Abstract
A large-time Eulerian–Lagrangian stochastic approach is employed to: (1) estimate centroid position uncertainty of contaminant plumes that originate from instantaneous point sources in statistically stationary and isotropic porous formations; (2) assess the time needed for achieving ergodic conditions, which would allow for the evaluation of local concentration values based on the only ensemble mean distribution; (3) derive the concentration coefficient of variation (CV) as a function of asymptotic macro-dispersion coefficients and centroid trajectory variances. The results indicate that the decay time of plume position uncertainty is so large that there is practically no chance for effective ergodicity. The concentration coefficient of variation is zero at the centroid but rapidly increases when moving away from it. The dissipative effect of local dispersion in the presence of relatively high Péclet numbers is considerably exalted by marked flow field heterogeneity, which confirms the previously postulated synergic, non-additive effect of advection and local dispersion in passive solute dilution. A further result from this study is the derivation of the power law that relates dimensionless concentration micro-scale to dimensionless local dispersive area. The exponent of this power law is the same that appears in the relationship between dimensionless Kolmogorov turbulent micro-scale and flow Reynolds number.
Highlights
Protecting groundwater resources and attempting to limit the damages deriving from their sometimes unavoidable deterioration is one of the main goals in the field of environmental engineering.For that reason, many scientists in the last few decades have addressed, using different approaches, all the issues related to water flow and solute transport in heterogeneous porous formations
Many scientists in the last few decades have addressed, using different approaches, all the issues related to water flow and solute transport in heterogeneous porous formations
Before analyzing the large-time behavior of coefficient of variation (CV) at a given short distance from the centroid and at a fixed point in space, it is useful to evaluate the time needed for S11 to decrease to some small percentage ω of X11 = 2σ2Y VIY t + 2Dt: τω =
Summary
Many scientists in the last few decades have addressed, using different approaches, all the issues related to water flow and solute transport in heterogeneous porous formations (among the benchmark treatises: [1,2,3]). As a matter of fact, the marked heterogeneity of groundwater flow fields considerably complicates the theoretical analysis, tying it to the choice of a well-defined reference space–time scale and to the introduction of a number of simplifying assumptions. When affording the study of porous media by the classic continuum theory (which averages the microscopic balance equations over a representative elementary volume), one implicitly admits the existence of a lower-limit natural scale. The geometrical organization of the porous formation will be considered only one of the infinite possible scenarios, all of them belonging to the same statistical population
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