Page:4SIGHT manual- a computer program for modelling degradation of underground low level waste concrete vaults (IA 4sightmanualcomp5612snyd).pdf/127

53. ADVECTION-DIFFUSION.

Ion Transport

At the core of 4SIGHT is the advection-diffusion equation to account for both diffusion of ions and Darcy flow of the pore solution due to hydrostatic head. The flux of ions due to both gradients in the ion concentration and to a volume average flow of pore solution is

$\mathbf {j} =-D\nabla c+c\mathbf {u}$

where $\mathbf {j}$ is the ion flux, $D$ is the diffusivity, $c$ the ion concentration, and $\mathbf {u}$ is is the volume-averaged velocity of the pore solution. The time dependent change in concentration is the negative divergence of the flux:

${\frac {\partial c}{\partial t}}=\nabla \cdot D\nabla c-\mathbf {u} -\nabla c\cdot \mathbf {u}$

Given a hydrostatic pressure head on a vertical column of porous media, the pore volume-averaged flow $\mathbf {v} _{D}$ is

$\mathbf {v} _{D}=-{\frac {k}{\mu }}(\nabla p-\rho g)$

For the hydrostatic heads considered here, the body force term, $\rho g$ is non-negligible. This equation can be cast into the more familiar form, assuming a constant density pore fluid, using a modified pressure potential:

$\psi =p-\rho gz$

This gives the more familiar Darcy equation

$\mathbf {v} _{D}=-{\frac {k}{\mu }}\nabla \psi$

The volume-averaged velocity $\mathbf {u}$ can be related to the Darcy flow velocity:

$\mathbf {v} _{D}=\phi \mathbf {u}$

where $\phi$ is the porosity.

Finally, the above equations can be combined to give

${\frac {\partial c}{\partial t}}=\nabla \cdot D\nabla c+{\frac {k}{\phi \mu }}\nabla \psi \cdot \nabla c+c\nabla \cdot {\frac {k}{\phi \mu }}\nabla \psi$

This equation gives the spatial and temporal behavior of the concentrations. To complete the calculations a means is needed to update the hydrostatic pressure potential, $\psi$.

Continuity Equation The temporal behavior of$\psi$ is calculated using the continuity equation:

${\frac {\partial p}{\partial t}}=-\nabla \cdot \rho \mathbf {v}$

where $\mathbf {v}$ is the intrinsic velocity of the pore solution. After averaging over the microstructure, the continuity equation becomes

${\frac {\partial p}{\partial t}}=-\nabla \cdot \mathbf {v} _{D}$

This can be related back to pressure using the Darcy equation once again:

${\frac {\partial p}{\partial t}}=-\nabla \cdot {\frac {k}{\mu }}\nabla \psi$