Assessing the impact of random media for eigenvalue problems plays a central role in nuclear reactor physics and criticality safety. In a recent work (Larmier et al., 2018a), we have applied a probabilistic model based on stochastic tessellations in order to describe fuel degradation following severe accidents with partial melting and rearrangement of the resulting debris. The distribution of the multiplication factor and of the kinetics parameters as a function of the mixing statistics model and of the typical correlation length of the tessellation were examined in detail for a benchmark configuration consisting in a fuel assembly with UOX or MOX fuel pins. In this paper, we extend our previous findings by including in the stochastic tessellation model the effects of anisotropy that might result from gravity and material stratification: for this purpose, we adopt the broad class of anisotropic Poisson geometries. We examine the evolution of the statistical properties of the tessellations, including the volume, surface and chord length of the cells for various anisotropy laws, and compare them to the case of isotropic Poisson geometries. Then, we discuss the behaviour of the key observables of interest for eigenvalue problems in anisotropic tessellations by revisiting the fuel assembly benchmark calculations proposed in (Larmier et al., 2018a). The effects of anisotropic random media on the multiplication factor and on the kinetics parameters will be carefully examined.

Markov media are often used as a prototype model in the analysis of linear particle transport in disordered materials. For this class of stochastic geometries, it is assumed that the chord lengths must follow an exponential distribution, with a direction-dependent average if anisotropy effects are to be taken into account. The practical realizability of Markov media in arbitrary dimension has been a long-standing open question. In this work we show that Poisson hyperplane tessellations provide an explicit construction for random media satisfying the Markov property and easily including anisotropy. The average chord length can be computed explicitly and is be shown to be intimately related to the statistical properties of the tessellation cells and in particular to their surface-to-volume ratio. A computer code that is able to generate anisotropic Poisson tessellations in arbitrary dimension restricted to a given finite domain is developed, and the convergence to exact asymptotic formulas for the chord length distribution and the polyhedral features of the tessellation cells is established by extensive Monte Carlo simulations in the limit of domains having an infinite size.