Solute transport in heterogeneous porous media with long-range correlations

Transport of finite plumes in three-dimensional, heterogeneous, and statistically isotropic aquifers is investigated where the log hydraulic conductivity is characterized by a fractional Gaussian noise (fGn) covariance structure of Hurst coefficient H. Leading-order analytical expressions for velocity autocovariance functions u(ij), one-particle displacement covariance X(ij),, and macrodispersivity tensor α(ij) are derived under ergodic conditions and mean-uniform steady state flow. Nonergodic transport is then discussed for a line source of finite length, either normal or parallel to the mean flow, by evaluating time-dependent ensemble averages of the second spatial moments, Z(ij) ≡ - A(ij)(0) = X(ij) - R(ij) and the effective dispersivity tensor, γ(ij) defined as (0.5/μ)(d/dt), where A(ij)(0) is the initial value of the second spatial moments of a plume A(ij), and R(ij) is the plume centroid covariance. The main finding is that in a fGn log K field the spreading of a solute plume is never ergodic; as H increases, effective dispersivity results differ more from their ergodic counterparts, since a larger H implies the medium is more correlated. The most interesting results are as follows: for a source parallel to flow, γ22 does not decrease below zero at large time but remains strictly positive, in variance with exponential or Gaussian covariance. For a source normal to flow, γ11 reaches a large-time asymptote, whose value depends on H as follows: it decreases with H for a small source, it increases, reaches a peak, and then decreases as H goes from 1/2 to 1 for intermediate and large sources; for H = 1, γ11 is zero irrespective of the source size.