1pncmin

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

Description

A single-phase, multi-component model considering mineralization processes.

This model implements one-phase n-component flow of a compressible fluid composed of the n components \(\kappa \) in combination with mineral precipitation and dissolution of the solid phases. The standard multi-phase 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,

\[ \frac{\partial ( \phi \varrho_f X^\kappa )} {\partial t} - \nabla \cdot \left\{ \varrho_f X^\kappa \frac{k_{r}}{\mu} \mathbf{K} (\nabla p - \varrho_{f} \mathbf{g}) \right\} - \nabla \cdot \left\{{\bf D_{pm}^\kappa} \varrho_{f} \nabla X^\kappa \right\} - q_\kappa = 0 \qquad \kappa \in \{w, a,\cdots \}, \]

where:

• \( \phi \) is the porosity,
• \( \varrho_f \) is the mass density of the fluid,
• \( X^\kappa \) is the mass fraction of component \( \kappa \) in the fluid,
• \( k_{r} \) is the relative permeability,
• \( \mu \) represents the dynamic viscosity,
• \( \mathbf{K} \) is the intrinsic permeability tensor,
• \( p \) is the pressure,
• \( \mathbf{g} \) is the gravitational acceleration vector,
• \( {\bf D_{pm}^\kappa} \) is the effective diffusivity in the porous medium,
• \( q_\kappa \) 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:

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

The primary variables are the pressure \(p\) and the mole fractions of the dissolved components \(x^k\). The primary variable of the solid phases is the volume fraction \(\phi_\lambda = \frac{V_\lambda}{V_{total}}\),

where:

• \( V_\lambda \) is the volume of phase \( \lambda \),
• \( V_{\text{total}} \) is the total volume of the system.

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, \( \phi = \phi_0 - \sum (\phi_\lambda),\)

where:

• \( \phi \) represents the remaining porosity in the system,
• \( \phi_0 \) is the initial porosity,
• \( \phi_\lambda \) denotes the volume fraction of phase \( \lambda \). Additionally, the permeability is updated depending on the current porosity.

Files
file	model.hh
	A single-phase, multi-component model considering mineralization processes.