A numerical model for the prediction of radon flux from uranium mill tailings at Jaduguda, India

Abstract

Solid process fine waste or tailings of a uranium mill is a potential source of release of radiologically significant gaseous radon (²²²Rn). A number of variables such as radium (²²⁶Ra) content, porosity, moisture content, and tailings density can affect the extent of emanation from the tailings. Further, if a cover material is used for remediation purposes, additional challenges due to changes in the matrix characteristics in predicting the radon flux can be anticipated. The uranium mill tailings impoundment systems at Jaduguda have been in use for the long-term storage of fine process waste (tailings). A pilot-scale remediation exercise of one of the tailings ponds has been undertaken with 30 cm soil as a cover material. For the prediction of the radon flux, a numerical model has been developed to account for the radon exhalation process at the remediated site. The model can effectively be used to accommodate both the continuous and discrete variable inputs. Depth profiling and physicochemical characterization for the remediated site have been done for the required input variables of the proposed numerical model. The predicted flux worked out is well below the reference level of 0.74 Bq m⁻² s⁻¹ IAEA (2004).

Appendix

Tri-diagonal matrix algorithm.

The tridiagonal matrix equation is:

$$A\left(i\right){C}_{a}\left(i+1\right)+B\left(i\right){C}_{a}\left(i-1\right)-F\left(i\right){C}_{a}\left(i\right)=Co\left(i\right)$$

(27)

Equation (27) can be written as:

$${C}_{a}\left(i\right)+Q\left(i\right){C}_{a}\left(i-1\right)+P\left(i\right){C}_{a}\left(i+1\right)=R\left(i\right)$$

(28)

where $P\left(i\right)=-\frac{A(i)}{F(i)}$, $Q\left(i\right)=-\frac{B(i)}{F(i)}$, and $R\left(i\right)=-\frac{Co(i)}{F(i)}$ are the modified coefficients of the tri-diagonal matrix equation. For 1st node, Q(1) = 0 and Eq. (28) for 1st node can be written as:

$${C}_{a}\left(1\right)+P\left(1\right){C}_{a}\left(2\right)=R\left(1\right)$$

(28)

Similarly for 2nd node:

$${C}_{a}\left(2\right)+Q\left(2\right){C}_{a}\left(1\right)+P\left(2\right){C}_{a}\left(3\right)=R\left(2\right)$$

(29)

Combining Eq. (28) and Eq. (29), one can obtain: 

(30)

where $\check{P}\left(2\right)=\frac{P(2)}{1-P\left(1\right)Q(1)}$ and $\check{R}\left(2\right)=\frac{R\left(2\right)-R\left(1\right)Q(1)}{1-P\left(1\right)Q(1)}$. Repeating the same process in subsequent equations, the derived recurrence relations are written in Eqs. (31.1) and (31.2):

(31.1)

(31.2)

where $\check{P}\left(1\right)={\text{P}}\left(1\right)$ and $\check{R}\left(1\right)={\text{R}}\left(1\right)$. The modified form of Eq. (28) using Eqs. (31.1) and (31.2) is shown in Eq. (32).

(32)

For the N^th node, with ${C}_{a}\left(N+1\right)=0$, one can obtain ${C}_{a}\left(N\right)= \check{R}\left(N\right)$, and reverse substitution in Eq. (32) in the order i = N-1, N-2, ……0.1 will give radon concentration at each node. Flux at the boundary has been estimated using Eq. (33):

$$flux=-{n}_{e}\left(N\right)D(N)\frac{{{\text{C}}}_{{\text{a}}}\left({\text{N}}+1\right)-{{\text{C}}}_{{\text{a}}}\left({\text{N}}\right)}{{\text{dZ}}/2}$$

(33)