Page:4SIGHT manual- a computer program for modelling degradation of underground low level waste concrete vaults (IA 4sightmanualcomp5612snyd).pdf/43

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From comparison to eqn. 4, the relation between $v_{D}$ and $u$ is:

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

(7)

7.3Continuity

Once the Dary equation is incorporated into eqn. 2, the continuity equation is needed in order to update the pressures. Upon execution of the computer program, the following sequence is continuously reiterated: transport ions, attain chemical equilibrium, adjust transport properties, update boundary conditions. As the porosity of the concrete changes, the transport properties also change. The change in the transport properties will effect the hydraulic pressure distribution in the concrete since the transport properties will not by uniform throughout the concrete. The pressure distribution will be updated using the continuity equation for porous media.

The continuity equation for a fluid is

{\frac {\partial \rho }{\partial t}}+\nabla \cdot \rho \mathbf {v} =0

(8)

To develop a continuity equation for porous media, find the average integral value of eqn. 8 over a representative volume, $V$ :

{\frac {1}{V}}\int _{V}{\frac {\partial \rho }{\partial t}}dV+{\frac {1}{V}}\int _{V}\nabla \cdot \rho \mathbf {v} dV=0

(9)

Rearranging the order of integration gives

{\frac {\partial }{\partial t}}{\frac {1}{V}}\int _{V}\rho dV+\nabla \cdot {\frac {1}{V}}\int _{V}\rho \mathbf {v} dV=0

(10)

Finally, assuming $\rho$ is constant and using the definition of Darcy velocity gives

{\frac {\partial \phi }{\partial t}}+\nabla \cdot \mathbf {v} _{D}=0

(11)

This is the result obtained by Slattery [4] for porous media. Substituting for $\mathbf {v} _{D}$ from eqn. 3 gives

{\frac {\partial \phi }{\partial t}}=\nabla \cdot {\frac {k}{\mu }}(\nabla p-\rho g)

(12)

7.4Dimensionless Variables

Eqn. 2, 3, and 11 can be combined to form a system of equations to propagate ions through a porous media. However, the system of equations can be condensed by a transformation into dimensionless variables. Consider the following definitions: