Olivier Smidts

Transport of radionuclides in stochastic media: 1. The quasi-asymptotic approximation

Abstract 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 Telegraphers equation as found by Williams in the particular context of fractured media. We obtain the Telegraphers 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 Telegraphers equation for constant parameters. This solution is then visualised for a source in the form of a square wave.

A Lagrangian Stochastic Model for the Transport in Statistically Homogeneous Porous Media

A new type of stochastic simulation models is developed for solving transport problems in saturated porous media which is based on a generalized Langevin stochastic differential equation. A detailed derivation of the model is presented in the case when the hydraulic conductivity is assumed to be a random field with a lognormal distribution, being statistically isotropic in space. To construct a model consistent with this statistical information, we use the well-mixed condition which relates the structure of the Langevin equation and the probability density function of the Eulerian velocity field. Numerical simulations of various statistical characteristics like the mean displacement, the displacement covariance tensor and the Lagrangian correlation function are presented. These results are compared against the conventional Direct Simulation Method.

Theoretical and experimental study of the correlation between forward and backward electron emissions induced by H0 or H+ projectiles incident on carbon foils

Abstract The emission statistics of secondary electrons emitted from solid targets under ion impact has been recently the object of much interest from a theoretical as well as experimental point of view since it provides very useful information about the electron excitation process in the solid. In the case of very thin targets, it is possible to measure simultaneously the number of electrons emitted from both sides of the target and the exit charge state of the projectile. The resulting two variable (number of electrons emitted in the backward and in the forward directions) statistical distribution shows in some cases a clear correlation between the numbers of electrons emitted from both sides of the target. This correlation between backward and forward emission has been studied from Monte Carlo simulations and from experiments for H0 and H+ projectiles incident on thin carbon foils in the MeV energy range. It is shown that the charge exchange processes of the incident projectiles in the target play an important role in the interpretation of the backward–forward correlation of the secondary electron emission.

A non-analog Monte Carlo simulation of transport of radionuclides in a porous medium

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 variational method for determining uncertain parameters and geometry in hydrogeology

Abstract 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.

Long-Term In-Package Spent Fuel Criticality Calculations

This paper presents MCNP criticality calculations for both UOX and MOX disrupted fuel assemblies canisters systems in the reference Belgian disposal concept and one of its variant. We examine the influence of different parameters (water moderation and geometry alteration) on the neutron multiplication factor, keff . In all the studied cases, the reference concept does not present criticality risks. The variant concept sometimes presents criticality risks. The present results only concern fresh UOX and MOX fuel assemblies. Further developments of this work will include irradiated (UOX and MOX) fuels.Copyright

Point and surface estimations by a non-analog Monte Carlo simulation for the tranport of radionuclide chains in porous media

In the context of contaminant transport in geological media, the estimation of concentration (or activity) in local areas is addressed in this paper. Recently, a non-analog Monte Carlo simulation based on the integral formulation of the advectiondispersion equation in a saturated porous medium has been obtained. The present work describes, firstly, the random walks constructed from this integral equation for the transport of radionuclide chains, secondly, how these random walks may be easily modified in order to assess cumulative activities (integrated over the simulation time) in local areas of the geological medium. The method is based on the calculation of transition probabilities by simple quadratures during a pre-processing step. The simulation itself is organized as a random walk with multiple branchings. The estimation of instantaneous activities at points is also treated. The methods are illustrated on numerical tests in one and three-dimensional problems.

Transport of radionuclides in stochastic media : 2. Evaluation of the parameters

Abstract This paper shows how we may evaluate in a hydrogeological context (i.e. with hypotheses and parameters commonly encountered in hydrogeology) the different coefficients introduced in the three-dimensional quasi-asymptotic equation found previously (see this issue). The evaluation is based on stochastic models developed in hydrogeology and based on field experiments. An analytical evaluation is made for a simple model of the velocity autocorrelation tensor. Numerical values of the coefficients are given. Next, we applied the model in planar geometry. Exact solutions of the quasi-asymptotic equation are shown in this case. In particular, we examined the differences between our model and the model of Williams in fractured rocks. We end this paper with an estimation of the validity limits of the model.

Adaptive adjoint Monte Carlo simulation for the uncertainty

In order to take into account uncertainties about the values of the hydro-geological parameters of the rock hosting a deep geological repository, probabilistic methods are used in the risk assessment of radioactive waste repositories. Random generators could be globally invoked twice in adjoint Monte Carlo (AMC) simulation. Once for sampling hydro-geological parameters from known probability density functions (pdf). Next, for each selected set of parameters, random walks could be simulated for the evaluation of concentration of contaminants. With a moderate number of random walks (batch size), AMC method is efficient for computing mean values of concentrations. However, the higher moments of the concentration distribution and the distribution tails are in general not evaluated with accuracy. To cope with these inconveniences, we propose an adaptive AMC method in which the batch size is dynamically increased. The new approach is applied for the accurate assessment of the probability of exceeding some imposed critical concentrations.

Sensitivity analysis in the migration of radionuclides: differential Monte Carlo versus double randomization

We address in this paper the efficient estimation of sensitivity coefficients by Monte Carlo simulations. In the context of geological performance for the risk assessment of radioactive waste repositories, a recent non-analog Monte Carlo simulation [1] based on an integral equation for the transport of radionuclides in porous media is examined in the view of sensitivity analysis. Two methods are compared: Differential Monte Carlo which requires special care when the integral kernel of the integral equation vanishes; and Double Randomization technique which is used to evaluate an effective sensitivity coefficient. Numerical results illustrate the methods for radionuclide migration and focus on the fraction reaching the upper surface of the medium.