For deeply located repository, a radionuclide released from a failed waste container moves through the engineered and natural barriers before it reaches the biosphere. Typically a radionuclide migration through a barrier is the form of solute, even though a particle in an engineering barrier can create a pseudo-colloid when a solute is absorbed on a moving natural colloid. The pseudo-colloid can be generated fractured porous matrix. The hydrochromatic effect and the size of a colloid, enhance the migration of a radionuclide so that we need a more precise model to assess the impact of a pseudo-colloid to the biosphere. We have to develop a mathematical model for a decay chain to observe the effects of actinides and a new model under the existence of a more realistic inlet boundary condition at the interface between an engineered barrier and a fracture. In this analysis, the Laplace transformed solution is derived for a pseudo-colloid in a fracture and a solute in a fracture and a surrounding rock with a realistic inlet boundary condition. The canonical form solutions are used for the FORTRAN based computational code which uses a special subroutine for the inversion of Laplace transform based on the Talbot theory.
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