Transient dispersion regimes in heterogeneous porous media: On the impact of spatial heterogeneity in permeability and exchange kinetics in mobile–immobile transport

Transport processes in the subsurface are coupled with the heterogeneity of the porous structure. Obtaining an accurate description of mass transport in such media at all time scales remains a crucial task, as the subsurface is strongly impacted by human activity. Spreading of pollutants in the transient but also in the asymptotic regime, as well as the time to reach a critical location (e.g. aquifer, well) significantly depend on the underlying heterogeneity of the permeability field. Moreover, in the case of solute retention, transport is also characterized by local exchange kinetics that depend on the local aquifer properties. Consequently, exchange (retention) times are expected to be spatially heterogeneous. In this work we focus on the influence of spatially heterogeneous permeability and exchange times on the transient and asymptotic transport regimes and provide a parametric study. To this goal, we simulate in a first part the transport in a two-dimensional heterogeneous medium under spatially varying permeability and mobile–immobile mass transfer parameters. Equations are solved using a Lattice-Boltzmann two-relaxation-time (TRT) algorithm. We assume the following relation between the local permeability K and the local exchange time: τ∝Kγ. Taking into account this relation, we investigate the impact of the Damköhler number (Da, ratio of the advection and exchange time scales), the disorder of the permeability field and the value of the exponent of the coupling function (γ) on the spatial evolution of the concentration field and breakthrough curve. We show that, depending on the parameters (Da, γ, etc.), we can observe transient, non-Fickian dispersion, which is characterized by non-exponential tails of the solute breakthrough curves and a non-linear evolution of the spatial variance of the solute distribution. In a second part, we present a new continuous time random walk (CTRW) model to upscale these transport behaviors. The model is based on a spatial Markov model for particle velocities that couples advective–dispersive transport and heterogeneous mass transfer through a compound Poisson process. The upscaled model can be fully parameterized by the statistics of permeability and the hydraulic gradient (with no fitting parameter), that is, in terms of medium and flow characteristics. The results of the CTRW model fully capture the non-Fickian transient transport regimes both for the breakthrough curves and spatial concentration variance. In the longtime limit, as expected from the central limit theorem, the CTRW model predicts normal, Fickian behavior. Finally we show, that the time to reach the asymptotic regimes in heterogeneous media (e.g. heterogeneous exchange times) is parameter dependent and in average is two orders of magnitude larger than for the respective homogeneous case. To summarize, the coupling between the heterogeneous permeability field and the local mass transfer properties can strongly influence transient and asymptotic transport regimes and potentially explain experimentally observed non-Gaussian behaviors.

M.D. acknowledges the support of the Spanish Research Agency (10.13039/501100011033) and the Spanish Ministry of Science and Innovation through the grants CEX2018-000794-S and PID2019-106887GB-C31 (HydroPore).

Peer reviewed