From Sub-Diffusion to Fickian Diffusion of Conservative Solute Tracers in Porous Rocks

Abstract. Molecular diffusion of conservative solute tracers is relevant to many hydrogeological scenarios, such as contaminant and radionuclide migration. Sub-diffusion—wherein the mean squared displacement grows sub-linearly with time due to prolonged trapping in the pore space—can occur in porous rocks but is expected to evolve asymptotically toward Fickian diffusion at long times. However, this transition has received little attention in the literature. Here, simulations of diffusion are conducted using particle-tracking methods in three digital porous-rock domains. The first two rocks are extracted from real rock samples, and periodic boundary conditions are imposed on both of them. The third rock is constructed by combining several rock samples and is therefore a long, artificial, and highly heterogeneous rock column. The survival probability computed from the simulations demonstrates clearly the transition from sub-diffusion to Fickian diffusion. We then derive an analytical survival probability model within the continuous time random walk (CTRW) framework; a truncated waiting-time distribution is used to introduce a finite characteristic time scale. For the first two rocks, the transition can be captured by the CTRW model, whereas the Fickian model is inadequate, when the observation time is comparable to or exceeds the finite time scale. For the third rock, neither the CTRW nor the Fickian model can adequately describe the transition due to its strongly heterogeneous nature. The implications of sub-diffusion are illustrated through an example in which a diffusive tracer plume arrives much earlier than predicted by the Fickian model. Both the numerical results and the analytical model highlight the importance of properly accounting for sub-diffusion in diffusion modeling of porous rocks.