Upscaling transport of a reacting solute through a peridocially converging–diverging channel at pre-asymptotic times

Abstract

In this study we extend the Spatial Markov model, which has been successfully used to upscale conservative transport across a diverse range of porous media flows, to test if it can accurately upscale reactive transport, defined by a spatially heterogeneous first order degradation rate. We test the model in a well known highly simplified geometry, commonly considered as an idealized pore or fracture structure, a periodic channel with wavy boundaries. The edges of the flow domain have a layer through which there is no flow, but in which diffusion of a solute still occurs. Reactions are confined to this region. We demonstrate that the Spatial Markov model, an upscaled random walk model that enforces correlation between successive jumps, can reproduce breakthrough curves measured from microscale simulations that explicitly resolve all pertinent processes. We also demonstrate that a similar random walk model that does not enforce successive correlations is unable to reproduce all features of the measured breakthrough curves.

Stochastic dynamics of intermittent pore‐scale particle motion in three‐dimensional porous media: Experiments and theory

“…Various pore‐scale upscaling models have been proposed to extract the relevant mechanisms that control transport in porous media, albeit with a trade‐off between computational efficiency and faithful representation of the real pore geometry. Take, for instance, the geometric simplification of the pore space through use of sinusoidal wavy channels in theoretical studies of pore‐scale reactive transport [ Bolster et al , ; Le Borgne et al , ; Sund et al , ]. Such idealized pore approaches approximate the transit time distribution and spatial correlation properties of more complex media, which permits modeling sizable samples with ease.…”

“…The continuous time random walk (CTRW) model has become a growingly popular framework for predicting anomalous transport in heterogeneous media [ Dentz and Berkowitz , ; Berkowitz et al , ; de Anna et al , ; Kang et al , ; Le Borgne et al , ; Holzner et al , ; Le Borgne et al , ; Sund et al , ; Tyukhova et al , ]. At its core, CTRW models describe effective transport by discretizing the solute into a large number of particles that move as a sequence of transitions in space and time.…”

“…Holzner et al [] implement correlation through a persistence of particle velocity parameter that allows a particle to change velocities at turning points based on a probabilistic value. The work by Le Borgne et al [] and the various others that follow it [ Kang et al , , ; Le Borgne et al , ; Sund et al , ; de Anna et al , ] effects a conditional correlation via velocity transition matrices, which condition the probability density of transition time on the observed value in the previous step. Meyer and Bijeljic [] more recently account for velocity correlations through a velocity‐direction‐angle process that reflects the recurrent focusing and defocusing of flow at the pore scale.…”

Trajectories as Training Images to Simulate Advective‐Diffusive, Non‐Fickian Transport

“…Spatial Markov models (SMMs; Bolster & Dentz, 2012;Kang et al, 2014;Sund et al, 2016) are a family of models that represent memory of speed and direction through the use of a transition matrix. While these models have been applied to a broad range of systems (e.g., Bolster et al, 2014;Kang et al, 2014Kang et al, , 2015aKang et al, , 2016Le Borgne et al, 2008b, 2008aSund et al, 2015bSund et al, , 2015aSund et al, , 2017bSund et al, , 2017a, they typically rely on parameterizing a transition matrix, which can be difficult to do, although approaches applicable to real data have emerged recently (Kang et al, 2015a;Sherman et al, 2017). Another approach has involved sampling of particle trajectories (Sund et al, 2017b), which in turn can be used for mixed upscaling and downscaling models.…”

Predicting Downstream Concentration Histories From Upstream Data in Column Experiments

“…Traditionally, independence is assumed between successive particle jumps. However, this assumption has been shown to be violated in a multitude of studies (Bolster et al, ; de Anna et al, ; Le Borgne et al, ; Morales et al, ; Sund et al, ). The spatial Markov model (SMM) accounts for velocity correlation between particle jumps and has proven particularly advantageous when predicting transport in systems displaying intermittent behavior (de Anna et al, ; Kang et al, ).…”