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 〉 − A ij (0) = X ij − R ij and the effective dispersivity tensor, γ ij defined as (0.5/ μ )( d 〈 A ij 〉/ 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, γ 22 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, γ 11 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, γ 11 is zero irrespective of the source size.