Nonlinear random-walk approach to concentration-dependent contaminant transport in porous media

Heterogeneous continuous-time random walks

We introduce a heterogeneous continuous-time random walk (HCTRW) model as a versatile analytical formalism for studying and modeling diffusion processes in heterogeneous structures, such as porous or disordered media, multiscale or crowded environments, weighted graphs or networks. We derive the exact form of the propagator and investigate the effects of spatiotemporal heterogeneities onto the diffusive dynamics via the spectral properties of the generalized transition matrix. In particular, we show how the distribution of first-passage times changes due to local and global heterogeneities of the medium. The HCTRW formalism offers a unified mathematical language to address various diffusion-reaction problems, with numerous applications in material sciences, physics, chemistry, biology, and social sciences.

Hydrodynamic instability in the transport of miscible reactive slices through porous media

Miscible displacement of a slice of a pollutant solution by a carrier solution in homogeneous porous media is examined. The carrier solution reacts with the slice solution to generate a chemical product, and as a result of differences in viscosities of the three species, a hydrodynamic instability known as viscous fingering is observed. The dynamics of the instability and the rate of consumption as well as spread of the pollutant are examined through numerical simulations. The study shows that the rate of consumption of the pollutant is the highest when the chemical product is the most or the least viscous solution in the system. It was also found that displacements in which the pollutant viscosity is the smallest or the largest of all three species lead to the widest spread of the pollutant in the porous media. In addition, the most complex finger structures are observed when the carrier solution has the smallest or largest viscosity in the flow. Furthermore, a mechanism of channeling whereby the carrier is able to break through the slice, therefore bypassing the pollutant, is found in cases where the chemical product is more viscous than the carrier solution. The dynamics of the displacement are analyzed and physical interpretations of their development are presented.

Continuous-time random-walk model of transport in variably saturated heterogeneous porous media

We propose a unified physical framework for transport in variably saturated porous media. This approach allows fluid flow and solute migration to be treated as ensemble averages of fluid and solute particles, respectively. We consider the cases of homogeneous and heterogeneous porous materials. Within a fractal mobile-immobile continuous time random-walk framework, the heterogeneity will be characterized by algebraically decaying particle retention times. We derive the corresponding (nonlinear) continuum-limit partial differential equations and we compare their solutions to Monte Carlo simulation results. The proposed methodology is fairly general and can be used to track fluid and solutes particles trajectories for a variety of initial and boundary conditions.

Mass transport subject to time-dependent flow with nonuniform sorption in porous media

We address the description of solutes flow with trapping processes in porous media. Starting from a small-scale model for tracer particle trajectories, we derive the corresponding governing equations for the concentration of the mobile and immobile phases within a fractal mobile-immobile model approach. We show that this formulation is fairly general and can easily take into account nonconstant coefficients and in particular space-dependent sorption rates. The transport equations are solved numerically and a comparison with Monte Carlo particle-tracking simulations of spatial contaminant profiles and breakthrough curves is proposed, so as to illustrate the obtained results.

Spatial and temporal features of density-dependent contaminant transport: experimental investigation and numerical modeling

We investigate the spatial and temporal features of variable-density contaminant plumes migration in porous materials. Our analysis is supported by novel experimental results concerning concentration profiles inside a vertical column setup that has been conceived at CEA to this aim. The experimental method relies on X-ray spectrometry, which allows determining solute profiles as a function of time at several positions along the column. The salient outcomes of the measurements are elucidated, with focus on miscible fluids in homogeneous saturated media. The role of the injected solution molarity is evidenced. As molarity increases, the solutes plume transport progressively deviates from the usual Fickian behavior, and pollutants distribution becomes skewed in the direction dictated by gravity. By resorting to a finite elements approach, we numerically solve the nonlinear equations that rule the pollutants migration: a good agreement is found between the simulated profiles and the experimental data. At high molarity, a strong dependence on initial conditions is found. Finally, we qualitatively explore the (unstable) interfacial dynamics between the dense contaminant plume and the lighter resident fluid that saturates the column, and detail its evolution for finite-duration contaminant injections.