Anomalous diffusion in regular heterogeneous media

Abstract

Contaminant transport in homogeneous medium I restricted in one dimension (plane-parallel layer, PPL) or in two dimensions (straight cylinder, SC) and surrounded by another medium II filling the rest of the space has been analyzed. Both media are regular. Diffusivity of the medium I is assumed to be significantly greater than that for the medium II, D » d. It has been found that in the time range of t1 « t « t2 [where t 1 = a2/d, a is characteristic dimension along the restriction directions of the medium I; t 2 = (D/d)2 t 1 for PPL and t 2 = (D/d) ln(D/d)t 1 for SC] the contaminant transport is of subdiffusion behavior with variance for PPL and ˜Dt 1 ln(t/t1) for SC. Classical diffusion in the time ranges t « t 1 and t » t 2 takes place with effective diffusivities D and d, respectively. When the ratio D/d is extremely large (e.g. in case of rock fractures), the time t2 might practically not be accessible resulting in the regime of anomalous diffusion (subdiffusion) becoming asymptotic. Therefore, in connection with contaminant transport in fractured rocks, we must deal with the effect of dispersion suppression in fractures. The results obtained may be helpful for the development of methods to assess reliability of radioactive waste storage in rock massifs.