Simulations of non ergodic transport induced by inhomogeneous flow fields in heterogeneous porous formations of hierarchical sedimentary architecture

Dealing with subsurface flow and transport in naturally heterogeneous formations neither statistical homogeneity nor ergodicity are usually met in real world applications. If the stationarity is obeyed by Lagrangian velocity field the ergodicity of transport can be ascribed to the only size of the injection area, but this is not the case in all situations where flow field doesn't meet statistical homogeneity. In this case not only the size of the area where the concentration is injected, but also its location as well as the spatial variability of mean flux play a relevant role in the expected plume evolution. Among other causes, the hierarchical sedimentary architecture of porous formations combined with the finite size of the domain may be considered as one of the main causes of the flow field inhomogeneity and of the subsequent evidences of the non ergodic transport in real cases. To give a better understanding of this phenomenon synthetic 2-D cases of isotropic log conductivity field with integral scale λ1 whose expected value is assumed as a periodic function of the space coordinates of prescribed amplitude are investigated. The wave length of the mean log conductivity fluctuation is λ2, properly chosen to satisfy the relationship λ1 < λ2 < L, being L the characteristic dimension of the finite domain. The time evolution of the spatial moments of the plume driven by a statistically inhomogeneous steady state random velocity field is analyzed by varying the amplitude of the periodic mean flow and its wave length λ2 and by taking into account different sizes of injection area. These moments are achieved by space- time integration of the velocity field covariance structure derived according to the first-order Taylor series expansion by the stochastic finite element method. The discussion of the results leads to a better understanding into the ergodicity lack that affects solute transport in heterogeneous aquifers, by giving a comparative measure of the relevance of two possible causes, that is the flow field inhomogeneity due to the hierarchical sedimentary architecture of porous formations combined with a limited domain and the injection area of finite size.