# 4SIGHT Manual: A Computer Program for Modelling Degradation of Underground Low Level Waste Concrete Vaults/Ion Transport

​ 7Ion Transport
A single transport equation is developed to propagate ions through the slab. This equation can be converted into a finite difference equation so that it can be implemented in a computer program. A result of this approach is that time will advance in discrete intervals. After every time interval each computational element is put in chemical equilibrium using solubility products and a charge balance. This step is achieved through dissolution/precipitation of available salts. Any change in the quantity of solid salts in the pore space will change the ​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) |
$v_{D}$
$u$
$\mathbf {v} _{D}=\phi \mathbf {u}$
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
${\frac {\partial \rho }{\partial t}}+\nabla \cdot \rho \mathbf {v} =0$
To develop a continuity equation for porous media, find the average integral value of eqn. 8 over a representative volume, $V$:
${\frac {1}{V}}\int _{V}{\frac {\partial \rho }{\partial t}}dV+{\frac {1}{V}}\int _{V}\nabla \cdot \rho \mathbf {v} dV=0$
Rearranging the order of integration gives
${\frac {\partial }{\partial t}}{\frac {1}{V}}\int _{V}\rho dV+\nabla \cdot {\frac {1}{V}}\int _{V}\rho \mathbf {v} dV=0$
Finally, assuming $\rho$ is constant and using the definition of Darcy velocity gives
${\frac {\partial \phi }{\partial t}}+\nabla \cdot \mathbf {v} _{D}=0$
This is the result obtained by Slattery [4] for porous media. Substituting for $\mathbf {v} _{D}$ from eqn. 3 gives
${\frac {\partial \phi }{\partial t}}=\nabla \cdot {\frac {k}{\mu }}(\nabla p-\rho g)$
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:
$X={\frac {x}{L}}$
$T={\frac {t}{k_{o}/D_{o}}}$
$P={\frac {p}{\mu D_{o}/k_{o}}}$
$X$
$T$
$P$
$D_{o}$
$k_{o}$
$L$
${\frac {\partial c}{\partial T}}=\nabla \cdot {\frac {D_{i}}{D_{o}}}\nabla P+{\frac {k}{k_{o}}}\mathbf {v} _{D}\cdot \nabla c$
The equation will be simplified further later using material properties related to porosity.
7.5Atkinson-Hearne Model
As yet, a dependable mechanistic model does not exist for sulfate attack. Therefore, to incorporate sulfate attack, the depth of sulfate degradation is calculated using the Atkinson and Hearne [5] model:
$R={\frac {E\beta C_{o}C_{E}D}{\alpha \gamma (1-{?})}}$
| $E$ | Youngs modulus |
| $C_{o}$ | External sulfate concentration |
| $C_{E}$ | Concentration of sulfate as ettringite |
| $D$ | Sulfate diffusion coefficient |
| $\alpha$ | Roughness factor |
| $\gamma$ | Fracture surface energy |
| ${?}$ | Poisson ratio |
Since the Atkinson-Hearne model gives the location of the sulfate front, 4SIGHT presumes that all of the concrete behind the sulfate front has been completely disintegrated, giving it the properties of the surrounding soil. Since the transport coefficients of soil are much larger than concrete, the external boundary conditions are advanced to the sulfate front, creating a moving boundary condition.
The Youngs modulus, roughness factor, fracture surface energy, and Poisson ratio must be determined from experimental measurements. The quantity $C_{E}$ can be either calculated from the cement composition, or more accurately from experiment. To experimentally determine $C_{E}$, the sulfate reacted per unit mass hydrated cement is plotted against the logarithm of time. This data is fit to the equation [5]
$m=m_{o}\log _{10}\left({\frac {tc}{t_{r}c_{k}}}\right)$
where $m$ is the moles of sulfate reacted in the cement, $m_{o}$ is the free parameter of the regression, $t$ is time, $c$ is the concentration of sulfate in liquid, $t_{r}$ is the characteristic time for reaction, and $c_{k}$ is the concentration in kinetic experiments. The maximum value of $m$ ​can be calculated from the initial quantity of C3A in the cement. If this maximum value is labelled $m_{c}$, the value of $t$ for $m=m_{c}$ is the time at which all of the C3A is consumed, $t=t_{\infty }$.
The model proposed by Atkinson and Hearne assumes that reaction is the controlling rate process. Therefore, the time to spalling should be greater than the time required for complete consumption of the C3A. The time to spalling is [5]
$t_{\textit {spall}}={\frac {X_{\textit {spall}}^{2}C_{E}}{2Dc_{o}}}$
where
$X_{\textit {spall}}^{2}={\frac {2\alpha \gamma (1-v)}{E(\beta C_{E})^{2}}}$
In the event that $t_{\textit {spall}}<t_{\infty }$, $C_{E}$ must be calculated in a self consistent manner using eqns. 16, 17, and 18. More complete details can be found in [5]. For ordinary portland cements, it is likely that $t_{\textit {spall}}>t_{\infty }$. Therefore, since it is assumed that the quantity of external sulfate is sufficiently great to act as an infinite reservoir (external concentration is constant), $C_{E}$ can be calculated from the C3A content of the cement. Since each mole of ettringite requires one mole of $Al_{2}O_{3}$[6], the molar concentration of $Al_{2}O_{3}$ will be the molar concentration of ettringite. Given a concrete mix have $x_{cem}$ kilograms of cement per cubic meter of concrete, and the cement having a weight fraction of $\phi _{Al_{2}O_{3}}$ of aluminum oxide, the moles of ettringite formed per cubic meter of concrete is
$C_{E}={\frac {x_{cem}\phi _{Al_{2}O_{3}}}{0.10196}}$
with the $Al_{2}O_{3}$ molar weight of 0.10196 kg per mole.
7.6Effects of Cracks and Joints
Although not a degradation mechanism per se, effects due to the presence of cracks and joints can be incorporated into the models for transport of the ions. Since the roof slab will likely be a supported member such that the bottom of the roof slab will be in tension, it is assumed that cracks will appear on the bottom of the slab and extend upwards to the neutral axis of the slab. In the case of joints, 4SIGHT assumes that the joint extends through the entire depth of the root slab. The joint is filled with a joint compound with a known permeability and service life.
The permeability of a cracked slab is calculated assuming the crack walls are smooth and parallel. Given the square slab with $L$ and depth $D$ having cracks with width $w$ penetrating the full depth $D$ of the slab, the permeability of the slab is a weighted sum of ​the permeability of the crack, $k_{c}$, and the permeability of the uncracked concrete, $k_{o}$:
| $k_{s}={\frac {w}{L+w}}k_{c}+{\frac {L}{L+w}}k_{o}$ | (20) |
| $={\frac {w}{L+2}}{\frac {w^{2}}{12}}+{\frac {L}{L+w}}k_{o}$ | (21) |
Since $w^{3}/12$ is typically far greater than $Lk_{o}$, the permeability of the slab can be approximated by the permeability due to the crack. Further, if each of the cracks of width $w$ are spaced a distance $a$ apart, the permeability of the slab, $k_{s}$, is
| $k_{s}={\frac {w^{3}}{12a}}$ | (22) |
Joints can be handled in a similar manner as cracks. However, joints will typically be very much wider than cracks. Since joints will presumably extend the entire thickness of the slab, once the joint fails, the flow through the joint would overwhelm the transport of ions through the central portion of the slab. In fact, the transport coefficients would be as great as, or greater than, those of the soil. Therefore, upon failure of the joint, 4SIGHT assumes that the roof fails to impede the flow of water into the vault, the transport properties of the concrete should be approximated by the transport properties of soil, and the calculation ceases.
This work is in the public domain in the United States because it is a work of the United States federal government (see 17 U.S.C. 105).
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