Solute transport in heterogeneous porous media with long‐range correlations

Abstract

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 uij, one‐particle displacement covariance Xij, and macrodispersivity tensor αij 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,Zij ≡ 〈Aij〉 − Aij(0) = Xij − Rij and the effective dispersivity tensor, γij defined as (0.5/μ)(d〈Aij〉/dt), where Aij(0) is the initial value of the second spatial moments of a plume Aij, and Rij 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.