mpnc

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 $N\ge M-1$ 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

Files
file	porousmediumflow/mpnc/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	porousmediumflow/mpnc/iofields.hh
	Adds I/O fields specific to the mpnc model.
file	porousmediumflow/mpnc/localresidual.hh
	MpNc specific details needed to approximately calculate the local defect in the fully implicit scheme.
file	porousmediumflow/mpnc/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	porousmediumflow/mpnc/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. More...

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 = typedef MPNCVolumeVariablesImplementation<Traits, Traits::ModelTraits::enableChemicalNonEquilibrium()>

Template Parameters

Traits

Class encapsulating types to be used by the vol vars

Enumeration Type Documentation

MpNcPressureFormulation

enum class Dumux::MpNcPressureFormulation

strong

Enumerator
mostWettingFirst
leastWettingFirst