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Random fields are becoming a mature tool sharing applications in many area of physics, mechanics and geosciences. In the latter, it is commonly used under the name of geostatistics. Continuous enrichment of geological/geostatistical models leads to manipulating hydrogeological models characterized by many parameters or hyperparameters corresponding to statistical aggregates that may be poorly estimated due to the scarcity of field data. Those parameters are generally support-scale-dependent and uncertain, so some inverse problem and uncertainty analysis must be carried out in practical applications that involve generally some forward calculation for example a fluid flow simulation if one in interested in transfers in the subsurface. Up scaling techniques are still required to find and to restrict in a controlled manner the more relevant parameters, allowing to lower the dimension of the parameter space. In the stochastic case, the interaction between the conductivity spatial distribution and the flow pattern can lead to non trivial behaviours that will be discussed. Fractured media will not be considered. That note does not present original results, but a selection of some potentially fruitful research avenues suggested by previous works.

In this paper, some applications of statistical physics (SP) concepts and techniques in applied geosciences are reviewed. The domain includes hydrology, oil and gas industry, nuclear or CO$_2$ waste geological disposal, massive energy storage, heat recovery from geothermal formations and many other applications. Several scales are considered: we start from applications of SP at the molecular scale to understand the effect of extreme confinements concerning the fluid transport in nanopores in clay rocks. The paper ends with coarse graining techniques that are employed to build operational models relevant at the practical scale of several kilometers, including strongly fractured geological environments. Perspectives are proposed regarding some issues about the practical use of these often over-parameterized models in connection to random matrix and graph theories and the associated quenched disorder problems and “big data” issues.

We present in this Note a stochastic approach to the matrix-fracture exchange in a heterogeneous fractured porous medium. We introduce an intermediate scale, called the unit-scale, between the local-scale (fracture-scale) and the large-scale characteristic of the reservoir mesh (reservoir block). This paper focuses on the problem of upscaling fluid exchange phenomena from the unit scale to the reservoir mesh or block scale. Simplifying the Darcian flow terms enables us to obtain a probabilistic solution of the dual continuum problem, in continuous time, in the case of a purely random exchange coefficient. This is then used to develop several upscaling approaches to the fluid exchange problem, and to analyze the so-called ‘effective’ exchange coefficient. The results are a first contribution to the more general problem of upscaling multidimensional flow-exchange processes in space and time, in randomly heterogeneous dual continua. To cite this article: M. Kfoury et al., C. R. Mecanique 332 (2004). [Copyright &y& Elsevier]