Theoretical Investigation of non-Ergodic Solute Dispersion in Natural Porous Formations Characterized by Persistent and Anti-persistent Power-law Log-Conductivity Correlations

Tracer dispersion within log-conductivity fields represented by power-law semivariograms is investigated by an analytical first-order Lagrangian approach that, in treating subsurface flow and transport, resorts to the superposition principle of an infinite double hierarchy of mutually independent scales of heterogeneity. The results of the investigation, which are corroborated by a preliminary field validation, and also interpreted in terms of probabilistic risk analysis, say that transport anomaly is intrinsically associated with evolving-scale heterogeneous porous formations, regardless of their semivariogram scaling exponent b. In contrast with what was found by previous studies that dealt with this subject in a Lagrangian framework, it is demonstrated that: (1) the magnitude of nonergodic dispersion is nonmonotonically related to b; (2) consistently assuming a characteristic advective-heterogeneity length-scale leads to a universal (and quadratic) dependence of the dimensionless macrodispersion coefficients on the dimensionless time. Additionally, it is demonstrated that, in the presence of fractal heterogeneity, and unlike what happens for short-range correlations, diffusion acts as an antagonist mechanism in terms of Fickian dispersion achievement. Finally, the reinterpretation of antipersistent and persistent correlations as a double hierarchy of oscillatory nonperiodic and periodic fields, respectively, besides allowing for a technical explanation of all the detected trends, envisions a possible alternative methodology for their numerical generation.