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

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:

• \( \phi \) is the porosity of the porous medium,
• \( S_\alpha \) represents the saturation of phase \( \alpha \),
• \( \rho_\alpha \) is the mass density of phase \( \alpha \),
• \( X_\alpha^\kappa \) is the mass fraction of component \( \kappa \) in phase \( \alpha \),
• \( x_\alpha^\kappa \) is the mole fraction of component \( \kappa \) in phase \( \alpha \),
• \( v_\alpha \) is the velocity of phase \( \alpha \),
• \( {\bf D_{\alpha, pm}^\kappa} \) is the diffusivity of component \( \kappa \) in phase \( \alpha \),
• \( \overline M_\alpha \) is the average molar mass of phase \( \alpha \)
• \( q_\alpha^\kappa \) is a source or sink term.

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:

• The component fugacities \(f^1, \dots, f^{N}\)
• The pressure of the first phase \(p_1\)
• The saturations of the first \(M-1\) phases \(S_1, \dots, S_{M-1}\)
• Temperature \(T\) if the energy equation is enabled

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
	The energy balance equation for a porous solid.
namespace	Dumux::Properties::TTag
	Type tag for numeric models.

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