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

Description

two-phase, multi-component Darcy flow

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.

Files
file	indices.hh
	Defines the indices required for the two-phase n-component model.
file	iofields.hh
	Adds I/O fields specific to the twop-nc model.
file	model.hh
	Adaption of the fully implicit scheme to the two-phase n-component fully implicit model.
file	primaryvariableswitch.hh
	The primary variable switch for the 2pnc model.
file	volumevariables.hh
	Contains the quantities which are constant within a finite volume in the two-phase, n-component model.

Classes
struct	Dumux::TwoPNCIndices
	The indices for the isothermal two-phase n-component model. More...
class	Dumux::TwoPNCIOFields
	Adds I/O fields specific to the TwoPNC model. More...
struct	Dumux::TwoPNCModelTraits< nComp, useMol, setMoleFractionForFP, formulation, repCompEqIdx >
	Specifies a number properties of two-phase n-component models. More...
struct	Dumux::TwoPNCVolumeVariablesTraits< PV, FSY, FST, SSY, SST, PT, MT >
	Traits class for the volume variables of the single-phase model. More...
class	Dumux::TwoPNCPrimaryVariableSwitch
	The primary variable switch controlling the phase presence state variable. More...
class	Dumux::TwoPNCVolumeVariables< Traits >
	Contains the quantities which are are constant within a finite volume in the two-phase, n-component model. More...

Description

Files

Classes