Stochastic modelling and upscaling of hydrodynamic transport in geological fractures

Characterizing hydrodynamic transport in fractured rocks is essential for carbon storage and geothermal energy production. Multiscale heterogeneities lead to anomalous solute transport, featuring breakthrough curve (BTC) tailing and nonlinear growth of plume spatial moments. We focus on purely advective transport within synthetic geological fractures with prescribed relative closure σa/⟨a⟩ and correlation length Lc. We adopt a stochastic approach with multiple fracture realizations for each set of geometric parameters. Steady-state depth-averaged Stokes flow is solved under the lubrication approximation. Flow heterogeneity is organized over the correlation length Lc. The ensemble-averaged velocity probability density functions (PDFs) are insensitive to Lc but strongly influenced by σa/⟨a⟩, particularly in their low-velocity power-law scaling. A time-domain random walk (TDRW) simulation is used to compute plume spatial moments and outlet BTCs. The mean longitudinal plume position scales linearly with time at both early and late stages. The variance shows ballistic scaling at early times and late-time behavior controlled by the low-velocity power law of the velocity PDF, with exponent α strongly influenced by σa/⟨a⟩. The properties of the BTCs are also controlled by α, including the broadening of the peak as σa/⟨a⟩ increases and the scaling of the power-law tails. Advective transport is also modeled using a one-dimensional continuous-time random walk (CTRW) that relies only on the velocity PDF, flow tortuosity, and flow correlation length. The CTRW reproduces the TDRW results and provides analytical predictions for the asymptotic transport scalings.