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3.2-git
DUNE for Multi-{Phase, Component, Scale, Physics, ...} flow and transport in porous media
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3.2-git
DUNE for Multi-{Phase, Component, Scale, Physics, ...} flow and transport in porous media
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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/spatialparams/fv.hh>#include <dumux/material/fluidmatrixinteractions/diffusivitymillingtonquirk.hh>#include <dumux/material/fluidmatrixinteractions/2p/thermalconductivitysimplefluidlumping.hh>#include <dumux/material/fluidmatrixinteractions/2p/thermalconductivitysomerton.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.
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.
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:
Namespaces | |
| namespace | Dumux |
| namespace | Dumux::Properties |
| namespace | Dumux::Properties::TTag |
| Type tag for numeric models. | |
1.9.3