Solutions for Radionuclide Transport From an Injection Well Into a Single Fracture in a Porous Formation

Abstract

Solutions are obtained with the Laplace transform technique for four cases related to radionuclide transport from an injection well to a fractured porous formation. The four cases are described by two models (models I and II). Each model is solved for two different boundary conditions at the well bore, namely, a constant concentration or an exponentially decaying concentration relationship. Model I assumes that radionuclides are transported through the fracture by radial advection and longitudinal dispersion, while model II assumes radial advection only. In both models, attenuation mechanisms, such as radioactive decay and adsorption of linear equilibrium isotherms, are considered in the fracture as well as in the porous rock. Transport from the fracture to the surrounding porous rock is accounted for by molecular diffusion. Analytical solutions of concentration distributions valid for small‐time periods and for steady state are obtained for model I; solutions at intermediate and large time intervals are determined by inverting the appropriate Laplace transform equations with the Stehfest method. Analytical solutions of concentration distributions for the transient (valid for any specific time) and steady state conditions are given for model II. These two boundary conditions at the well bore may lead to significant discrepancies in calculated concentration distributions for large injection time periods, but they yield essentially the same results for short injection periods. As shown by a sample problem, computational results of models I and II converge at large injection time periods, indicating that the effect of longitudinal dispersion on the transport of radionuclides may be unimportant at long injection periods. When subjected to Taylor's dispersion theory, it is found that model I can be reduced to the conventional one‐dimensional advection‐dispersion model involving radionuclide transport through a fracture in which the groundwater velocity is constant.