# 4SIGHT Manual: A Computer Program for Modelling Degradation of Underground Low Level Waste Concrete Vaults/Calculated Material Properties

​10Calculated Material Properties
All of the concrete physical parameters (e.g., diffusivity, permeability, etc.) are user specified inputs to 4SIGHT. However, in cases where not all properties are available, missing quantities must be approximated using existing correlations. The physical properties must be established due to both hydration and leaching. The physical properties due to hydration are the initial conditions. However, as the porosity changes due to leaching, corrected values of the physical parameters are needed.
10.1Hydration
The hydration of cement can be approximated by a reaction between tri-calcium silicate (C3S) and water, forming a calcium silicate hydrate (CSH). A more elaborate model incorporating multiple mineral phases would require chemical analysis of the cement and yield relatively little additional information concerning degree of hydration. The weight ratio of water to cement, ${\frac {w}{c}}$, is the oft reported quantity to characterize the concrete mix. After some period of hydration, the fraction of the initial C3S which has hydrated is the degree of hydration, $\alpha$. The relation between these two properties and porosity can be determined stoichiometrically [7],
| $\phi =1-{\frac {1+1.31\alpha }{1+3.2{\frac {w}{c}}}}$ | (25) |
and is valid for ${\frac {w}{c}}$ values used in practice.
The diffusivity can be related to either ${\frac {w}{c}}$ or $\phi$. After the first 100 days of hydration the transport properties of most cement pastes are near their asymptotic values. Although the values are still changing after 100 days, these changes are small compared to the accuracy with which these transport measurements can be made. Due to this apparently steady state, an empirical relation between $D_{o}$, the chloride diffusivity in $m^{2}/s$, and ${\frac {w}{c}}$ was developed for cement paste by Atkinson, Nickerson, and Valentine[8] and Walton, Plansky, and Smith[9]
| $\log _{10}D_{o}=6.0{\frac {w}{c}}-9.84$ | (26) |
${\frac {w}{c}}$
​$D_{o}$ for cement paste can also be related to $\phi$ using the universal relation for the formation factor due to hydration, $\vartheta$, [10]:
| $\vartheta ={\frac {D_{o}}{D_{Cl-}^{f}}}=0.001+.07\phi ^{2}+1.8(\phi -.18)^{2}H(\phi -.18)$ | (27) |
where $D_{o}$ is the concrete chloride diffusivity, $D_{f}$ is the free ion diffusivity of chloride ions, and $H(x)$ is the Heaviside function. The quantity $\vartheta$ is a constant for all ions.
The above equations relate the diffusivity of cement paste to water:cement ratio or porosity. A relationship is now needed between cement paste diffusivity and concrete diffusivity. Experimental results of Luping and Nilsson[11] for cement paste and mortar suggest that their diffusivities are approximately equal. Additionally, results from numerical experiments by Garboczi, Schwartz, and Bentz[12] investigating the effects of aggregate-paste interfacial zone diffusivity upon bulk diffusivity suggest that concrete diffusivity is approximately equal to the paste diffusivity.
Given ${\frac {w}{c}}$, the permeability can be approximated from the data in Hearn, et al.[13]:
| $k=10^{5.0{\frac {w}{c}}}\times 10^{-21}m^{2}$ | (28) |
for ${\frac {w}{c}}$ in the range (0.35,0.80).
10.2Leaching
Once $D$ and $k$ have been established, changes due to leaching can be calculated from $\vartheta (\phi _{o})$, where $\phi _{o}$ is the initial porosity due to hydration. As the $Ca(OH)_{2}$ is leached, the porosity increases. Let the value of porosity after leaching be $\phi ^{\prime }$, and the diffusivity be $D^{\prime }$. Unfortunately, the ratio $D^{\prime }/D^{f}$ is not simply $\vartheta (\phi ^{\prime })$ because as calcium hydroxide is leached from the paste the ratio $D^{\prime }/D^{f}$ does not retrace eqn. 27. Rather, the ratio $D^{\prime }/D^{f}$ is greater than $\vartheta (\phi ^{\prime })$, as demonstrated by the NIST microstructural model.
The NIST cement microstructural model[14] was used to determine the relation for $D^{\prime }/D_{o}$ upon leaching. Results for a ${\frac {w}{c}}=0.35$ paste are shown in Figure 2. The formation factor decreases with decreasing porosity due to hydration, denoted by circles. Upon leaching of the calcium hydroxide, the formation factor follows the curve denoted by squares. Given the following definitions:
| $\vartheta _{o}=\vartheta (\phi _{o})$$\vartheta ^{\prime }=\vartheta (\phi ^{\prime })$ | (29) |
An empirical relation was developed to relate the leached pore structure to the undamaged pore structure.
| $\xi =\vartheta _{o}+2.0(\vartheta ^{\prime }-\vartheta _{o})$ | (30) |
and is shown by the solid curve in Figure 2. Therefore, the ratio of the leached value of diffusivity, $D^{\prime }$, to the initial value $D_{o}$ is
| ${\frac {D^{\prime }}{D_{o}}}={\frac {\xi }{\vartheta _{o}}}$ | (31) |
Figure 2: Formation factor, $D/D^{f}$, due to hydration (circles) and to leaching (squares) for w/c=0.35 (after [14]). The solid line represents the approximation used by 4SIGHT for the leaching formation factor.
THe relative change in permeability can be calculated using $\xi$. The Katz-Thompson equation relates permeability to $\vartheta$ [15]:
$k={\frac {d_{c}^{2}}{226}}\vartheta$
where $d_{c}$ is the diameter of the largest sphere which can pass through the pore space of the sample. Also, given that $d_{c}\propto \vartheta$ [16], the relative change in permeability becomes
${\frac {k^{\prime }}{k_{o}}}=\left({\frac {\xi }{\vartheta _{o}}}\right)^{3}$
The final dimensionless advection-diffusion equation is a combination of eqns. 14, 31, and 33:
${\frac {\partial c_{i}}{\partial T}}=\nabla \cdot {\frac {\xi _{i}}{\vartheta _{o}}}\nabla P+\left({\frac {\xi _{i}}{\vartheta _{o}}}\right)^{3}\mathbf {v} _{D}\cdot \nabla c_{i}$
There is one such equation for chemical species $i$.
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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