Effective Models for Transport in Complex Heterogeneous Hydrologic Systems

Abstract

Time and time again, transport behaviors that cannot be described by traditional simple advection dispersion equations (ADE) are observed in hydrologic systems, both in surface and subsurface environments. Such departures from classical predictions have been termed anomalous. In all cases, the presence of processes and heterogeneities spanning a broad range of temporal and spatial scales leads to a breakdown in assumptions inherent to classical ADEs and complicates modeling efforts. Developing approaches that account for these complexities naturally leads to the use of upscaling and stochastic modeling frameworks. We briefly introduce these ideas from both Lagrangian and Eulerian perspectives. We start with classical ADE models, highlighting many of the assumptions they are built on. By relaxing these assumptions, models capable of reproducing observed anomalous behaviors have been developed. Here, we briefly review three of the most popular models: continuous time random walks (CTRW), fractional advection dispersion equations (fADE), and multirate mass transfer (MRMT). While all three have enjoyed success, they too have their limitations. We conclude this article with a discussion of these shortcomings and present three very recent models that aim to address them as well as a brief discussion on future challenges.

The chemical continuous time random walk framework for upscaling transport limitations in fluid–solid reactions

“…From both a theoretical and a numerical perspective, it is convenient to adopt a conceptualization of transport in terms of Lagrangian tracer particles. Each Lagrangian particle represents a macroscopic number of physical reactant particles undergoing advection-diffusion [61,62] and subject to reaction near the interface. Disregarding reaction for the moment, particle positions X(t) as a function of time t are described by the Langevin equation (see, e.g., [63])…”

“…Numerically, we implement these dynamics using particle tracking random walk (PTRW) simulations, which discretize the Langevin equation (1) (see, e.g., [61,62] and Appendix A for further details). If a fluid-reactant particle is in the reactive region during a time step of duration ∆t, its mass evolves according to…”

The diffusing-velocity random walk: a spatial-Markov formulation of heterogeneous advection and diffusion

“…Transport processes in heterogeneous media are determined by the sampling of the underlying heterogeneous flow field through advection and diffusion, leading to rich dynamical behaviour and departure from classical Fickian dynamics (Berkowitz et al 2006;Klages, Radons & Sokolov 2008). The evolution of Lagrangian velocities along particle trajectories can be modelled as a stochastic process, taking into account the statistical properties of the underlying flow field and diffusion (Pope 2011; Sund, Aquino & Bolster 2019). Such approaches are greatly simplified if the changes in velocity may be conceptualized as a Markov process, that is, if their evolution depends only on the current state and not on past history (Meyer & Tchelepi 2010;Meyer & Saggini 2016).…”

“…Stochastic Lagrangian methods describe transport in terms of random particle displacements and associated transit times (Sund et al 2019). The stochastic character of these models reflects the statistical properties of the underlying heterogeneity, which can be conceptualized in different manners.…”

Impact of velocity correlations on longitudinal dispersion in space-Lagrangian advective transport models

“…Lagrangian random walk methods attempt to predict anomalous transport features, such as faster or slower plume dispersion and a preponderance of late arrivals at a control plane, by discretizing transported solute masses into a collection of particles undergoing movement according to stochastic rules. These rules encode the statistical variability of medium and flow properties in terms of the distribution of the duration and length of particle jumps [4,5].…”