Handling subsurface transport in nonstationary velocity fields

In the stochastic subsurface analysis most of transport solutions available in literature are obtained under the assumption of statistical homogeneity of velocity field even though in real-world applications this hypothesis is not always verified. Several causes may induce statistical inhomogeneity of flow. This may derive from the influence of boundaries in a limited domain, complex flow configurations (related to pumping and/or injecting wells), spatial nonstationarity of the properties of formations (distinct geological layers and zones) as well as from conditioning on measurements. While several contributes discusses the effects on the flow statistic, only few attempts to find a general solution of transport problems suitable in absence of statistical homogeneity of velocity were carried out. A method is here proposed to handle different and concurrent causes of the flow field nonstationarity and to develop a plume evolution in a domain of finite size. The goal is reached by expanding the steady state flow equation in Taylor series limited to first-order and by the recursive application of finite element method. The unknowns are the piezometric head mean values and its derivatives respect to the fluctuating porous media hydraulic conductivity. Spatial moments of a solute plume are a posteriori obtained by a consistent Lagrangian analysis starting from the knowledge of the velocity field covariance matrices without any restriction regarding the statistical homogeneity of flow and/or ergodicity conditions. By this approach any combination of finite complex boundary, internal sources or sinks, fully inhomogeneous hydraulic conductivity characteristics can be handled. The application of the proposed method to some test cases in domains of finite extension gives results in agreement with literature findings. To give a measure of the capability of the method the examples here developed take into account the spatial velocity nonstationarity coming from inhomogeneous hydraulic conductivity also. A Monte Carlo analysis expressly developed gives a term of comparison.