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 \(N \geq M - 1\) 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 effective 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

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