Evolution and persistence of cross-directional statistical dependence during finite-Peclet transport through a real porous medium

Transport of passive, dissolved compounds in fully-saturated complex porous media frequently exhibits non-Fickian characteristics. One of the most interesting questions is to ascertain the time scales at which it is possible to describe transport as a statistically independent process. Therefore, we study the mechanisms for evolution and then the decrease of non-Fickianity as a function of increasing time. Adopting the Lagrangian perspective, we provide a nonlinear copula analysis of advective-diffusive processes by analyzing particle trajectories in a real porous media, as provided by direct numerical simulations on the three-dimensional image of Doddington sandstone. First, we analyze the memory effects between time-consecutive particle position increments and cross dependence between longitudinal and transversal particle position increments as a function of given time increments and time lags between consecutive time increments. Second, we investigate the influence of the Peclet regime on the temporal evolution of dependence. Our main findings are: (a) Cross dependence between longitudinal and transversal particle position increments is persistent over the investigated range of time increments, even though this aspect has been neglected up to date. (b) Lower Peclet numbers lead to a weaker dependence that is, however, more persistent over time than in higher-Peclet transport regimes. We confirm that non-Fickianity comes from spatial coherence associated with heterogeneities of the velocity field that introduce cross dependence and memory into the transport process. Overall, we show that memory and cross dependence are persistent in and among all directions, that the dependence is highly-nonlinear, occurs at different temporal scales, and is dependent on the Peclet number.