Numerical Solution of the Fokker-Planck Equation for the Overflow Probability of a Radioactive Near Surface Repository by the Crank-Nicolson Method: Preliminary Results

In an earlier paper we discussed the analytical solution of the Fokker-Planck (FP) equation for evaluating the overflow probability for the near surface repository of Abadia de Goiás, Brazil. Some preliminary considerations were approached in this reference. However, a formal approach for validating the results needed to be formulated. In this sense, we face the problem in this paper by approaching the solution for the overflow probability by numerical methods. An implicit numerical method was used, namely, the Crank-Nicolson method, which is known to be numerically stable. A discussion is performed on the initial and boundary conditions to solve the FP equation in order to obtain the probability density needed for calculating the overflow probability (in this sense, contrary to the analytical solution, the numerical solution does not need to be truncated because it starts at t = 0 with the defined initial condition). This latter depends on the repository institutional control period and, as in earlier published work on this subject, the institutional control period is varied from 5 years to 60 years. This wide range variation interval is justified by the fact that the repository design considers initially an institutional control period equal to 50 years. The numerical results for the probability density agree in terms of magnitude orders with the analytical ones published earlier. The overflow probabilities obtained are in good agreement with the ones obtained by the analytical methods.

Keywords: Near surface repository, Abadia de Goiás, Overflow probability, Fokker-Planck equation, Crank-Nicolson method.