Modeling mixed retention and early arrivals in multidimensional heterogeneous media using an explicit Lagrangian scheme

Abstract

This study develops an explicit two-step Lagrangian scheme based on the renewal-reward process to capture transient anomalous diffusion with mixed retention and early arrivals in multidimensional media. The resulting 3-D anomalous transport simulator provides a flexible platform for modeling transport. The first step explicitly models retention due to mass exchange between one mobile zone and any number of parallel immobile zones. The mobile component of the renewal process can be calculated as either an exponential random variable or a preassigned time step, and the subsequent random immobile time follows a Hyper-exponential distribution for finite immobile zones or a tempered stable distribution for infinite immobile zones with an exponentially tempered power-law memory function. The second step describes well-documented early arrivals which can follow streamlines due to mechanical dispersion using the method of subordination to regional flow. Applicability and implementation of the Lagrangian solver are further checked against transport observed in various media. Results show that, although the time-nonlocal model parameters are predictable for transport with retention in alluvial settings, the standard timenonlocal model cannot capture early arrivals. Retention and early arrivals observed in porous and fractured media can be efficiently modeled by our Lagrangian solver, allowing anomalous transport to be incorporated into 2-D/3-D models with irregular flow fields. Extensions of the particle-tracking approach are also discussed for transport with parameters conditioned on local aquifer properties, as required by transient flow and nonstationary media.

Can a Time Fractional‐Derivative Model Capture Scale‐Dependent Dispersion in Saturated Soils?

“…This is because the usage of the ADE model in our work is to identify whether column transport exhibits scale‐dependent dispersion, not to prove that the SRMT or MRMT model can fit the data. The t‐fADE is an MRMT model with an upper‐truncated, power‐law rate coefficient, as demonstrated in our previous work (Zhang et al ).…”

“…There may be three possible solutions for promising fractional‐derivative models. First, one can add a spatial fractional‐derivative term to the t‐fADE (1), resulting in a spatiotemporal fADE model (Zhang et al ). The spatiotemporal fADE can capture both retention and the early arrivals of solutes; therefore it may characterize scale‐dependent dispersion due to mobile particles gradually sampling a wider distribution of velocities.…”

Introducing “The Integrator”: A novel technique to monitor environmental flow systems

“…Eulerian and Lagrangian monitoring allow estimates of control‐volume, average responses of flow systems to upflow events (e.g., runoff from precipitation and plume propagation from spills) or tracer injections and can be done using grab or semi‐continuous sampling (in situ automated sensors). Due to logistical simplicity, the use of Eulerian monitoring to collect solute breakthrough curves (concentration vs. time) is much more common in environmental studies despite known challenges and faults (Nordin and Sabol ; Drummond et al ; González‐Pinzón et al ) (Table ), and Lagrangian approaches tend to be more used in modeling studies (e.g., particle‐tracking analyses; Berkowitz and Scher ; Schumer et al ) because dispersion and transient storage processes create wide distributions of travel times that are difficult to predict and respond to when tracking fluids in motion (Zhang et al , ; Goeury et al ).…”

Identification of Pollutant Source for Super‐Diffusion in Aquifers and Rivers with Bounded Domains

“…The FF‐ADE model (a) contains three unknown parameters: the scale index α , velocity v , and the dispersion coefficient D . Zhang et al () found that α varies from 1.10 to 1.30, resulting in an average value of 1.20. For simplicity, we also assume that the space‐dependent dispersion coefficient is a linear function of the travel distance ( l ).…”

“…This technical note develops a backward super‐diffusive model on bounded domains for source identification in both surface and subsurface waters. Super‐diffusive transport is considered here because super‐diffusion due to, for example, preferential flow paths has been documented for tracer transport in heterogeneous porous or fractured media at all scales; see the review in Zhang et al () and examples in this study. The space FADE is superior to standard time‐nonlocal transport models by efficiently capturing early pollutant arrivals in groundwater and rivers (Zhang et al, ).…”