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/thermalconductivityaverage.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 $N\ge M-1$ 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 $\kappa$,

$$\sum _{\kappa }(\frac{\partial \left(\varphi {\varrho }_{\alpha }{x}_{\alpha }^{\kappa }{S}_{\alpha }\right)}{\partial t}+\mathrm{div}\left\{v_{\alpha }\frac{{\varrho }_{\alpha }}{{\bar{M}}_{\alpha }}{x}_{\alpha }^{\kappa }\right\})=q^{\kappa }$$

with ${\bar{M}}_{\alpha }$ being the average molar mass of phase $\alpha$:

$${\bar{M}}_{\alpha }=\sum _{\kappa }{M}^{\kappa }{x}_{\alpha }^{\kappa }$$

Additionally:

• $\varphi$ 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 $\alpha$,
• ${\mathbf{D}}_{\alpha \mathbf{,}\mathbf{p}\mathbf{m}}^{\kappa }$ is the effective diffusivity of component $\kappa$ in phase $\alpha$,
• ${\bar{M}}_{\alpha }$ 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.

$$\mathrm{\forall }\alpha :S_{\alpha }=0\Rightarrow \sum _{\kappa }{x}_{\alpha }^{\kappa }\le 1$$

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

$$\mathrm{\forall }\alpha :\sum _{\kappa }{x}_{\alpha }^{\kappa }=1\Rightarrow S_{\alpha }\ge 0$$

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.

$$\mathrm{\forall }\alpha :S_{\alpha }(\sum _{\kappa }{x}_{\alpha }^{\kappa }-1)=0$$

always holds.

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

$$\mathrm{\Phi }(a,b)=0⟺a\ge 0\wedge b\ge 0\wedge a\cdot b=0$$

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

$$\mathrm{\Phi }(a,b)=a+b-\sqrt{a^2+b^2}.$$

This model uses

$$\mathrm{\Phi }(a,b)=min\{a,b\},$$

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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