Radionuclides migration in fractured porous rock: Analytical solutions for various boundary conditions

Analytical solutions based on Laplace transform are developed for the problem of radionuclides transport along a discrete planar fracture in porous rock. The solutions take into account advective transport in the fracture, longitudinal hydrodynamic dispersion along the fracture axis, molecular diffusion from the fracture into the rock matrix, sorption within the rock matrix, sorption onto the surface of the fracture and radioactive decay. The initial concentration in both the fracture and the rock matrix is assumed to be zero. Four boundary conditions, constant concentration, exponentially decaying concentration, exponentially decaying flux and kinetic solubility-limited dissolution are assumed. All these analytical solutions are in a form of a single integral that is evaluated by a Gauss-Legendre quadrature for each point in space and time. A comparison between the concentration profiles with a constant concentration inlet boundary condition and those with a decaying concentration inlet boundary condition shows that the concentration profile is strongly influenced by the inlet boundary condition when the retardation factor of the matrix is high. As the dissolution rate constant approaches infinity, the inlet boundary condition of the kinetic solubility-limited dissolution model can be replaced by the boundary condition of constant concentration