## Application to the case of radionuclide escape from a near-surface disposal facility

This approach to safety assessment is demonstrated using an example with the environment as a protected object. The dangerous object is the RAW contained in a near-surface disposal facility. Initially, we need to identify which engineering and physical barriers prevent escape of radio nuclides into the environment. In the case of processing plants, barriers include: piping, valves, pumps, physical or radiological protection, etc. In the case of the disposal facility, the multi-barrier protective shield consists of three systems:

1. the matrix (container), which contains the RAW,

2. the repository, as an engineering and construction structure, and

3. the geological environment in which the repository is located.

To formalize the numerical calculations, the functioning of the individual protection system is described by the binary relation of вҖҳfailureвҖҷ to вҖҳworkвҖҷ. Decomposition of the protective shield to the individual systems is an essential part of the safety analysis. An emergency event which may be an incident, accident, occurrence, or situation, etc., is defined by a sequence of failures of the systems and components leading to adverse effects such as a loss of control over the ionizing radiation sources or uncontrolled escape of energy and matter from a dangerous object into the environment. This sequence of events is called a scenario. If a complex of the protective shield is divided into N systems, there are only M = 2N scenarios. In the case of near-surface disposal facility, N = 3 and M = 8. Functioning of the complex is displayed graphically in Fig. 9.4. Such a diagram is called an event tree.

In this diagram, the systems are presented in the form of columns. The rows represent the state of these systems. Armed with the event tree, it is easy to calculate the probability of each scenario from the general formula:

pm=n stqnm )sqnm, [9.2]

n=1         9.4 Emergent event tree for activity of disposal complex.

where Sn, n = 1, 2, … , N, is the failure probability of the n-th system, and Sn = 1 — Sn is the probability of the work state of this system; m = 1, 2, … , M is the scenario number; qnm is the state indicator of the system; qnm = 0 if in scenario m the system n is in a failure state, and qnm = 1 if the system is in a working state.

The probabilities Sn are calculated by analysis of the failure trees for each system. The failure tree is a logical connection between the system elements, connected by symbols вҖҳORвҖҷ, вҖҳANDвҖҷ, corresponding to addition or multipliВӯcation of the random failure events. The symbol вҖҳORвҖҷ links together the group of elements, failure of at least one of which leads to failure of the entire group. The failure probability of such a group is calculated from:

Jor

Sjor = 1 — Рҹ(1 — Ej), [9.3]

j=1

where Ej is the failure probability of the j-th element from the group; Jor is the number of elements in the group. The symbol вҖҳANDвҖҷ combines the group of elements, only the joint failure of which leads to failure of the whole group. The failure probability of such a group is calculated from:

Jand

Sjand = Рҹ Ej, [9.4]

j=1

where Jand is the number of elements in the group.

The failure probability of elements is calculated from:

Ej = 1 — exp(-Xjt), [9.5]

where Xj is the failure rate of a given element; and t is the time from start of observation or operation of the disposal facility.

We now consider analysis of alarm events associated with possible escape of radionuclides into the environment from the complex for disposal of RAW. Since the complex consists of three systems, event three is the same as in Fig. 9.4 . Since not all scenarios can be realized, we consider only the four scenarios of the alarm events that have physical meaning:

вҖў Scenario 1. Failure of all systems that make up the disposal complex.

вҖў Scenario 2. Joint failure of the matrices and the disposal facility.

вҖў Scenario 3. Only failure of system 2 (the repository). Physical represenВӯtation of this scenario is to destroy the structural elements of the reposiВӯtory with probable leakage of radionuclides beyond.

вҖў Scenario 4. Only joint failure of systems 2 (the repository) and 3 (the geological environment). Physical representation of this scenario is the escape of radionuclides from the disposal facility and their migration into the geological environment.

As the scenarios are interdependent, the sum of probabilities of all sceВӯnarios is 1. Therefore the probability of accident Pac = P1 + P2 + P3 + P4 and the probability of the work Pwork = 1 — Pac.

In the model representation, system 1 consists of two elements. Element 1 is the RAW itself, contained in a matrix or container. Element 2 is the body of the matrix. System 1 failure occurs when a failure occurs in element

1 or element 2 or both. Hence, these two elements are working on an вҖҳORвҖҷ scheme. The relevant failure tree is shown in Fig. 9.5. Physical representaВӯtion of element 1 failure is the radionuclide escape from the matrix body as a result of diffusion and leaching. Physical representation of the element

2 failure is the matrix degradation during its aging, corrosion, and cracking. Although the physical processes of failure of these two elements are interВӯrelated, from a model point of view it is convenient to present them as independent. Conditionally we can accept the failure rate of element 1 as 9.5 Failure tree of matrix with radioactive waste.

X1 = 9.1 x 10-11 1/year and of element 2as РҗРі = 1.6 x 10-81/year. From Eq. [9.5] we calculate the failure probability of elements 1 and 2. Then the failure probability of system 1 can be calculated from Eq. [9.3].

We now discuss the physical barrier in the RAW repository (system 2). From the system analysis standpoint, this system can be regarded as consistВӯing of the following elements: covering slabs, walls, a bottom, waterproofing. In turn, these elements may be composed of elementary units: concrete slabs, cement joints, beams, etc. The failure tree of system 2 is shown in Fig. 9.6 .

I t should be noted that, depending on whether the composition of a structural element contains slabs or joints, they may have different performВӯance parameters. Therefore in Fig. 9.6 the same names of the elements are presented with different numbers. The logical scheme of calculation can be easily understood from Fig. 9.6. System 2 failure occurs when the events are implemented by the probabilities denoted in Fig. 9.6 by symbols P1, P2 and P3. Then the system 2 failure probability is calculated from formula: S2 = 1 — (1 — P1)(1 — P2)(1 — P3) in accordance with Eq. [9.3]. Probabilities P1, P2, and P3 are calculated using Eq. [9.4]: P1 = P4E4, P2 = P5E4 E7, P3 = P6E4. 9.6 Failure tree of repository.

Probabilities P4, P5. and P6 are calculated in accordance with Eq. [9.3]: P4 = 1 — (1 — £0(1 — £2X1 — E3),P5 = 1 — (1 — EsXl — E) and P6 = 1 — (1 — E8) (1 — E9). Probabilities Ej are calculated from Eq. [9.5].

Physical representation of the system 3 failure is the escape of radionuВӯclides beyond the sanitary protection zone, or their penetration into the aquifer. From the standpoint of the system analysis, the geological environВӯment is composed of two elements. The first element prevents the horizontal migration of radio nuclides on their way to the border of the sanitary proВӯtective zone. The second element retains radionuclides during their vertical migration towards the aquifer. Obviously, these two elements are working on an вҖҳORвҖҷ scheme, so that the system 3 failure tree is the same as in Fig. 9.5. The failure rate of these elements is assumed to be equal to the reciproВӯcal of mean time between failures. This is the radionuclide migration time, which may be calculated from known geometric data and the speed of horizontal and vertical migration. Migration rate can also be obtained from the results of field measurements or from calculations by deterministic models of the radionuclides transport. With this information, Eq. [9.3] calВӯculates the system 3 failure probability. Then Eq. [9.2] is used to calculate the probability of each scenario. As an illustration, the results of such calВӯculations are shown in Fig. 9.7 . On the graph, ordinate is logarithm of the scenario probability, and the horizontal axis gives the time period during which this scenario can be realized. 