Continuous time random walks for the evolution of Lagrangian velocities

Abstract

We develop a continuous time random walk (CTRW) approach for the evolution of Lagrangian velocities in steady heterogeneous flows based on a stochastic relaxation process for the streamwise particle velocities. This approach describes persistence of velocities over a characteristic spatial scale, unlike classical random walk methods, which model persistence over a characteristic time scale. We first establish the relation between Eulerian and Lagrangian velocities for both equidistant and isochrone sampling along streamlines, under transient and stationary conditions. Based on this, we develop a space continuous CTRW approach for the spatial and temporal dynamics of Lagrangian velocities. While classical CTRW formulations have non-stationary Lagrangian velocity statistics, the proposed approach quantifies the evolution of the Lagrangian velocity statistics under both stationary and non-stationary conditions. We provide explicit expressions for the Lagrangian velocity statistics, and determine the behaviors of the mean particle velocity, velocity covariance and particle dispersion. We find strong Lagrangian correlation and anomalous dispersion for velocity distributions which are tailed toward low velocities as well as marked differences depending on the initial conditions. The developed CTRW approach predicts the Lagrangian particle dynamics from an arbitrary initial condition based on the Eulerian velocity distribution and a characteristic correlation scale.

Linking Structural and Transport Properties in Three‐Dimensional Fracture Networks

“…Under ergodic conditions, this means for a sufficiently large injection volume and flow domain, the steady space Lagrangian PDF and the Eulerian velocity PDFs are related through flux‐weighting as shown in (Comolli & Dentz, ; Dentz et al, ; Kang et al, ): …”

“…Focusing on the macroscale network structure, we quantify the differences in fracture geometric, for example, size and aperture, and topological, for example, connectivity and position in the network relative to inflow and outflow boundaries, properties between the networks using a graph‐based approach (Hyman et al, ). Flow and transport through the networks is interpreted using a stochastic convective streamtube model and a Bernoulli continuous time random walk model (Dentz et al, ), both of which provide insights into the link between fracture structure and transport properties.…”

An Analytical Model of Fickian and Non-Fickian Dispersion in Evolving-Scale Log-Conductivity Distributions

“…Due to the linearity implied by Equations (2), (24), and (25), the derivation of D Mij for log-conductivity fields characterized by evolving scales of heterogeneity and power-law semi-variograms can be pursued by computing the corresponding coefficient for any stationary field of the continuous hierarchy (exponential covariance for 0 < b <1 and Gaussian covariance for 1 ≤ b < 2) and by integrating the result over the truncated frequency domain. See Appendix A for the derivation of D Mij (∞, Λ) in the case of 3-D stationary exponential and Gaussian log-K covariance.…”

“…The moments of the flux-weighted curves were persistently lower than those characterizing the uniform resident injection case, although their propagation rates were converging toward some common value. Dentz et al [24] developed a continuous time random walk approach for the evolution of Lagrangian velocities in steady heterogeneous flows based on a stochastic relaxation process for the streamwise particle velocities. They predicted Lagrangian particle dynamics starting from an arbitrary initial condition based on the Eulerian velocity distribution and a characteristic correlation scale.…”

“…The main result of the study can be synthesized in the detection of strong Lagrangian correlation and anomalous dispersion for velocity distributions that are tailed toward low values, as well as in pronounced differences depending on the initial conditions. Note that all mentioned studies ( [20][21][22][23][24]) investigate the relationship between injection modes and advective heterogeneity on solute dispersion without discussing the effect of local dispersion magnitude. This effect is expected to be definitely more pronounced for transport in porous media, which is the focus of the present study.…”