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3.6-git
DUNE for Multi-{Phase, Component, Scale, Physics, ...} flow and transport in porous media
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3.6-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/fluidmatrixinteractions/diffusivitymillingtonquirk.hh>#include <dumux/material/fluidmatrixinteractions/2p/thermalconductivity/simplefluidlumping.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.
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 \(\kappa\)
\[ \sum_{\kappa} \left( \phi \frac{\partial \left(\varrho_\alpha x_\alpha^\kappa S_\alpha\right)}{\partial t} + \mathrm{div}\; \left\{ v_\alpha \frac{\varrho_\alpha}{\overline M_\alpha} x_\alpha^\kappa \right\} \right) = q^\kappa \]
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.
\[ \forall \alpha: S_\alpha = 0 \Rightarrow \sum_\kappa x_\alpha^\kappa \leq 1 \]
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:
Namespaces | |
| namespace | Dumux |
| Adaption of the non-isothermal two-phase two-component flow model to problems with CO2. | |
| namespace | Dumux::Properties |
| namespace | Dumux::Properties::TTag |
| Type tag for numeric models. | |
1.9.3