This talk introduces numerical methods and applications for fractional-order advection-dispersion equations(fADEs).A general Lagrangian solver is developed to approximate various fADEs using a three-step fractional adjoint method.This leads to a fully Lagrangian scheme which is a continuous time random walk with Lévy motion in space or time,providing discrete stochastic approximations for the fADEs.We will also show practical applications of the models in capturing hydrological dynamics observed in geological media.