Upscaling Mixing in Highly Heterogeneous Porous Media via a Spatial Markov Model

Abstract

In this work, we develop a novel Lagrangian model able to predict solute mixing in heterogeneous porous media. The Spatial Markov model has previously been used to predict effective mean conservative transport in flows through heterogeneous porous media. In predicting effective measures of mixing on larger scales, knowledge of only the mean transport is insufficient. Mixing is a small scale process driven by diffusion and the deformation of a plume by a non-uniform flow. In order to capture these small scale processes that are associated with mixing, the upscaled Spatial Markov model must be extended in such a way that it can adequately represent fluctuations in concentration. To address this problem, we develop downscaling procedures within the upscaled model to predict measures of mixing and dilution of a solute moving through an idealized heterogeneous porous medium. The upscaled model results are compared to measurements from a fully resolved simulation and found to be in good agreement.

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

“…2019; Wright et al. 2019). Despite the popularity and practical success of spatial-Markov methods over the last decade, a mechanistic model of the role of diffusion in this type of framework remains unavailable.…”

“…In some cases, simple Markov processes such as Bernoulli relaxation or Ornstein-Uhlenbeck for the spatial evolution of Lagrangian velocities have been shown to capture the key features of purely advective transport, leading to efficient methods for predicting larger-scale transport properties such as longitudinal dispersion (Comolli, Hakoun & Dentz 2019;Puyguiraud, Gouze & Dentz 2019a). In recent years, spatial-Markov models have also been extensively employed to describe conservative transport in porous media (Kang et al 2011(Kang et al , 2014, fractured media (Kang et al 2015(Kang et al , 2017, surface flows (Sherman et al 2017), and inertial and turbulent flows (Bolster et al 2014;Sund et al 2015;Kim & Kang 2020), as well as mixing and reaction (Sund et al 2017a;Sund, Porta & Bolster 2017b;Sherman et al 2019;Wright et al 2019). Despite the popularity and practical success of spatial-Markov methods over the last decade, a mechanistic model of the role of diffusion in this type of framework remains unavailable.…”

Upscaling of Anomalous Pore-Scale Dispersion

“…The quantification of pre-asymptotic dispersion and its causes in the medium and flow properties is a critical issue for upscaling hydrodynamic transport from the pore to the Darcy scale. Pre-asymptotic (non-Fickian) dispersion on the pore and Darcy scales have been modeled by a variety of non-local approaches (Neuman and Tartakovsky, 2009), such as the multirate mass transfer (MRMT) approach (Haggerty and Gorelick, 1995;Carrera et al, 1998), volume averaging and two-equation formulations for transport (Cherblanc et al, 2007;Davit et al, 2010;Porta et al, 2013), the continuous time and time-domain random walk approaches (Berkowitz and Scher, 1995;Dentz and Berkowitz, 2003;Berkowitz et al, 2006;Bijeljic and Blunt, 2006;Wright et al, 2019;Sund et al, 2015Sund et al, , 2017Sherman et al, 2019), see also the recent review by Noetinger et al (2016). A critical step for implementing these non-local models concerns the relation between the velocity statistics that are controlled by the pore-scale structure, and the macroscopic transport process.…”

Upscaling of Solute Plumes in Periodic Porous Media Through a Trajectory‐Based Spatial Markov Model

“…The advantages of employing a spatial Markov approach to obtain the solute breakthrough curve (or first passage time) at a given longitudinal distance has been demonstrated in a number of previous works, relying on both numerical and laboratory‐scale experimental data sets (e.g., Bolster et al, 2014; Le Borgne et al, 2011; Sherman, Bianchi Janetti, et al, 2020; Sherman et al, 2018). Several recent works have discussed methodologies that employ Lagrangian SMM‐like approaches to predict solute particles' space‐time locations at various scales of observations (Russian et al, 2016; Wright et al, 2019). Yet, to the best of our knowledge, this approach has not been applied to the explicit space‐time reconstruction of solute plumes starting from pore‐scale properties.…”