Methods are developed for solving the transport equation for radionuclides moving in porous rock by hydrodynamic dispersion and advection. The unique nature of the problem arises from the long time interval over which the solutions are required, e.g., 106 yr, during which geological and climatic changes can radically alter the system properties, such as the retardation factor and the water velocity. In order to solve this problem, we have developed eigenfunction expansion methods which eliminate the spatial variable and thereby enable the time dependence to be incorporated explicitly. Various problems are considered, each based on two simple boundary conditions: (a) concentration is fixed at both ends of the layer and (b) a delta function impulsive source at one end. The convergence of the solutions is improved by a technique based on the Poisson sum formula which makes them readily tractable numerically over a wide range of practically interesting parameters.Some exact solutions are obtained for purely advective transport which are particularly useful as they are very general and lend themselves to a variety of analytical averaging techniques.Of considerable importance is the development of a stochastic averaging procedure to account for uncertainties in the parameters and onset of climatic changes. We have illustrated the effects of averaging by application to a single layer with a delta input and one climatic change (switchtime). The switchtime is regarded as a random variable and averaged over lognormal and uniform distributions. In the same way, we have considered the retardation factor as uniformly distributed between upper and lower bounds and give graphical results for the concentration as a function of time. Finally, we consider various developments of the method to multinuclide chains and multilayer systems.