Page:4SIGHT manual- a computer program for modelling degradation of underground low level waste concrete vaults (IA 4sightmanualcomp5612snyd).pdf/41

porosity of the concrete, hence changing the transport coefficients. This is how the synergism of degradation mechanisms is achieved.

7.1Advection-Diffusion

Ath the core of 4SIGHT is the advection-diffusion equation which establishes the transport of ions through the slab. The advection-diffusion equation is simply the diffusion equation with an extra term to account for Darcy flow. The development of the equation starts most simply from the relation between flux, $j$, and concentration, $c$:

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

(1)

The parameter $D$ is the effective diffusion coefficient, and the quantity $\mathbf {u}$ is the average pore fluid velocity. The rate of change in concentration is the negative divergence of eqn. 1:

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

(2)

after neglecting the divergence of the volume averaged velocity, since the fluid is virtually incompressible and the rate change in porosity is small.

7.2Darcy Flow

The average pore fluid velocity can be calculated from the Darcy velocity, which is the volume-averaged pore fluid velocity. Given a porous media with permeability $k$ and pore fluid viscosity $\mu$, the Darcy velocity, $\mathbf {v} _{D}$, is proportional to the pressure gradient, $\nabla p$ and the density of the fluid [3]:

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

(3)

The Darcy velocity $\mathbf {v} _{D}$ can be related to the average pore velocity, $\mathbf {u}$, from a geometrical argument. Let $\mathbf {v} ({x})$ represent the velocity of the pore fluid at some point $x$ in the slab. Assume that the pore fluid completely fills the available pore space. Representing the porosity by $\phi$, $\mathbf {u}$ is defined as:

$\mathbf {u} ={\frac {1}{\phi V}}\int _{\phi }\mathbf {v} (x)dV$

(4)

The Darcy velocity is the average over the entire volume $V$:

$\mathbf {v} _{D}={\frac {1}{V}}\int _{V}\mathbf {v} (x)dV$

(5)

Since $\mathbf {v} (x)=0$ outside of the porosity, eqn. 5 can be limited to the pore space:

$\mathbf {v} _{D}={\frac {1}{V}}\int _{\phi }\mathbf {v} (x)dV$

(6)

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