Adaption of the fully implicit scheme to the two-phase n-component fully implicit model with additional solid/mineral phases. More...

#include <dumux/porousmediumflow/2pnc/model.hh>
#include <dumux/porousmediumflow/2pnc/volumevariables.hh>
#include <dumux/material/solidstates/compositionalsolidstate.hh>
#include <dumux/porousmediumflow/mineralization/model.hh>
#include <dumux/porousmediumflow/mineralization/localresidual.hh>
#include <dumux/porousmediumflow/mineralization/volumevariables.hh>
#include <dumux/porousmediumflow/mineralization/iofields.hh>
#include <dumux/porousmediumflow/nonisothermal/indices.hh>
#include <dumux/porousmediumflow/nonisothermal/iofields.hh>
#include <dumux/material/fluidmatrixinteractions/2p/thermalconductivitysomerton.hh>

Description

Adaption of the fully implicit scheme to the two-phase n-component fully implicit model with additional solid/mineral phases.

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:

Both phases are present: The saturation is used (either \(S_n\) or \(S_w\), dependent on the chosen Formulation), as long as \( 0 < S_\alpha < 1\).
Only wetting phase is present: The mole fraction of, e.g., air in the wetting phase \(x^a_w\) is used, as long as the maximum mole fraction is not exceeded ( \(x^a_w<x^a_{w,max}\))
Only non-wetting phase is present: The mole fraction of, e.g., water in the non-wetting phase, \(x^w_n\), is used, as long as the maximum mole fraction is not exceeded ( \(x^w_n<x^w_{n,max}\))

For the other components, the mole fraction \(x^\kappa_w\) is the primary variable. The primary variable of the solid phases is the volume fraction \(\phi_\lambda = \frac{V_\lambda}{V_{total}}\).

The source an sink terms link the mass balances of the n-transported component to the solid phases. The porosity \(\phi\) is updated according to the reduction of the initial (or solid-phase-free porous medium) porosity \(\phi_0\) by the accumulated volume fractions of the solid phases: \( \phi = \phi_0 - \sum (\phi_\lambda)\) Additionally, the permeability is updated depending on the current porosity.

Classes
struct	Dumux::Properties::TTag::TwoPNCMin

struct	Dumux::Properties::TTag::TwoPNCMinNI

struct	Dumux::Properties::LocalResidual< TypeTag, TTag::TwoPNCMin >

struct	Dumux::Properties::VolumeVariables< TypeTag, TTag::TwoPNCMin >
	use the mineralization volume variables together with the 2pnc vol vars More...

struct	Dumux::Properties::IOFields< TypeTag, TTag::TwoPNCMin >
	Set the vtk output fields specific to this model. More...

struct	Dumux::Properties::ModelTraits< TypeTag, TTag::TwoPNCMin >
	The 2pnc model traits define the non-mineralization part. More...

struct	Dumux::Properties::SolidState< TypeTag, TTag::TwoPNCMin >
	The two-phase model uses the immiscible fluid state. More...

struct	Dumux::Properties::ModelTraits< TypeTag, TTag::TwoPNCMinNI >
	Set non-isothermal model traits. More...

struct	Dumux::Properties::IOFields< TypeTag, TTag::TwoPNCMinNI >
	Non-isothermal vtkoutput. More...

struct	Dumux::Properties::ThermalConductivityModel< TypeTag, TTag::TwoPNCMinNI >
	Use the effective thermal conductivities calculated using the Somerton method. More...

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

namespace	Dumux::Properties

namespace	Dumux::Properties::TTag
	Type tag for numeric models.

Include dependency graph for porousmediumflow/2pncmin/model.hh:

Description

Classes

Namespaces