mpnc

Generalized multi-phase, multi-component Darcy flow. More...

Description

Generalized multi-phase, multi-component Darcy flow.

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.

\[ \forall \alpha: S_\alpha = 0 \implies \sum_\kappa x_\alpha^\kappa \leq 1 \]

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

\[ \forall \alpha: \sum_\kappa x_\alpha^\kappa = 1 \implies S_\alpha \geq 0 \]

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.

These equations are then discretized using a fully-implicit vertex centered finite volume scheme (often known as 'box'-scheme) for spatial discretization and the implicit Euler method as temporal discretization.

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	dumux/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.

Namespaces
namespace	Dumux
	make the local view function available whenever we use the grid geometry

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::MPNCVolumeVariablesTraits< PV, FSY, FST, SSY, SST, PT, MT >
	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()>

Contains the quantities which are constant within a finite volume in the MpNc model.

Template Parameters

Traits

Class encapsulating types to be used by the vol vars

Enumeration Type Documentation

MpNcPressureFormulation

enum class Dumux::MpNcPressureFormulation

strong

Enumerates the formulations which the MpNc model accepts.

Enumerator
mostWettingFirst
leastWettingFirst