2pncmin

Two-phase, multi-component Darcy flow with mineralization. More...

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. For details on Darcy's law see dumux/flux/darcyslaw.hh.

By inserting Darcy's law into the equations for the conservation of the components, one gets one transport equation for each component,

\begin{eqnarray*} && \frac{\partial (\sum_\alpha \phi \varrho_\alpha X_\alpha^\kappa S_\alpha )} {\partial t} - \sum_\alpha \nabla \cdot \left\{ \varrho_\alpha X_\alpha^\kappa \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} (\nabla p_\alpha - \varrho_{\alpha} \mathbf{g}) \right\} \nonumber \\ \nonumber \\ &-& \sum_\alpha \nabla \cdot \left\{{\bf D_{\alpha, pm}^\kappa} \varrho_{\alpha} \nabla X^\kappa_{\alpha} \right\} - \sum_\alpha q_\alpha^\kappa = 0 \qquad \kappa \in \{w, a,\cdots \} \, , \alpha \in \{w, g\}, \end{eqnarray*}

where:

• \( \phi \) is the porosity of the porous medium,
• \( S_\alpha \) represents the saturation of phase \( \alpha \),
• \( \rho_\alpha \) is the mass density of phase \( \alpha \),
• \( X_\alpha^\kappa \) is the mass fraction of component \( \kappa \) in phase \( \alpha \),
• \( x_\alpha^\kappa \) is the mole fraction of component \( \kappa \) in phase \( \alpha \),
• \( v_\alpha \) is the velocity of phase \( \alpha \),
• \( {\bf D_{\alpha, pm}^\kappa} \) is the effective diffusivity of component \( \kappa \) in phase \( \alpha \),
• \( q_\alpha^\kappa \) is a source or sink term.

The solid or mineral phases are assumed to consist of a single component. Their mass balance consists of only a storage and a source term, \(\frac{\partial ( \varrho_\lambda \phi_\lambda )} {\partial t} = q_\lambda,\)

where:

• \( \varrho_\lambda \) mass density of the solid phase \( \lambda \),
• \( \phi_\lambda \) is the porosity of the solid,
• \( q_\lambda \) is a source or sink term.

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 nonwetting phase is present: The mole fraction of, e.g., water in the nonwetting 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.

Files
file	porousmediumflow/2pncmin/model.hh
	Adaption of the fully implicit scheme to the two-phase n-component fully implicit model with additional solid/mineral phases.