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Two distinct but interconnected approaches can be used to model diffusion in fluids; the first focuses on dynamics of an individual particle, while the second deals with collective (effective) motion of (infinitely many) particles. We review both modeling strategies, starting with Langevin’s approach to a mechanistic description of the Brownian motion in free fluid of a point-size inert particle and establishing its relation to Fick’s diffusion equation. Next, we discuss its generalizations which account for a finite number of finite-size particles, particle’s electric charge, and chemical interactions between diffusing particles. That is followed by introduction of models of molecular diffusion in the presence of geometric constraints (e.g., the Knudsen and Fick–Jacobs diffusion); when these constraints are imposed by the solid matrix of a porous medium, the resulting equations provide a pore-scale representation of diffusion. Next, we discuss phenomenological Darcy-scale descriptors of pore-scale diffusion and provide a few examples of other processes whose Darcy-scale models take the form of linear or nonlinear diffusion equations. Our review is concluded with a discussion of field-scale models of non-Fickian diffusion.

Natural porous systems, such as soil, membranes, and biological tissues comprise disordered structures characterized by dead-end pores connected to a network of percolating channels. The release and dispersion of particles, solutes, and microorganisms from such features is key for a broad range of environmental and medical applications including soil remediation, filtration and drug delivery. Yet, owing to the stagnant and opaque nature of these disordered systems, the role of microscopic structure and flow on the dispersion of particles and solutes remains poorly understood. Here, we use a microfluidic model system that features a pore structure characterized by distributed dead-ends to determine how particles are transported, retained and dispersed. We observe strong tailing of arrival time distributions at the outlet of the medium characterized by power-law decay with an exponent of 2/3. Using numerical simulations and an analytical model, we link this behavior to particles initially located within dead-end pores, and explain the tailing exponent with a hopping across and rolling along the streamlines of vortices within dead-end pores. We quantify such anomalous dispersal by a stochastic model that predicts the full evolution of arrival times. Our results demonstrate how microscopic flow structures can impact macroscopic particle transport. Most porous systems comprise structures characterized by dead-end and transmitting pores. Here, authors show that macroscopic transport through such porous medium is controlled by structure-induced laminar vortices inside each dead-end pore, and such cannot be explained by diffusion alone.

Fractures provide pathways for preferential flow, whereas porous rock acts as storage that delays fluid propagation through matrix imbibition. These dual‐porosity mechanisms are investigated in laboratory experiments of partially saturated fracture infiltration. We analyze flow dynamics in terms of the fluid penetration depth in the fracture and delineate fracture‐ and matrix‐dominated flow regimes at different flow rates. We compare wetting front propagation in fracture and matrix and examine the interference of matrix‐wetting fronts with the lateral system boundary. The experimental data are interpreted using the analytical model of Nitao (1991), which accounts for the impact of fracture‐matrix interactions on fluid propagation in the fracture. We find that matrix imbibition affects the observed discontinuous, partially saturated fracture flow to behave, on average, like plug flow. Consequently, and within the range of applied flow rates above a critical threshold, the model’s plug flow assumption is not a relevant precondition for its applicability. Fluid propagation in the fracture exhibits three characteristic scaling regimes (FP1‐3) corresponding to the matrix imbibition state. Only two scaling regimes are established for flow rates below a critical threshold, hence required to recover bulk infiltration for the chosen geometry. Furthermore, wetting fronts switch from fracture‐to matrix‐dominated at moderate to high flow rates, indicating a flow‐rate‐dependent limitation of fracture‐dominated infiltration depth. While the scaling regimes agree with experiments for applied flow rates above the critical threshold, the model underestimates the initial penetration depth below. Here, we observe the direct onset of flow regime FP2 and the delayed transition into FP3. Key Points Laboratory experiments of infiltration dynamics in a vertical fracture between two sandstone blocks are compared to analytical solutions Complex fracture flow dynamics can be simplified as plug flow due to the strong effects of matrix imbibition during the wetting process Above critical inflow rates, only two of three conceptual flow periods are required to recover bulk infiltration in our experiments

We study the temporal evolution of solute dispersion in three‐dimensional porous rocks of different heterogeneity and pore structure. To this end, we perform direct numerical simulations of pore‐scale flow and transport in a sand pack, which exhibits mild heterogeneity, and a Berea sandstone, which is characterized by strong heterogeneity as measured by the variance of the logarithm of the flow velocity. The impact of medium structure and pore‐scale mass transfer mechanisms is probed by effective and ensemble dispersion coefficients. The former is a measure for the typical width of a plume, while the latter for the deformation, that is, the spread of a mixing front. Both dispersion coefficients evolve from the molecular diffusion coefficients toward a common finite asymptotic value. Their behavior is governed by the interplay between diffusion, pore‐scale velocity fluctuations and medium structure, which determine the characteristic mass transfer time scales. We find distinctly different dispersion behaviors in the two media, which can be traced back to how particles sample pore‐scale velocity variability and how this depends on the medium structure. These are key elements for the upscaling of transport, mixing and reaction from the pore to the continuum scale. Key Points Pore‐scale simulations of temporal evolution of solute dispersion in three‐dimensional porous rocks The impact of medium structure and diffusion on solute transport is studied in terms of effective and ensemble dispersion coefficients The evolution of dispersion coefficients indicates how particles sample pore‐scale velocities and how this depends on the medium structure

We show that the return‐point memory of cyclic macroscopic trajectories enables the derivation of a thermodynamic framework for quasistatically driven dissipative systems with multiple metastable states. We use this framework to sort out and quantify the energy dissipated in quasistatic fluid‐fluid displacements in disordered media. Numerical computations of imbibition–drainage cycles in a quasi‐2D medium with gap thickness modulations (imperfect Hele‐Shaw cell) show that energy dissipation in quasistatic displacements is due to abrupt changes in the fluid‐fluid configuration between consecutive metastable states (Haines jumps), and its dependence on microstructure and gravity. The relative importance of viscous dissipation is deduced from comparison with quasistatic experiments. Plain Language Summary Fluid flow into a porous material filled with another is not only an everyday process (gardening, stains in fabrics, or printing) but is also a key process affecting the water cycle, contamination in soils and storage of energy or hazardous waste in the subsurface. These flows are controlled by the energy of the fluids, and its dissipation during their advancement, making the knowledge of energy dissipation crucial to our ability to predict these phenomena. However, to date there is no rigorous way to evaluate this energy. This paper describes a novel method that overcomes this challenge, explaining how the properties of the medium affect dissipation and showing why even for very slow flows the viscous energy (i.e., related to rapid fluid motion) still makes a difference. Key Points Rigorous account of the microscopic physics allows to compute the energy dissipated between consecutive two‐phase configurations We link the microscopic origins of hysteresis and dissipation to the macroscopic pressure‐saturation behavior Quasistatic pressure‐driven experiments point to a secondary contribution of viscous dissipation during Haines jumps

We study the upscaling of advective pore-scale dispersion in terms of the Eulerian velocity distribution and advective tortuosity, both flow attributes, and of the average pore length, a medium attribute. The stochastic particle motion is modeled as a time-domain random walk, in which particles move along streamlines in equidistant spatial steps with random velocities and thus random transition times. Particle velocities describe stationary spatial Markov processes, which evolve along streamlines on the mean pore length. The streamwise motion is projected onto the mean flow direction using tortuosity. This upscaled stochastic particle model predicts accurately the (non-Fickian) transport dynamics obtained from direct numerical simulations of particle transport in a three-dimensional digitized Berea sandstone sample. It captures all aspects of transport and sheds light on the dependence of the upscaled transport behavior on the flow heterogeneity and the initial particle distribution, which are critical for the accurate modeling of dispersion from the pre-asymptotic to asymptotic regimes.

In fractured and stress‐sensitive reservoirs and aquifers, hydromechanical coupling is important, in connection with their heat and solute transport properties, and because the fluid production or extraction leads to land subsidence and potentially to induced seismicity. Classical dual‐porosity poroelasticity (DPP) models cannot upscale pressure diffusion and deformation in fractured porous media, which are characterized by anomalous behaviors that manifest in strong tailing in the temporal evolution of flow rate and subsidence. We study these behaviors using detailed numerical simulations of fluid production in naturally fractured formations characterized by multi‐Gaussian distributions of the matrix permeability. We find that the tailing behaviors depend on the permeability contrast between fracture and matrix, on the permeability distribution in the matrix, and on the correlation length. We use a non–equilibrium, multi‐porosity model to quantify the coupled behaviors of anomalous pressure diffusion, fluid flow and deformation. The model is parameterized by medium and fluid properties, which set the characteristic pressure diffusion time scales. It allows to identify the emerging scaling regimes and scaling behaviors of flow rate and subsidence. We propose a model implementation that captures the full anomalous evolution of flow rates and displacements observed in the detailed numerical simulations in terms of the permeability distribution and matrix length scales. The presented results shed new light on the controls of medium heterogeneity and geometry on pressure diffusion, fluid production and subsidence in highly heterogeneous fractured media. Key Points Fluid production and subsidence are studied for three‐dimensional fractured poroelastic media Strong tailing in the temporal evolution of the fluid production rate and displacement is observed A multi‐porosity model explains scaling regimes and behaviors, and quantifies the full evolution of coupled flow and deformation

Spatial Markov models (SMM) are an efficient approach to simulate transport in heterogeneous media across scales. They represent particle transport by equidistant spatial transitions with correlated random velocities, which renders the associated transition times correlated random variables. While SMM perform excellently for modeling purely advective transport, incorporating diffusion remains challenging. So far, applying SMM to advective‐diffusive transport in porous media has been mostly restricted to using empirical transition matrices based on numerical simulations. Using a transition matrix for advective‐diffusive transport obliterates the fundamental differences between the two transport processes and is conflicting with the goal of replacing explicit transport simulations by a SMM. Here, we present an advective‐diffusive SMM that conceptualizes diffusion as jumps between advective trajectories, that is, diffusion competes with advection for changing the particle velocity. At each particle transition, a random diffusion time is compared to the current advection time. If the advection time is shorter than the diffusion time, the particle remains on its current SMM trajectory and the longitudinal velocity correlation is kept. If the diffusion time is shorter, the particle velocity is reset. Breakthrough curves and their first and second moments calculated with the advective‐diffusive SMM are in agreement with three‐dimensional numerical simulations in heterogeneous log‐conductivity fields with isotropic, exponential covariance function with variances up to five.

We investigate the occurrence of anomalous (non‐Fickian) transport in an hydrological catchment system at kilometer scales and over a 36‐year period. Using spectral analysis, we examine the fluctuation scaling of long‐term time series measurements of a natural passive tracer (chloride), for rainfall and runoff. The scaling behavior can be described by a continuous time random walk (CTRW) based on a power‐law distribution of transition times, which indicates two distinct power‐law regimes in the distribution of overall travel times in the catchment. The CTRW provides a framework for assessing anomalous transport in catchments and its implications for water quality fluctuations. Plain Language Summary Rain falling on an hydrological catchment, and chemicals dissolved in the rain, can follow circuitous pathways below the ground surface until they reach a stream outlet that drains the catchment. Dissolved chemicals can diffuse into lower conductivity regions within the subsurface, and chemicals can also be transported in relatively fast pathways. We investigate a unique data set that monitors chemical transport over kilometer scales, and over a long, 36‐year duration. We develop a mathematical framework to describe the transport and retention of chemical tracers in a catchment, and their arrival times to a draining outlet. Solutions of the equations exhibit characteristic features of tracer concentration variations, and offer a means to characterize and quantity catchment response to chemical inputs. Key Points An hydrological catchment system at kilometer scales is shown to exhibit anomalous (non‐Fickian) transport over a 36‐year period A continuous time random walk suggests two distinct power‐law regimes in the distribution of overall catchment travel times In the catchments considered here, preferential flow appears to occur at all length and time scales

Gas migration in coal is strongly controlled by surface diffusion of adsorbed gas within the coal matrix. Surface diffusion coefficients are obtained by inverse modelling of transient gas desorption data from powdered coals. The diffusion coefficient is frequently considered to be dependent on time and initial pressure. In this article, it is shown that the pressure dependence can be eliminated by performing a joint inversion of both the diffusion coefficient and adsorption isotherm. A study of the log–log slope of desorbed gas production rate against time reveals that diffusion within the individual coal particles is a multi-rate process. The application of a power-law probability density function of diffusion rates enables the determination of a single gas diffusion coefficient that is constant in both time and initial pressure.