 |
3.2-git
DUNE for Multi-{Phase, Component, Scale, Physics, ...} flow and transport in porous media
|
| |
3.2-git
DUNE for Multi-{Phase, Component, Scale, Physics, ...} flow and transport in porous media
|
Adaption of the fully implicit scheme to the two-phase n-component fully implicit model. More...
#include <dune/common/fvector.hh>#include <dumux/common/properties.hh>#include <dumux/material/spatialparams/fv.hh>#include <dumux/material/fluidmatrixinteractions/diffusivitymillingtonquirk.hh>#include <dumux/material/fluidmatrixinteractions/2p/thermalconductivitysomerton.hh>#include <dumux/porousmediumflow/properties.hh>#include <dumux/porousmediumflow/compositional/localresidual.hh>#include <dumux/porousmediumflow/compositional/switchableprimaryvariables.hh>#include <dumux/porousmediumflow/nonisothermal/model.hh>#include <dumux/porousmediumflow/nonisothermal/indices.hh>#include <dumux/porousmediumflow/nonisothermal/iofields.hh>#include <dumux/porousmediumflow/2p/formulation.hh>#include "volumevariables.hh"#include "iofields.hh"#include "indices.hh"Go to the source code of this file.
Adaption of the fully implicit scheme to the two-phase n-component fully implicit model.
This model implements two-phase n-component flow of two compressible and partially miscible fluids \(\alpha \in \{ w, n \}\) composed of the n components \(\kappa \in \{ w, n,\cdots \}\) in combination with mineral precipitation and dissolution. The solid phases. The standard multiphase Darcy approach is used as the equation for the conservation of momentum:
\[ v_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mbox{\bf K} \left(\text{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g} \right) \]
By inserting this into the equations for the conservation of the components, one gets one transport equation for each component
\begin{eqnarray*} && \frac{\partial (\sum_\alpha \varrho_\alpha X_\alpha^\kappa \phi S_\alpha )} {\partial t} - \sum_\alpha \text{div} \left\{ \varrho_\alpha X_\alpha^\kappa \frac{k_{r\alpha}}{\mu_\alpha} \mbox{\bf K} (\text{grad}\, p_\alpha - \varrho_{\alpha} \mbox{\bf g}) \right\} \nonumber \\ \nonumber \\ &-& \sum_\alpha \text{div} \left\{{\bf D_{\alpha, pm}^\kappa} \varrho_{\alpha} \text{grad}\, X^\kappa_{\alpha} \right\} - \sum_\alpha q_\alpha^\kappa = 0 \qquad \kappa \in \{w, a,\cdots \} \, , \alpha \in \{w, g\} \end{eqnarray*}
The solid or mineral phases are assumed to consist of a single component. Their mass balance consist only of a storage and a source term: \(\frac{\partial \varrho_\lambda \phi_\lambda )} {\partial t} = q_\lambda\)
All equations are discretized using a vertex-centered finite volume (box) or cell-centered finite volume scheme as spatial and the implicit Euler method as time discretization.
By using constitutive relations for the capillary pressure \(p_c = p_n - p_w\) and relative permeability \(k_{r\alpha}\) and taking advantage of the fact that \(S_w + S_n = 1\) and \(X^\kappa_w + X^\kappa_n = 1\), the number of unknowns can be reduced to number of components.
The used primary variables are, like in the two-phase model, either \(p_w\) and \(S_n\) or \(p_n\) and \(S_w\). The formulation which ought to be used can be specified by setting the Formulation property to either TwoPTwoCIndices::pwsn or TwoPTwoCIndices::pnsw. By default, the model uses \(p_w\) and \(S_n\).
Moreover, the second primary variable depends on the phase state, since a primary variable switch is included. The phase state is stored for all nodes of the system. The model is uses mole fractions. Following cases can be distinguished:
Formulation), as long as \( 0 < S_\alpha < 1\). For the other components, the mole fraction \(x^\kappa_w\) is the primary variable.
Namespaces | |
| namespace | Dumux |
| namespace | Dumux::Properties |
| namespace | Dumux::Properties::TTag |
| Type tag for numeric models. | |
1.9.3