Dispersion in porous media, continuous-time random walks, and percolation

Structural and hydrodynamic controls on fluid travel time distributions across fracture networks

Fracture networks are preferential flow paths playing a critical role in a wide range of environmental and industrial problems. Their complex multiscale structure leads to broad distributions of fluid travel times, affecting many biogeochemical processes. Yet, the relationship between the fracture network structures, their hydrodynamic properties, and the resulting anomalous transport dynamics remains unclear. We use a large database of fracture network models to investigate the factors controlling fluid velocity and travel-time distributions across a wide range of networks, from synthetic to field-calibrated models, with aperture variability at both fracture and network scales. Analysis reveals that transport statistics have generic properties across investigated networks, including notably heavy-tailed travel time distributions. Networks of increasing complexity and heterogeneity lead to broader velocity distributions and more channeled velocity fields, where flow concentrates in a narrow channel web in the three-dimensional (3D) fracture structure. While heterogeneity in point-velocity statistics increases travel-time variability, channeling tends to reduce it. This counterintuitive phenomenon challenges current theories, which assume that long travel time power law exponents are determined solely by point-velocity statistics. By analyzing velocity and travel time statistics for different flow structures, we develop a coupled Continuous Time Random Walk framework capturing the unexpected control of the velocity field's spatial structure on anomalous transport in fracture networks. This leads to a unique class of random walk models capturing the respective roles of velocity heterogeneity and spatial structure on transport in networks. These findings open a prospective for characterizing, modeling, and predicting transport dynamics in complex networks, with potential applications to geological, biological, and engineered networks.

Relation between critical exponent of the conductivity and the morphological exponents of percolation theory

A central unsolved problem in percolation theory over the past five decades has been whether there is a direct relationship between the critical exponents that characterize the power-law behavior of the transport properties near the percolation threshold, particularly the effective electrical conductivity σ_{e}, and the exponents that describe the morphology of percolation clusters. The problem is also relevant to the relation between the static exponents of percolation clusters and the critical dynamics of spin waves in dilute ferromagnets, the elasticity of gels and composite solids, hopping conductivity in semiconductors, solute transport in porous media, and many others. We propose an approach to address the problem by showing that the contributions to σ_{e} can be decomposed into several groups representing the structure of percolation networks, including their mass and tortuosity, as well as constrictivity that describes the fluctuations in the driving potential gradient along the transport paths. The decomposition leads to a relationship between the critical exponent t of σ_{e} and other percolation exponents in d dimensions, t/ν=(d-D_{bb})+2(D_{op}-1)+d_{C}, where ν, D_{bb}, D_{op}, and d_{C} are, respectively, the correlation length exponent, the fractal dimensions of the backbones and the optimal paths, and the exponent that characterizes the constrictivity. Numerical simulations in two and three dimensions, as well as analytical results in d=1 and d=6, the upper critical dimension of percolation, validate the relationship. We, therefore, believe that the solution to the 50-year-old problem has been derived.

Solute transport in porous media studied by lattice Boltzmann simulations at pore scale and x-ray tomography experiments

With the aid of nondestructive microfocus x-ray computed tomography (CT), we performed three-dimensional (3D) tracer dispersion experiments on randomly unconsolidated packed beds. Plumes of nonreactive sodium iodide solution were point injected into a sodium chloride solvent as a tracer for the evaluation of the dispersion process. The asymptotic dispersion coefficient was obtainable within the experimental scale and was summarized over Péclet numbers from 11.7 to ∼860. Then, the lattice Boltzmann method and moment propagation method were used to elucidate the mechanisms embedded in the dispersion phenomenon. The methods were rigorously verified against the classical Taylor dispersion problem and extended to simulate fluid flow and tracer dispersion in high-resolution 3D digital porous structures from CT. The method of moments, Lagrangian velocity correction function, and dilution index were thoroughly analyzed to evaluate the dispersion behaviors. Numerical simulations revealed ballistic and superdiffusive regimes at the transient times, whereas asymptotic dispersion behaviors appear at longer characteristic times. Besides, the observed transient times unanimously persist over convective length scales of around 12 particles transversely and 16 particles longitudinally. The estimated dispersion coefficients from simulation are in consistence with the experimental result. Furthermore, the simulation also enabled the identification of regimes, including diffusive, power law, and mechanical dispersion. Thus, the proposed experimental and computational schemes are of practical means to study dispersion behaviors by direct pore scale imaging and modeling.

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.

Reactive transport in the complex heterogeneous alluvial aquifer of Fortymile Wash, Nevada

Yucca Mountain, Nevada, had been extensively investigated as a potential deep geologic repository for storing high-level nuclear wastes. Previous field investigations of stratified alluvial aquifer downstream of the site revealed that there is a hierarchy of sedimentary facies types. There is a corresponding log conductivity and reactive surface area subpopulations within each facies at each scale of sedimentary architecture. Here we use a Lagrangian-based transport model in order to analyze radionuclide dispersion in the saturated alluvium of Fortymile Wash, Nevada. First, we validate the Lagrangian model using high-resolution flow and reactive transport simulations. Then, we used the validated model to investigate how each scale of sedimentary architecture may affect long-term radionuclide transport at Yucca Mountain. Results show that the reactive solute dispersion developed by the Lagrangian model matches the ensemble average of numerical simulations well. The link between the alluvium spatial variability and reactive solute dispersion at different spatiotemporal scales is demonstrated using the Lagrangian model. The longitudinal dispersivity of the reactive plume can be on the order of hundreds to thousands of meters, and it may not reach its asymptotic value even after 10,000 years of travel time and 2-3 km of travel distance.

Recirculation zones induce non-Fickian transport in three-dimensional periodic porous media

In this work, the influence of pore space geometry on solute transport in porous media is investigated performing computational fluid dynamics pore-scale simulations of fluid flow and solute transport. The three-dimensional periodic domains are obtained from three different pore structure configurations, namely, face-centered-cubic (fcc), body-centered-cubic (bcc), and sphere-in-cube (sic) arrangements of spherical grains. Although transport simulations are performed with media having the same grain size and the same porosity (in fcc and bcc configurations), the resulting breakthrough curves present noteworthy differences, such as enhanced tailing. The cause of such differences is ascribed to the presence of recirculation zones, even at low Reynolds numbers. Various methods to readily identify recirculation zones and quantify their magnitude using pore-scale data are proposed. The information gained from this analysis is then used to define macroscale models able to provide an appropriate description of the observed anomalous transport. A mass transfer model is applied to estimate relevant macroscale parameters (hydrodynamic dispersion above all) and their spatial variation in the medium; a functional relation describing the spatial variation of such macroscale parameters is then proposed.

Pruning to Increase Taylor Dispersion in Physarum polycephalum Networks

How do the topology and geometry of a tubular network affect the spread of particles within fluid flows? We investigate patterns of effective dispersion in the hierarchical, biological transport network formed by Physarum polycephalum. We demonstrate that a change in topology-pruning in the foraging state-causes a large increase in effective dispersion throughout the network. By comparison, changes in the hierarchy of tube radii result in smaller and more localized differences. Pruned networks capitalize on Taylor dispersion to increase the dispersion capability.

Crossover from anomalous to normal diffusion in porous media

Random walks (RW) of particles adsorbed in the internal walls of porous deposits produced by ballistic-type growth models are studied. The particles start at the external surface of the deposits and enter their pores in order to simulate an external flux of a species towards a porous solid. For short times, the walker concentration decays as a stretched exponential of the depth z, but a crossover to long-time normal diffusion is observed in most samples. The anomalous concentration profile remains at long times in very porous solids if the walker steps are restricted to nearest neighbors and is accompanied with subdiffusion features. These findings are correlated with a decay of the explored area with z. The study of RW of tracer particles left at the internal part of the solid rules out an interpretation by diffusion equations with position-dependent coefficients. A model of RW in a tube of decreasing cross section explains those results by showing long crossovers from an effective subdiffusion regime to an asymptotic normal diffusion. The crossover position and density are analytically calculated for a tube with area decreasing exponentially with z and show good agreement with numerical data. The anomalous decay of the concentration profile is interpreted as a templating effect of the tube shape on the total number of diffusing particles at each depth, while the volumetric concentration in the actually explored porous region may not have significant decay. These results may explain the anomalous diffusion of metal atoms in porous deposits observed in recent works. They also confirm the difficulty in interpreting experimental or computational data on anomalous transport reported in recent works, particularly if only the concentration profiles are measured.

Flow intermittency, dispersion, and correlated continuous time random walks in porous media

We study the intermittency of fluid velocities in porous media and its relation to anomalous dispersion. Lagrangian velocities measured at equidistant points along streamlines are shown to form a spatial Markov process. As a consequence of this remarkable property, the dispersion of fluid particles can be described by a continuous time random walk with correlated temporal increments. This new dynamical picture of intermittency provides a direct link between the microscale flow, its intermittent properties, and non-Fickian dispersion.

Saturation dependence of dispersion in porous media

In this study, we develop a saturation-dependent treatment of dispersion in porous media using concepts from critical path analysis, cluster statistics of percolation, and fractal scaling of percolation clusters. We calculate spatial solute distributions as a function of time and calculate arrival time distributions as a function of system size. Our previous results correctly predict the range of observed dispersivity values over ten orders of magnitude in experimental length scale, but that theory contains no explicit dependence on porosity or relative saturation. This omission complicates comparisons with experimental results for dispersion, which are often conducted at saturation less than 1. We now make specific comparisons of our predictions for the arrival time distribution with experiments on a single column over a range of saturations. This comparison suggests that the most important predictor of such distributions as a function of saturation is not the value of the saturation per se, but the applicability of either random or invasion percolation models, depending on experimental conditions.

Evaluating transport in irregular pore networks

A general approach for investigating transport phenomena in porous media is presented. This approach has the capacity to represent various kinds of irregularity in porous media without the need for excessive detail or computational effort. The overall method combines a generalized effective medium approximation (EMA) with a macroscopic continuum model in order to derive a transport equation with explicit analytical expressions for the transport coefficients. The proposed form of the EMA is an anisotropic and heterogeneous extension of Kirkpatrick's EMA [Rev. Mod. Phys. 45, 574 (1973)] which allows the overall model to account for microscopic alterations in connectivity (with the locations of the pores and the orientation and length of the throat) as well as macroscopic variations in transport properties. A comparison to numerical results for randomly generated networks with different properties is given, indicating the potential for this methodology to handle cases that would pose significant difficulties to many other analytical models.