Transport in Heterogeneous Sediments with Multimodal Conductivity and Hierarchical Organization Across Scales

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We consider here the Lagrangian approach for stochastic modeling of the transport of inert solutes in porous media. A general global covariance function of log conductivity in sediments with hierarchical organization has been developed by combining proportions, transition probabilities, and covariances of log conductivity. The global integral scale is derived from the global covariance function with two types of correlation lengths: integral scales of local log conductivity and correlation scale of indicator space functions. The macrodispersion coefficients have been derived for the solute transport in two- and three-dimensional domains. An example is used to illustrate the time evolution trends and the relative contributions of the auto and cross terms. Sensitivity analysis indicates that the values of macrodispersion coefficients are positively related to the changes of indicator correlation scale, integral scale and the difference of the mean log conductivity between different units. But, in this example the macrodispersion coefficients are more sensitive to the indicator correlation scale than to the integral scale. The cross term in the macrodispersion coefficients has an increasing contribution when the contrast of the mean log conductivity increases. Under the condition of high contrast of log conductivity between different units, only the cross terms contribute to the macrodispersion coefficients and the auto terms can be ignored. At the large time limit, the longitudinal coefficient shows clearly a linear dependence on the global variance of log conductivity.