file.

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.

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

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

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

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

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:

Classes

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...
 
struct  Dumux::Properties::TTag::MPNC
 
struct  Dumux::Properties::TTag::MPNCNI
 
struct  Dumux::Properties::TTag::MPNCNonequil
 
struct  Dumux::Properties::LocalResidual< TypeTag, TTag::MPNC >
 Use the MpNc local residual for the MpNc model. More...
 
struct  Dumux::Properties::ModelTraits< TypeTag, TTag::MPNC >
 Set the model traits property. More...
 
struct  Dumux::Properties::FluidState< TypeTag, TTag::MPNC >
 This model uses the compositional fluid state. More...
 
struct  Dumux::Properties::VolumeVariables< TypeTag, TTag::MPNC >
 Set the volume variables property. More...
 
struct  Dumux::Properties::ReplaceCompEqIdx< TypeTag, TTag::MPNC >
 Per default, no component mass balance is replaced. More...
 
struct  Dumux::Properties::UseMoles< TypeTag, TTag::MPNC >
 Use mole fractions in the balance equations by default. More...
 
struct  Dumux::Properties::EffectiveDiffusivityModel< TypeTag, TTag::MPNC >
 Use the model after Millington (1961) for the effective diffusivity. More...
 
struct  Dumux::Properties::PressureFormulation< TypeTag, TTag::MPNC >
 Set the default pressure formulation to the pressure of the (most) wetting phase. More...
 
struct  Dumux::Properties::IOFields< TypeTag, TTag::MPNC >
 Set the vtk output fields specific to this model. More...
 
struct  Dumux::Properties::ModelTraits< TypeTag, TTag::MPNCNI >
 set the non-isothermal model traits More...
 
struct  Dumux::Properties::VolumeVariables< TypeTag, TTag::MPNCNI >
 Set the volume variables property. More...
 
struct  Dumux::Properties::ThermalConductivityModel< TypeTag, TTag::MPNCNI >
 Somerton is used as default model to compute the effective thermal heat conductivity. More...
 
struct  Dumux::Properties::EquilibriumLocalResidual< TypeTag, TTag::MPNCNonequil >
 
struct  Dumux::Properties::EquilibriumIOFields< TypeTag, TTag::MPNCNonequil >
 Set the vtk output fields specific to this model. More...
 
struct  Dumux::Properties::ModelTraits< TypeTag, TTag::MPNCNonequil >
 
struct  Dumux::Properties::EquilibriumModelTraits< TypeTag, TTag::MPNCNonequil >
 set equilibrium model traits More...
 
struct  Dumux::Properties::ThermalConductivityModel< TypeTag, TTag::MPNCNonequil >
 in case we do not assume full non-equilibrium one needs a thermal conductivity More...
 
struct  Dumux::Properties::VolumeVariables< TypeTag, TTag::MPNCNonequil >
 use the mineralization volume variables together with the 2pnc vol vars More...
 

Namespaces

namespace  Dumux
 
namespace  Dumux::Properties
 
namespace  Dumux::Properties::TTag
 Type tag for numeric models.
 
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