On transport in porous formations characterized by heterogeneity of evolving scales
Solute transport in natural formations at the regional scale is influenced by several scales of heterogeneity which correspond to the presence of several geological units called facies. As customarily assumed in stochastic theories, inside the facies the transport can be characterized by a single scale of heterogeneity. At the regional scale several geological units are present such that a hierarchy of relevant scales needs to be defined. A possible model for this spatial variability assumes the log conductivity as a random space function of stationary increments characterized by a power law semivariogram. With this hypothesis the ergodic dispersion coefficient grows unbounded as time increases, leading to the phenomenon called anomalous dispersion. An alternative approach considers the plume in nonergodic conditions and assumes the effective dispersion coefficient, which is defined through differentiation in time of the expected value of the spatial second-order plume moment, as representative of macrodispersion. Large differences have been observed in the resulting plume spreading while approaching the problem using the above alternative definitions. In this paper we provide first-order analytical solutions for the longitudinal effective dispersion coefficient, D(L), as well as for the expected value of the longitudinal spatial plume moment, , that complement semianalytical expressions recently proposed in literature. Furthermore, we provide a semianalytical expression for the standard deviation of the longitudinal second-order moment which is important in assessing the interval of confidence of the estimation provided by . Suitable numerical simulations are performed to validate analytical and semianalytical expressions as well as to assess the impact of the cutoff in the log conductivity power spectrum imposed by choosing a finite domain dimension. We conclude that according to recently published results, the dispersion is anomalous when the Hurst coefficients, H, is larger than 0.5 while it is Fickian for H < 0.5. This is in contrast with the ergodic analysis which concludes that the dispersion is anomalous irrespective of the Hurst coefficient. Hence the effective dispersion coefficient is more effective than the ergodic dispersion coefficient to represent the plume spreading. However, the standard deviation of the longitudinal spatial second-order moment is of the same order of magnitude as the expected value leading to the conclusion that the estimations provided by D(L) and are affected by large uncertainties. Numerical results are in good agreement with the analytical solutions, and under some hypotheses they are not influenced by the cutoff. This is not the case for the ergodic second-order longitudinal moment, which strongly depends on the imposed cutoff.