Mechanisms of dispersion in a porous medium

Abstract

This paper studies the mechanisms of dispersion in the laminar flow through the pore space of a 3-dimensional porous medium. We focus on pre-asymptotic 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 characterized by a broad distribution of advective residence times in single pores. The second regime is characterized 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, this means 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 parameterized with the characteristics of the porous medium under consideration and captures both pre-asymptotic 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 characterized by hydrodynamic dispersion.

Molecular Information Delivery in Porous Media

“…To this end, we first analyze the (cumulative) breakthrough curve, i.e., the cumulative density function (CDF) of the first arrival time at the outlet of any molecule released from the inlet, which is used for characterizing molecular transport in the PM. This is given by [10]…”

“…Proof: The proofs of Corollaries 1 and 2 are given in In this section, we present numerical results to investigate the channel response and communication performance of MC via the PM. We consider the 3D sand-like PM described in [8], [10]. The medium was generated according to the characteristics of standard sand samples.…”

“…The other parameters are given in Table I. With these parameters, [10] numerically solved (1) and (2), obtaining the values of F (t) for Pe = 3, 30, 300, 1000. The results in the following figures are obtained based on this simulation data.…”

“…The molecules undergo complex trajectories and experience heterogeneous advection as they propagate through pores of different sizes and lengths, causing so-called mechanical dispersion [6], [8], which is an augmented effective diffusion caused by velocity fluctuations. More importantly, particles may become trapped in immobile or re-circulation zones in the vicinity or the wake of solid grains [9], [10], therefore taking some time to exit, and causing non-trivial anomalous transport phenomena, such as long tails Y. Fang and N. Yang are with the Research School of Electrical, Energy and Materials Engineering, The Australian National University, Canberra, ACT 2600, Australia (e-mail: {yuting.fang, nan.yang}@anu.edu.au).…”

Generalized multirate models for conjugate transfer in heterogeneous materials

“…While transport in weakly heterogeneous media can be accurately described using stochastic perturbative approaches [13] (see [14,15,16] for an extensive review), typical flow structures and exchange phenomena arising from strong heterogeneities (see for example [17,18]) can not be captured by low order expansions. In fact, predictions from these methods show significant discrepancies when compared against observations from field experiments [19,20], numerical simulations (for example [21]) and laboratory experiments [22].…”