Nonergodic solute transport in heterogeneous porous media: Influence of multiscale structure

Analysis of ergodic transport in multiscale log conductivity fields with truncated power variograms is extended to nonergodic transport. Under the first-order approximation and uniform mean flow condition, the ensemble averages of the second spatial moments, Zii, and associated effective dispersivities, Γii, were evaluated as functions of the dimensionless time for a line source of the dimensionless lengths, 1, 5, 10, and 20, either normal or parallel to the mean flow direction. The multiscale results are compared with those obtained previously for a single-scale medium. It is found that the multiscale results of Zii and γii follow a similar pattern of the single-scale results but they are strongly influenced by the value of the Hurst coefficient H (0 < H < 0.5): the larger the value of H, the smaller the difference between the results for the multiscale and those for the single-scale medium. The main differences between the results for the two media are: (1) the longitudinal spreading is weakened while the asymptotic transverse spreading is enhanced by the multiscale structure of log K field, (2) the ergodic limits are reached slower in the multiscale medium than in the single-scale medium as the dimensionless source length ι2 increases, and (3) the preasymptotic period is longer in a multiscale medium than in a single-scale medium. These differences can be attributed to the longer time taken by solute particles of a finite plume to sample all the velocity variation in a multiscale medium than in a single-scale medium. In the multiscale medium the longitudinal moment approaches ergodic condition slower in three dimensions than in two dimensions as ι2 increases. The negative effective transverse dispersivity, which was discovered for a single-scale medium, also exists for the case of a line source parallel to the mean flow in the multiscale medium, which calls for further study.