Upscaling and Prediction of Lagrangian Velocity Dynamics in Heterogeneous Porous Media

Abstract

The understanding of the dynamics of Lagrangian velocities is key for the understanding and upscaling of solute transport in heterogeneous porous media. The prediction of large‐scale particle motion in a stochastic framework implies identifying the relation between the Lagrangian velocity statistics and the statistical characteristics of the Eulerian flow field and the hydraulic medium properties. In this paper, we approach both challenges from numerical and theoretical points of view. Direct numerical simulations of Darcy‐scale flow and particle motion give detailed information on the evolution of the statistics of particle velocities both as a function of travel time and distance along streamlines. Both statistics evolve from a given initial distribution to different steady‐state distributions, which are related to the Eulerian velocity probability density function. Furthermore, we find that Lagrangian velocities measured isochronally as a function of travel time show intermittency dominated by low velocities, which is removed when measured equidistantly as a function of travel distance. This observation gives insight into the stochastic dynamics of the particle velocity series. As the equidistant particle velocities show a regular random pattern that fluctuates on a characteristic length scale, it is represented by two stationary Markov processes, which are parametrized by the distribution of flow velocities and a correlation distance. The velocity Markov models capture the evolution of the Lagrangian velocity statistics in terms of the Eulerian flow properties and a characteristics length scale and shed light on the role of the initial conditions and flow statistics on large‐scale particle motion.

Anomalous Transport in Three‐Dimensional Discrete Fracture Networks: Interplay Between Aperture Heterogeneity and Injection Modes

“…Many studies in the literature (Benke & Painter, 2003; Hyman et al., 2019b; Kang, Brown, et al., 2016; Le Borgne et al., 2008; Sund et al., 2016) have shown that the Lagrangian velocity series { v n }, if sampled in space, can be modeled as Markov processes. These observations have led to the development of CTRW formulations based on velocity Markov models (Dentz et al., 2016; Hakoun et al., 2019a; Kang et al., 2017; Le Borgne et al., 2008) that are able to predict anomalous transport in porous and fractured media (Comolli et al., 2019; Hyman et al., 2019b; Kang, Brown, et al., 2016; Kang et al., 2014; Le Borgne et al., 2008; Sherman et al., 2020; Sund et al., 2016). The empirical estimation of the transition matrix, equation (), from numerical or experimental data can be a challenging issue in practice (Sherman et al., 2017).…”

Transport Upscaling in Highly Heterogeneous Aquifers and the Prediction of Tracer Dispersion at the MADE Site

“…The temporal variation patterns of particle velocities are in general rather complex. They are intermittent because their evolution is governed by a broad distribution of times scales, which are determined by the flow speeds and characteristic length scales imprinted in the medium structure (Dentz et al., 2016; Hakoun et al., 2019). The complexity of intermittent temporal series of particle velocities can be removed by adopting an equidistant point of view.…”

Transport under advective trapping

“…The methodology is based on a Lagrangian approach that allows to identify and quantify the stochastic rules of advective particle motion in disordered media. Similar approaches have been used in previous works for the analysis and upscaling of pore-scale transport (Morales et al 2017;Puyguiraud et al 2019) and for transport in multi-Gaussian hydraulic conductivity fields (Hakoun et al 2019) and fractured media (Hyman et al 2019). Here, we use a Lagrangian approach to gain understanding of the stochastic principles of transport in random composite media through the analysis of advective trapping events in low conductivity inclusions, and the distribution of flow speeds sampled between them.…”