Asymptotic transport parameters in a heterogeneous porous medium: Comparison of two ensemble-averaging procedures

We analyze the movement of a solute cloud in a saturated aquifer, resulting from a point-like instantaneous solute injection. Physical heterogeneities of the medium due to spatial variations of the hydraulic conductivity, as well as the chemical heterogeneities due to variations in the linear adsorp tion coefficient and the degradation rate, are modeled as spatial stochasti c processes with exponential autocorrelation functions. Furthermore, cross- correlations between the chemical properties and the conductivity are taken into account. For large transport times, the movement of the solute cloud is characterized by its center-of-mass velocity, by the macroscopic dispers ion constant, and the macroscopic degradation rate. These quantities are ev aluated using perturbation theory and two different averaging procedures. T he first procedure derives the large-scale properties from the central mome nts of the concentration distribution in a given aquifer realization, and a verages over the ensemble afterwards. The second method which is mathematic ally less demanding obtains large scale transport coefficients from the cen tral moments of the ensemble-averaged concentration distribution. Under the assumption that both prescriptions lead to the same macro-scale quantities , the second approach is usually preferred in literature. The present paper is an extension of the work of Metzger et al. (1996). We show that the two averaging procedures lead to different results in one-dim ensional systems, whereas the difference vanishes for higher dimensions. Ta king into account the influence of small scale dispersion, we give explicit results for the macroscopic parameters characterizing the solute plume. We analyze the various contributions to these parameters and show how the phy sical origin of these contributions can be traced back uniquely to fluctuat ions in the retardation factor, in the flow field, and in the degradation r ate, and to the cross-correlations between these inhomogeneities, respectiv ely.