3.1-git
DUNE for Multi-{Phase, Component, Scale, Physics, ...} flow and transport in porous media
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Two-phase, multi-component Darcy flow with mineralization. More...

Description

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

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:

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.