Exact transverse macro dispersion coefficients for transport in heterogeneous porous media

We study transport through heterogeneous media. We derive the exact large scale transport equation. The macro dispersion coefficients are determined by additional partial differential equations. In the case of infinite Peclet numbers, we present explicit results for the transverse macro dispersion coefficients. In two spatial dimensions, we demonstrate that the transverse macro dispersion coefficient is zero. The result is not limited on lowest order perturbation theory approximations but is an exact result. However, the situation in three spatial dimensions is very different: The transverse macro dispersion coefficients are finite - a result which is confirmed by numerical simulations we performe

Transport under advective trapping

Advective trapping occurs when solute enters low velocity zones in heterogeneous porous media. Classical local modelling approaches combine the impact of slow advection and diffusion into a hydrodynamic dispersion coefficient and many temporally non-local approaches lump these mechanisms into a single memory function. This joint treatment makes parameterization difficult and thus prediction of large-scale transport a challenge. Here, we investigate the mechanisms of advective trapping and their impact on transport in media composed of a high conductivity background and isolated low permeability inclusions. Breakthrough curves show that effective transport changes from a streamtube-like behaviour to genuine random trapping as the degree of disorder of the inclusion arrangement increases. We upscale this behaviour using a Lagrangian view point, in which idealized solute particles transition over a fixed distance at random advection times combined with Poissonian advective trapping events. We discuss the mathematical formulation of the upscaled model in the continuous time random walk and mobile-immobile mass transfer frameworks, and derive a model for large-scale solute non-Fickian dispersion. These findings give new insight into transport in highly heterogeneous media. © 2020 BMJ Publishing Group. All rights reserved

Intermittent Lagrangian velocities and accelerations in three-dimensional porous medium flow

Intermittency of Lagrangian velocity and acceleration is a key to understanding transport in complex systems ranging from fluid turbulence to flow in porous media. High-resolution optical particle tracking in a three-dimensional (3D) porous medium provides detailed 3D information on Lagrangian velocities and accelerations. We find sharp transitions close to pore throats, and low flow variability in the pore bodies, which gives rise to stretched exponential Lagrangian velocity and acceleration distributions characterized by a sharp peak at low velocity, superlinear evolution of particle dispersion, and double-peak behavior in the propagators. The velocity distribution is quantified in terms of pore geometry and flow connectivity, which forms the basis for a continuous-time random-walk model that sheds light on the observed Lagrangian flow and transport behaviors

Coupling of mass transfer and reactive transport for nonlinear reactions in heterogeneous media

Fast chemical reactions are driven by mixing‐induced chemical disequilibrium. Mixing is poorly represented by the advection‐dispersion equation. Instead, effective dynamics models, such as multirate mass transfer (MRMT), have been successful in reproducing observed field‐scale transport, notably, breakthrough curves (BTCs) of conservative solutes. The objective of this work is to test whether such effective models, derived from conservative transport observations, can be used to describe effective multicomponent reactive transport in heterogeneous media. We use a localized formulation of the MRMT model that allows us to solve general reactive transport problems. We test this formulation on a simple three‐species mineral precipitation problem at equilibrium. We first simulate the spatial and temporal distribution of mineral precipitation rates in synthetic hydraulically heterogeneous aquifers. We then compare these reaction rates to those corresponding to an equivalent (i.e., same conservative BTC) homogenized medium with transport characterized by a nonlocal in time equation involving a memory function. We find an excellent agreement between the two models in terms of cumulative precipitated mass for a broad range of generally stationary heterogeneity structures. These results indicate that mass transfer models can be considered to represent quite accurately the large‐scale effective dynamics of mixing controlled reactive transport at least for the cases tested here, where individual transport paths sample the full range of heterogeneities represented by the BTC

Variable density flow in porous media

We review the state of the art in modeling of variable-density flow and transport in porous media, including conceptual models for convection systems, governing balance equations, phenomenological laws, constitutive relations for fluid density and viscosity, and numerical methods for solving the resulting nonlinear multifield problems. The discussion of numerical methods addresses strategies for solving the coupled spatio-temporal convection process, consistent velocity approximation, and error-based mesh adaptation techniques. As numerical models for those nonlinear systems must be carefully verified in appropriate tests, we discuss weaknesses and inconsistencies of current model-verification methods as well as benchmark solutions. We give examples of field-related applications to illustrate specific challenges of further research, where heterogeneities and large scales are important

Mechanisms of dispersion in a porous medium

This paper studies the mechanisms of dispersion in the laminar flow through the pore space of a three-dimensional porous medium. We focus on preasymptotic transport prior to the asymptotic hydrodynamic dispersion regime, in which solute motion may be described by the average flow velocity and a hydrodynamic dispersion coefficient. High-performance numerical flow and transport simulations of solute breakthrough at the outlet of a sand-like porous medium evidence marked deviations from the hydrodynamic dispersion paradigm and identify two distinct regimes. The first regime is characterised by a broad distribution of advective residence times in single pores. The second regime is characterised by diffusive mass transfer into low-velocity regions in the wake of solid grains. These mechanisms are quantified systematically in the framework of a time-domain random walk for the motion of marked elements (particles) of the transported material quantity. Particle transitions occur over the length scale imprinted in the pore structure at random times given by heterogeneous advection and diffusion. Under globally advection-dominated conditions, i.e., Péclet numbers larger than 1, particles sample the intrapore velocities by diffusion and the interpore velocities through advection. Thus, for a single transition, particle velocities are approximated by the mean pore velocity. In order to quantify this advection mechanism, we develop a model for the statistics of the Eulerian velocity magnitude based on Poiseuille’s law for flow through a single pore and for the distribution of mean pore velocities, both of which are linked to the distribution of pore diameters. Diffusion across streamlines through immobile zones in the wake of solid grains gives rise to exponentially distributed residence times that decay on the diffusion time over the pore length. The trapping rate is determined by the inverse diffusion time. This trapping mechanism is represented by a compound Poisson process conditioned on the advective residence time in the proposed time-domain random walk approach. The model is parameterised with the characteristics of the porous medium under consideration and captures both preasymptotic regimes. Macroscale transport is described by an integro-differential equation for solute concentration, whose memory kernels are given in terms of the distribution of mean pore velocities and trapping times. This approach quantifies the physical non-equilibrium caused by a broad distribution of mass transfer time scales, both advective and diffusive, on the representative elementary volume (REV). Thus, while the REV indicates the scale at which medium properties like porosity can be uniquely defined, this does not imply that transport can be characterised by hydrodynamic dispersion

Probabilistic risk analysis of groundwater remediation strategies

Heterogeneity of subsurface environments and insufficient site characterization are some of the reasons why decisions about groundwater exploitation and remediation have to be made under uncertainty. A typical decision maker chooses between several alternative remediation strategies by balancing their respective costs with the probability of their success or failure. We conduct a probabilistic risk assessment (PRA) to determine the likelihood of the success of a permeable reactive barrier, one of the leading approaches to groundwater remediation. While PRA is used extensively in many engineering fields, its applications in hydrogeology are scarce. This is because rigorous PRA requires one to quantify structural and parametric uncertainties inherent in predictions of subsurface flow and transport. We demonstrate how PRA can facilitate a comprehensive uncertainty quantification for complex subsurface phenomena by identifying key transport processes contributing to a barrier's failure, each of which is amenable to uncertainty analysis. Probability of failure of a remediation strategy is computed by combining independent and conditional probabilities of failure of each process. Individual probabilities can be evaluated either analytically or numerically or, barring both, can be inferred from expert opinio

Some Insights in Superdiffusive Transport

In this paper we deal with high-order corrections for the Fractional Derivative approach to anomalous diffusion, in super-diffusive regime, which become relevand whenever one attempts to describe the behavior of particles close to normal diffusion.Comment: 14 pages, 7 figure

Monte Carlo evaluation of FADE approach to anomalous kinetics

In this paper we propose a comparison between the CTRW (Monte Carlo) and Fractional Derivative approaches to the modelling of anomalous diffusion phenomena in the presence of an advection field. Galilei variant and invariant schemes are revised.Comment: 13 pages, 6 figure