In the risk assesment of highly radioactive waste in deep geological repositories, the migration of radionuclide chains in porous media by advection and dispersion processes are of main concern. The computation of activities (or cumulated activities over time) in confined zones of the medium are required in most of the contamination scenarios (e.g.: radionuclides extracted from wells, discharged to rivers or accumulated on the upper surface of the geological medium). In this context, we address here the problem of computing doses by Monte Carlo at specific points or on specific surfaces with low statistical uncertainties.The random walks of the Monte Carlo simulation are constructed from an integral equation applied recently to the migration of a radionuclide chain [1]. The simulation uses the solution of an adjoint reference problem to improve the efficiency of the calculations [2].During a pre-processing step, transition probabilities from any location of the domain to specific points or surfaces are computed and memorized. These additional transitions are taken into account in a re-organized simulation with new banches in the walks. This method allows to force, at each step, transitions of random walkers to specific zones where their presence is expected with a very low probability.The organization of the simulations will be described and the computation of point and surface scores will be illustrated by numerical results.

The P.R.A, of a dynamical system with parametric uncertainty is a problem that can be treated semianalytically in the frame of dynamic reliability. The probability of damage is obtained explicitly either by solving a direct or an adjoint problem, assuming a general elliptic distribution for the parameter vector.

Monte Carlo simulation has become an important tool for the estimation of reliability characteristics, since conventional numerical methods are no more efficient when the size of the system to solve increases. However, evaluating by a simulation the probability of occurrence of very rare events means playing a very large number of histories of the system, which leads to unacceptable computation times. Acceleration and variance reduction techniques have to be worked out. We show in this paper how to write the equations of Markovian reliability as a transport problem, and how the well known zero-variance scheme can be adapted to this application. But such a method is always specific to the estimation of one quantity, while a Monte Carlo simulation allows to perform simultaneously estimations of diverse quantities. Therefore, the estimation of one of them could be made more accurate while degrading at the same time the variance of other estimations. We propound here a method to reduce simultaneously the variance for several quantities, by using probability laws that would lead to zero-variance in the estimation of a mean of these quantities. Just like the zero-variance one, the method we propound is impossible to perform exactly. However we show that simple approximations of it may be very efficient.

In Dynamic Reliability, the parametric uncertainty may have a strong influence on the results. The preliminary results of a method allowing to evaluate this influence are presented in this paper. The application is a reactivity transient in a nuclear reactor with an uncertain slope of reactivity. The numerical results give the probability that the power or the energy crosses a given level during a reactivity accident with a gaussian distributed slope of reactivity.

A new Monte Carlo (M.C.) method is presented for the risk assessment of deep radioactive waste repositories in clay. The method relies on a biased M.C. simulation for the migration of radionuclide chains combined with the M.C. sampling of uncertain parameters. The M.C; simulations are applied to a radionuclide chain with a band-release source. The uncertainties considered are mainly those due to the values of the retardation factors in clay.

The methodologies employed in the probabilistic safety assessment (PSA) of hazardous engineering systems have reached a high level of maturity. However, a few issues are still worth of further investigation in order to increase the confidence in the results obtained. In this view, in the past 5–10 years, researchers in the field of reliability have proposed a more `dynamic' approach to PSA with the aim of addressing issues concerning the possible mutual interactions between the hardware system states and the plant physical evolution. Nonetheless, some objections have been raised against such a dynamic approach, especially against its practical complexity and the lack of a clear definition of the domain of its applicability. In this paper, an attempt to define more precisely the field of application for a dynamic approach is propounded on the basis of the concept of accident duration. The qualitative discussion is supported with examples of postulated severe accidents in nuclear power plants like those investigated in level-2 PSA. The application of Monte Carlo simulation as a tool capable, in principle, of handling all the features of dynamic PSA is illustrated. Monte Carlo algorithms are illustrated aiming at improving the stochastic part of the analysis and the deterministic integration as well, so as to allow for a considerable reduction in the computation times.

In this paper, a non-analog Monte Carlo methodology is applied to the transport of radionuclide chains in a geological medium. A Monte Carlo simulation is first constructed from an integral equation for the concentration of one radionuclide. This integral equation depends on the solution of an adjoint problem which brings more efficiency to the simulations. The simulation techniques are described. They are then generalized for the transport of several elements of a radionuclide chain. Finally, the method is illustrated by numerical tests.

A biased Monte Carlo methodology is presented for solving the transport of radionuclide chains through a porous medium in the context of the risk assessment of radioactive waste repositories. It is based on the construction of random walks from an integral equation. This leads to a biased Monte Carlo simulation because it uses the solution of an adjoint reference problem to improve the efficiency of the calculations. The transport of a radionuclide chain is modeled by introducing the notion of a radionuclide "state." The consequence is that only one integral equation has to be considered for the simulation in a continuous - discrete space (r,t;i), where r is the radionuclide position vector, t is time, and i is the radionuclide state. Transport in a random velocity field is also considered by using double randomization techniques. The methodology is illustrated by numerical results on test problems; the score of the simulations being the quantity of radionuclides transferred, during the mission time, to the upper surface of the geological domain. Validations of the simulations are first realized by comparison with analytical solutions, and the influence of biasing techniques is put in evidence. Finally, simulations conducted simultaneously with the generation of a large number of random velocity fields illustrate the feasibility of the method for the transport of radionuclides in a stochastic medium.

The problem of model uncertainty versus model inaccuracy is examined in the light of the concept of the ‘probability of correctness of a model under a given context’ introduced by Apostolakis. To avoid possible difficulties linked with this concept, a distinction is introduced between ‘predictive’ models and ‘constitutive’ models, the former being generic in the sense that they can host the latter as submodels. A metric or distance between linear models as well as an objective of the model are introduced, from which we can give an operational definition of ‘model uncertainty’ (with respect to distribution of parameters of the associated constitutive models) and of ‘model accuracy’ with respect to a reference model. Finally the choice of a predictive model is linked to a loss function and a cost of using or defining a model.

Hydraulic potential, solution of the groundwater flow equation, is submitted to several types of uncertainties. These uncertainties are found not only in the parameters of the flow equation and in the boundary conditions, but also in the structure of the geological medium. This 'structure uncertainty' comes from the necessary interpolation between borehole information or seismic information at different locations of the geological site.In the case of parameter uncertainties, sensitivity coefficients are usually evaluated in order to solve an inverse problem based on the knowledge of measures of hydraulic potential at locations sparsely distributed in the flow region.We propose, in this paper, to apply the same principle to structure uncertainty, that is, to calculate explicitly (under some hypotheses) the sensitivity coefficients of hydraulic potential for displacements of interfaces delimiting successive geological layers. Therefore, we have derived a first order variational formula which is used for the solution of the inverse problem in an iterative algorithm. Some iterations are used to reduce the effects of approximations made in the variational formula. The algorithm is described and numerical results are discussed for test problems. (C) 1997 Elsevier Science Limited.

Recently, measurements on the statistics of the number of electrons emitted due to MeV light ion impact on solid targets have been fitted adequately by negative binomial distributions. We present here Monte Carlo simulations for backward and forward emissions induced by protons incident on thin carbon foils in the energy range 250 keV –2 MeV, and compare them with experimental results. A general discussion on the statistical distribution of the number of emitted electrons is given. A transport model giving the general form of the probability generating function (pgf) of the distribution of the number of emitted electrons is also proposed.

It is well known that the electron emission yield from insulators is larger than from metals. Most of the previous works on electron emission have been on metals, and very few works have been done on insulators, either experimentally or theoretically. We present in this paper a microscopic description of proton induced electron emission from alumina. Elastic and inelastic interactions for the incident protons and excited electrons are considered as well as the charging-up of the target. The characteristics of proton induced electron emission from alumina (especially the yield γ) are calculated by a Monte Carlo simulation method. A comparison with aluminum targets is done, especially with respect to the influence of the electron inelastic mean free path and of the work function.

In this paper we prove that classical event trees can be derived rigorously from the theory of Probabilistic Dynamics only if we assume setpoint transitions, i.e., transitions that depend only on algebraic combinations of instantaneous values of the process variables (setpoints). Approximate formulae are also given in the more general case where the time of actuation after a setpoint is reached is stochastic, extending the classical event tree approach. The paper also reviews the theory of Probabilistic Dynamics and shows some important simplifications that can be used under certain common situations.

Aggregation is a way to reduce the size of Markovian reliability and availability problems. Since exact aggregation is only possible in exceptional cases, we introduce a canonical partition for the Markovian states for systems with 2-state components that allows a systematic study of approximate aggregation. Restriction and prolongation operators are introduced to define aggregation. Four methods are proposed to define approximate systems of birth and death type. Examples are given for k-out-of-4 systems and for a 7 component system.

This paper introduces a new finite element approximation for multi-dimensional transport problems in piecewise homogeneous media. The transport equation is solved using a Galerkin technique with polynomial basis functions in space-angle variables derived from asymptotic transport theory. The phase space is partitioned into cells consistent with the geometry and having each an elemental expansion which is not a tensor product. improved accuracy may be obtained by multiplying the number of cells or/and increasing the polynomial degree. Numerical results on 1D and 2D reference problems in square geometry show a good agreement with other approximate methods.

A three-dimensional quasi-asymptotic approximate equation is developed for the transport of radionuclides in a stochastic velocity field. This approximation is derived from an integro-differential equation of transport in stochastic media, commonly encountered in hydrogeology. The quasi-asmptotic equation turns out to be a generalised Telegrapher's equation as found by Williams in the particular context of fractured media.We obtain the Telegrapher's equation without specifying the causes responsible for the random velocity field. Our model may thus be applied in porous media as well as in fractured media.We give the developments leading to the analytical solution of the three-dimensional Telegrapher's equation for constant parameters. This solution is then visualised for a source in the form of a square wave.

Markovian modelling is a widespread framework to solve PRA problems [1, 2]. Though this assumption implies that the system is memoryless and thus that a large number of circumstances cannot be modelled in this fashion, addition of supplementary variables [3] or of physical variables [4] allows to handle a much wider range of applications than those satisfying stricto senso the Markovian hypothesis. The drawback of this simple modelling is the large number of states that have to be considered in a realistic application, and thereby the size of the system to solve. Monte Carlo simulation [5, 6] appears to be a practicable way to circumvent this problem.

The computation of availability or reliability in a Markovian approach involves the solution of O.D.E.: dp/dt = L(t)p, where the transition matrix is time dependent, at least when some aggregation of the states is introduced to reduce the size of the problem. Methods of solution like uniformization are in this case unapplicable and we investigate here four explicit and six implicit R.K. methods from the point of view of stability, amount of numerical work and accuracy. The test problem chosen allows an analytical solution, a uniformization method (when L(t) is constant) and a fortiori all R.K. methods. The implicit trapezoidal rule used with a variable step scheme appears to be the best compromise between accuracy and computational work.

In this paper we will develop a new method of solution of large Markovian problems by means of auxiliary problems of smaller size. The influence graph of the physical components is built, each edge of the digraph symbolizing the influence (if any) of the composant (node of the graph) on the rate of failure or repair of another component. The analysis of the digraph introduces a natural ordering and the buildup of approximate differential systems which are solved recursively for the conditional probabilities. Two examples are given to illustrate the method.