Mathematical modeling of the anomalous transport of radioactive materials in a porous medium

Abstract

The subsurface nuclear waste repositories have several engineered and natural barriers that isolate the radioactive material from the human's environment until the radio-toxicity of the waste decays to insignificance. One of the major natural insulating barriers is rock formation. If due to the various reasons the leakage of the radioactive waste would take place, the groundwater reservoirs in the vicinity of the repository can be seriously contaminated by the radioactive elements transferred through the cracks and fractures within the insulating barriers. Aquifer contamination by contaminants radioactive elements is an actual environmental problem for all developed countries. Analysis of mass transport in a complex environment shows that the conventional diffusion equation based on Fick's law fails to model the anomalous character of the diffusive mass transport observed in the field and laboratory experiments. Two regimes of anomalous diffusion are identified. One regime, which is called sub-diffusion, is characterized by the slower propagation of the concentration front, so that the squared distance of the front passage requires longer time than in the case of the classical Fickian diffusion. The second regime (called super-diffusion) is characterized by the higher diffusion rate. Both regimes can be modelled by non-local diffusion equation with temporal and spatial fractional derivatives. In the present paper fractional differential equations are used for modeling the transport of radioactive materials in fractured porous medium. New form of fractional equation for modeling migration of the radioactive elements is proposed and justified. Solutions of particular boundary value problems for these equations were found by application of the Laplace transform method. As an example, a mathematical model of the radioactive contaminant transport in a confined, porous, fractured aquifer is derived and analysed. Through the use of fractional derivatives, the model accounts for contaminant exchange between fissures and randomly distributed porous blocks of fractal geometry and non-local character of radioactive decay of the contaminant trapped by the porous medium. For the case of an arbitrary time-dependent source of radioactive contamination located at the inlet of the aquifer, closed-form solutions for solute concentration in the aquifer and in the confining rock is obtained.