## Deterministic and probabilistic analysis methods

The essence of deterministic safety analysis (DSA) is the solution of differential equations simulating the processes of radionuclide transfer and ionizing radiation from the source to the environment. Solving the equations gives the concentration or the volume activity of certain radionuclides at a given distance from the source at a given time. Comparing these results with regulatory requirements enables conclusions to be drawn about the safety of the environment. At the next stage of the analysis, calculated concentrations are transferred to the projected radiation doses to control groups of the population, using different scenarios and pathways of radionuclides in the body. Comparing the predicted and normative values enables conclusions to be drawn about the safety of the population. The equations used contain deterministic functions and coefficients, so giving the name to the method.

Solving the equations requires many simplifying basic assumptions including: use of a point source, the uniformity of the environment, fixing in space and time the parameters and coefficients of the equation and baseline data. In the case of an extended source, such as a near-surface disposal facility, account must be taken of its heterogeneity. To solve the equations, depending on the model, 10-16 parameters must be set that describe the source properties, the engineered barriers and the surrounding geological environment. These settings are heterogeneous in the physical sense, and so difficult to define and with a large scatter in numerical values. Equations are solved by the usual finite difference method using readymade software products such as MathCard Enterprise Editoria V11.A, AMBER from Quantisci (UK), ModFlow (USA), or MT3D (USA).

Many parameters in the equations vary greatly in both space and time. This is a consequence of the stochastic nature of the environment and the changing external conditions. Therefore, solutions to the equations must be a random variable. It is well known empirically that the concentration of radionuclides in different, even neighboring areas, is substantially different. Such a property of the distribution of radioactive contamination is particularly pronounced after an accident with a significant release of radioactive substances into the environment such as at Chernobyl or Fukushima. Once an accident occurs, the migration of radionuclides is governed by largely random processes, so it is natural to use the methods of probabilistic safety analysis (PSA). The purpose of PSA is to estimate the probabilities of certain accident scenarios over a given period of time and to identify the weakest elements of the complex in the disposal of RAW. The basis of the PSA methodology is a systematic analysis of the radiation-dangerous object, the selection of systems and components that make up the protective shield, and making event tree and failure trees with subsequent calculation of the probabilities of various scenarios of accident events. To perform the calculations it is necessary to access fundamental homogeneous data on the physical properties of elements of the physical barriers to ensure retention of radionuclides in the bulk of the medium. This data includes the following interdependent and replaceable parameters: the intensity of the element failures, mean time between failures, and failure probability. Such information can be obtained from technical regulations, manuals and handbooks, as well as the results of physical and field experiments, mathematical modeling or calculations by deterministic models.