14.3.1 Simple analytical model of crack generation

The effects of the stress generated by gas pressurisation were studied using a simple model of a spherical gas source in spherical repository. The model is illustrated schematically in Fig. 14.1. The ultimate effect of stress will be a failure of the backfill resulting in the formation of a crack. If the region of gas generation shown in Fig. 14.1 is considered to be a void then backfill failure at the void surface will occur if the tangential hoop stress exceeds the tensile strength of the backfill. Hence, in the model the crucial stress for determining the failure of the material is considered to be the tangential hoop stress at the surface of the void.

The relationship between the tangential hoop strain at a particular radius r and the radial and tangential hoop stresses acting at that position is given by Roark and Young (1975):

$\text{tangential hoop strain}=\frac{1}{E}\left[\left(1-\nu \right)S_{\text{θ}}-\nu S_{\text{r}}\right]$ [14.1]

[14.1]

where:

S_r and S_θ are the radial and tangential hoop stresses, respectively, and ν and E are the Poisson’s ratio and Young’s modulus for the backfill material.

The tangential hoop strain at a particular radius is also equal to the ratio of the radial displacement at that point to the radius.

Analytical solutions exist for the tangential hoop stress and radial stress at a given radius for a spherical pressure vessel (Roark and Young, 1975). The repository model can be considered to be equivalent to such a situation and hence the equations for the stresses at radius r and the radial displacement of the inner surface y are:

$S_{\text{r}}=\frac{-\text{Δ}P{R}^3\left({\xi }^3-r^3\right)}{r^3\left({\xi }^3-R^3\right)}$ [14.2]

[14.2]

$S_{\text{θ}}=\frac{\text{Δ}P{R}^3\left({\xi }^3+2r^3\right)}{2r^3\left({\xi }^3-R^3\right)}$ [14.3]

[14.3]

$y=\frac{\text{Δ}P}{E}\left[\frac{\left(1-\nu \right)\left({\xi }^3+2R^3\right)}{2\left({\xi }^3-R^3\right)}+\nu \right]$ [14.4]

[14.4]

where:

ΔP is the excess pressure in the void (compared to the hydrostatic pressure in the medium surrounding the repository) and

R and ξ are the radii of the void and the repository, respectively.

The above equations are derived for the particular case where the porosity of the repository construction material is zero.

The radial displacement of the inner surface of the repository, that is the surface of the void, can be derived from equations (14.2)–(14.14) by putting the radius at which the stress is to be calculated r equal to the void radius R. It can be shown that if the outer radius ξ is more than an order of magnitude greater than the void radius. Equations (14.2)–(14.14) can be further simplified by assuming that the outer radius is effectively infinite. The dimensions of a repository are likely to be of the order of tens of metres whilst the voids will be of the order of a few tens of centimetres. Hence, the assumption that the outer radius is effectively infinite is reasonable for this model of stress generation. The radial displacement of the void surface y_v is then given by:

$y_{\text{v}}=\frac{\text{Δ}PR\left(1+\nu \right)}{2E}$ [14.5]

[14.5]

Dividing the radial displacement of the void surface, given by equation (14.5), by the radius of the void and equating to the tangential hoop strain given in equation (14.1) produces an expression for the tangential hoop stress at the void surface:

$S_{\text{θ}}=\frac{1}{\left(1-\nu \right)}\left[\frac{\text{Δ}P\left(1+\nu \right)}{2}+\nu S_{\text{r}}\right]$ [14.6]

[14.6]

For the zero porosity case, the radial stress can be derived from the equations given previously. This gives a radial stress equal in magnitude to the excess pressure in the void, although negative in sign. The tangential hoop stress obtained from equation (14.6) is then equal to half the excess pressure. This is a standard result for a pressure vessel.

The effect of the porosity of the backfill material has been included into the model by modifying the value used for the radial stress:

$S_{\text{r}}=-\left(1-\epsilon \right)\text{Δ}P$ [14.7]

[14.7]

where ε is the fractional porosity of the material. This approach is rather simplistic but, as discussed below, it can be shown to be a good approximation for the conditions applicable to the behaviour of a repository. Substituting the equation for the radial stress into the expression for the tangential hoop stress given in equation (14.6) gives:

$S_{\text{θ}}=\frac{\text{Δ}P}{2}\left[1+\frac{2\nu \epsilon }{\left(1-\nu \right)}\right]$ [14.8]

[14.8]

This expression can be used to calculate the hoop stress using appropriate parameters. The backfill is considered to fail if the calculated stress exceeds the tensile strength of the material.

14.3.2 Numerical solution for non-zero porosities

The analytical solution for the tangential hoop stress at the surface of the void is an approximation which takes limited account of the effect of the backfill porosity. The validity of this approximation was tested using a numerical approach.

The numerical model explicitly considered the inward and outward forces acting on a spherical shell within the backfill, centred on the gas-generating inclusion, with an inner radius r and an outer radius r + dr. The inward force F_i and outward force F₀ are, respectively:

$F_{\text{i}}=4\pi r^2\left(\epsilon \text{d}P-S_{\text{r}}\right)$ [14.9]

[14.9]

and

$F_0=4\pi {\left(r+\text{d}r\right)}^2\left[\frac{2S_{\text{θ}}\text{d}r}{r}-\left(S_{\text{r}}+\text{d}{S}_{\text{r}}\right)\right]$ [14.10]

[14.10]

where:

dP is the pressure difference across the shell and

dS_r is the change in radial stress across the shell.

In a stable situation these forces will be equal. Equating the two forces gives:

$\text{d}{S}_{\text{r}}=\frac{S_{\text{r}}{r}^2}{{\left(r+\text{d}r\right)}^2-1}+\frac{2S_{\text{θ}}\text{d}r}{r}-\epsilon \text{d}P$ [14.11]

[14.11]

The radial strain is equal to the rate of change of radial displacement with radius, that is strain is given by dy/dr. The standard definition of radial strain gives:

$\text{radial strain}=\frac{1}{E}\left(S_{\text{r}}-2\nu S_{\text{θ}}\right)$ [14.12]

[14.12]

Equations (14.9)–(14.14) were used to determine the radial displacement of the void surface by iterating until a constant value of radial stress was obtained. The boundary conditions imposed on the procedure were that the radial stress at the void surface was equal to the excess pressure, modified by porosity, and that at infinite radius was zero. The radial displacement was then used to calculate the tangential hoop stress.

The numerical solution was used to obtain the hoop stress at the void surface and the value compared to that obtained using the analytical approximation. It was found that the difference between the two solutions was negligible and the analytical equation, combined with a description of the pressurisation of the void given in Section 14.3.3, was adopted for subsequent calculations.

14.3.3 Pressurisation of the void

Gas is considered to be generated within the void in the material. The excess pressure produced by the gas generation will depend on the rate at which the gas can flow through the material. The volumetric flow rate within the backfill material at a radius r, denoted dV/dt, is determined by the pressure at that position P_r and the gas generation rate Q, measured at the hydrostatic pressure P:

$\frac{\text{d}V}{\text{d}t}=\frac{\text{QP}}{P_{\text{r}}}$ [14.13]

[14.13]

The volumetric flow rate is also given by the Darcy equation (equation 1.9), provided that the value of the pressure gradient used is that at the appropriate radius. Combining equations (1.9) and (14.13) and rearranging gives an equation for the pressure gradient at radius r:

$\frac{\text{d}p}{\text{d}r}=\frac{\text{eQP}}{\text{AK}{P}_{\text{r}}}$ [14.14]

[14.14]

where A is the area of the spherical shell of radius r. This expression can be integrated between the void radius R and the outer radius of the repository ξ. This gives an expression for the pressure produced in the void P_v:

$P_{\text{v}}^2-P^2=\left[\frac{1}{R}-\frac{1}{\zeta }\right]\frac{Qe{P}_{\text{atm}}}{2\pi K}$ [14.15]

[14.15]

If it is assumed that the outer radius is effectively infinite, equation (14.15) can be simplified and rearranged to yield an equation for the difference between the pressure in the void and the hydrostatic; pressure, the excess pressure ΔP used in previous equations:

$\text{Δ}P=\frac{Qe{P}_{\text{atm}}}{2\pi KR\left(P_{\text{v}}+P\right)}$ [14.16]

[14.16]

Substituting the expression for ΔP given in equation (14.16) into equation (14.8) gives an expression for the tangential hoop stress generated at the void surface by a given void pressure P_v:

$S_{\text{θ}}=\frac{1+2\nu \epsilon }{1-\nu }\frac{Qe{P}_{\text{atm}}}{2\pi KR\left(P_{\text{v}}+P\right)}$ [14.17]

[14.17]

This equation is used to calculate the tangential hoop stresses in the discussion in Section 14.4.

14.4.1 Gas generation rate

The anticipated maximum gas generation rate is approximately one repository volume of gas per year, measured at standard temperature and pressure. The model is intended to represent the venting of gas from a single waste drum with an approximate volume of 1 m³. Hence, the typical value of the generation rate was taken to be 1 m³ per year at standard temperature and pressure. The range of generation rates was specified as 0–5 m³ per year. The variation in stress with gas generation rate is shown in Fig. 14.2. The stress is increased as the gas generation is increased.

14.4.2 Hydrostatic pressure

The majority of current radioactive waste disposal strategies envisage the placement of a repository at a depth of up to 1000 m below ground level. This indicates a potential range of hydrostatic pressures of between 100 kPa and 10 MPa if the repository is fully saturated by the groundwater. Figure 14.2 demonstrates that the stress is inversely dependent on the hydrostatic pressure and hence the lower pressure of 100 kPa was adopted as the typical value to represent a plausible worst case. The reduction in stress with increasing hydrostatic pressure is attributed to the compressibility of the gas. As the pressure increases, the volumetric flow rate is decreased. This behaviour implies that a repository will be more susceptible to gas-generated cracking before resaturation by groundwater after closure as the effective hydrostatic pressure will be lower.

14.4.3 Fractional porosity

The data given in Table 6.2 indicates a range of fractional porosities of between about 0.1 and 0.6 for the experimental materials. The relationship between the measured values of fractional porosity and that available for gas flow is not known. The typical value was taken as 0.5, close to the values given for the backfill grouts in Table 6.2. The analytical solution demonstrates that gas-generated stress is linearly dependent on the fractional porosity of the material, assuming constant permeability. The stress increases by a factor of 0.5 as the porosity is varied between zero and unity. The effect of porosity on the stress may not be adequately modelled by the analytical approximation and the dependence of stress on porosity may not be as simple as indicated here.

14.4.4 Void radius

The model is intended to simulate the effects of gas vented from waste drums. It is expected that any vent will be of the order of the size of the drum; approximately 0–1 m in dimension. The typical value of 0.05 m was selected to represent a vent of 100 mm diameter in the drum. Figure 14.2 demonstrates that the stress generated at the surface of the void is strongly dependent on the void radius. The requirement for a void dimension is a limitation in the overall model, and Fig. 14.2 shows that the value of the void radius is crucial in determining the level of stress.

14.4.5 Poisson’s ratio

The expected value of the Poisson’s ratio for an ideal material is 0.5. The values for real materials tend to be between about 0.2 and 0.4 and hence a range of 0–0.5 was investigated. The typical value was taken to be 0.2. The stress increased by a factor of two as the Poisson’s ratio was varied between 0 and 0.5. Variation in this parameter is not of primary importance in determining the stress.

14.4.6 Permeability coefficient

As has been discussed previously, the repository is likely to become water saturated at some stage after closure and hence the gas permeability will be influenced by the presence of water. Figure 14.2 demonstrates that the stress is strongly dependent on permeability for permeability coefficients below about 10⁻¹⁶ m². Table 6.5 shows that the permeability coefficients for water-saturated materials are likely to lie below this value and hence will be crucial in determining the stress. The value of the permeability coefficient used as the typical value was 10⁻¹⁸ m², somewhat below the measured permeability coefficient for the reference backfill grout in the 100 % relative humidity (RH) conditioned state.