Non-ergodic transport of kinetically sorbing solutes

摘要

The fate of contaminants in groundwater is controlled by many factors; the most relevant are the formation heterogeneities, the geochemical properties of the aquifer and the source size which is a limiting factor for the spreading of the contaminant. When the source size is much larger than the characteristic scale of the formation heterogeneities, the transport develops under ergodic conditions and the spatial moments of the plume are not influenced by the source size. In this case, the plume samples all the relevant scales of heterogeneity and the spreading of the solute reaches its maximum. To date, a number of solutions are available for transport under ergodic conditions and only few studies address the non--ergodic case, which develops when the source size is of the same order of magnitude of the formation integral scale. In this paper, we study how the source size affects the transport of linear kinetically sorbing solutes, characterized by constant reaction parameters. First, the general expressions for the effective plume moments in natural formations are obtained by using the Lagrangian approach. Then, closed form solutions are provided for instantaneous injection over a plane normal to the mean flow direction and in the limiting case of large travel times. In addition, we analyze the plume moments at small and moderate times in an isotropic two-dimensional formation characterized by the Gaussian log--con- ductivity covariance function. We found that the source size is the key factor controlling the relative impact on transport of the formation heterogeneities and the geochemical properties of the aquifer. In addition, we found that the relative importance of the contribution to dispersion stemming from the formation heterogeneities increases with the source size. Conversely, when the source size is comparable with the Darcy's scale, the formation heterogeneities do not contribute to solute spreading, which is controlled by the reaction only. The results show also that the ergodic conditions for the longitudinal second order moment are obtained with a progressively smaller source size as the reaction rates increases. Hence, the presence of reaction facilitates the occurrence of ergodic transport, which is obtained for plumes significantly smaller than for nonreactive solutes transport.