Incorporating Super‐Diffusion due to Sub‐Grid Heterogeneity to Capture Non‐Fickian Transport

Baeumer, Boris; Zhang, Yong; Schumer, R.

doi:10.1111/gwat.12267

Cited by 16 publications

(20 citation statements)

References 41 publications

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“…These studies confirm that mechanical dispersion in the governing equation is negligible if the numerical model can capture the layered distribution of hydraulic conductivity or channeling velocity. For regional‐scale transport processes with limited subsurface information or intermediate‐scale repacked sand columns without a layering structure, mechanical dispersion might still be needed to explain the local deviation from mean velocity (Baeumer et al ).…”

Section: Discussionmentioning

confidence: 99%

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

et al. 2017

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Time nonlocal transport models such as the time fractional advection-dispersion equation (t-fADE) were proposed to capture well-documented non-Fickian dynamics for conservative solutes transport in heterogeneous media, with the underlying assumption that the time nonlocality (which means that the current concentration change is affected by previous concentration load) embedded in the physical models can release the effective dispersion coefficient from scale dependency. This assumption, however, has never been systematically examined using real data. This study fills this historical knowledge gap by capturing non-Fickian transport (likely due to solute retention) documented in the literature (Huang et al. 1995) and observed in our laboratory from small to intermediate spatial scale using the promising, tempered t-fADE model. Fitting exercises show that the effective dispersion coefficient in the t-fADE, although differing subtly from the dispersion coefficient in the standard advection-dispersion equation, increases nonlinearly with the travel distance (varying from 0.5 to 12 m) for both heterogeneous and macroscopically homogeneous sand columns. Further analysis reveals that, while solute retention in relatively immobile zones can be efficiently captured by the time nonlocal parameters in the t-fADE, the motion-independent solute movement in the mobile zone is affected by the spatial evolution of local velocities in the host medium, resulting in a scale-dependent dispersion coefficient. The same result may be found for the other standard time nonlocal transport models that separate solute retention and jumps (i.e., displacement). Therefore, the t-fADE with a constant dispersion coefficient cannot capture scale-dependent dispersion in saturated porous media, challenging the application for stochastic hydrogeology methods in quantifying real-world, preasymptotic transport. Hence improvements on time nonlocal models using, for example, the novel subordination approach are necessary to incorporate the spatial evolution of local velocities without adding cumbersome parameters.

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Section: Discussionmentioning

confidence: 99%

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

et al. 2017

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“…Baeumer et al . [] proposed the following model to capture super‐diffusion along flow lines using subordination:

\frac{\partial C}{\partial t} = - \nabla_{\vec{v}} C + σ {true(\nabla_{\vec{v}} true)}^{α} C + \nabla true[D (\overset{x}{true}, t) \nabla C true],

where C

[M L^{- 3}]

denotes the solute concentration;

\vec{v}

[L T^{- 1}]

is the velocity vector; σ [dimensionless] is a scalar factor;

1 < α < 2

[dimensionless] is the order of the space fractional derivative; and

D (\overset{x}{true}, t)

[L^{2} T^{- 1}]

is the diffusion coefficient. The advection operator

\nabla_{\vec{v}}

is defined via

\nabla_{\vec{v}} C = \nabla (\overset{v}{true} C)

.…”

Section: Time and Space Nonlocal Transport Modelsmentioning

confidence: 99%

“…This transformation is known as subordination [ Baeumer et al ., ; Schumer et al ., ; Sokolov and Metzler , ] and can capture mechanical dispersion due to the local variation from the mean velocity; see Baeumer et al . [] for a detailed explanation.…”

Section: Time and Space Nonlocal Transport Modelsmentioning

confidence: 99%

“…Laboratory experiments also revealed that the desorption rates of radionuclides from volcanic rocks are very widely distributed, yielding a

3 \sim 4

order of magnitude range of desorption rate constants [ Dean and Reimus , ]. Second, early arrivals can be observed for tracer transport through highly permeable preferential flow paths, where the solute particle's trajectory can certainly follow the streamlines [ Baeumer et al ., ]. An extension of the standard vector fADE can therefore be practically important, as will be discussed further in the following.…”

Section: Introductionmentioning

confidence: 99%

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

Zhang

Meerschaert

Baeumer

et al. 2015

Water Resources Research

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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.

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“…Following equation in Baeumer et al . [], we randomize the dynamics of particles undergoing the above continuity equation; i.e., we assume that there are particles that are moving much faster (relatively speaking) than the average and some are moving slower. This randomization is done by assuming that the resulting distribution (for particle displacements deviating from the local average) is a tempered stable distribution; i.e., the subordinator satisfies

\frac{\partial}{\partial t} g (t, τ) = - \frac{\partial}{\partial τ} g (t, τ) + σ {(\frac{\partial}{\partial τ})}^{α, λ_{τ}} g (t, τ); g (0, τ) = δ (τ),

where σ (dimensionless) is a scalar factor, and the symbol

{false(\partial / \partial τ false)}^{α, λ_{τ}}

denotes the subordination operator with an index

0 < α \leq 2

and a truncation parameter

λ_{τ}

[L^{- 1}]

.…”

Section: Methodology Developmentmentioning

confidence: 99%

A fully subordinated linear flow model for hillslope subsurface stormflow

Zhang

Baeumer

Chen

et al. 2017

Water Resources Research

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Hillslope subsurface stormflow exhibits complex patterns when natural soils with multiscale heterogeneity impart a spatiotemporally nonlocal memory on flow dynamics. To efficiently quantify such nonlocal flow responses, this technical note proposes a fully subordinated flow (FSF) equation where the time‐ and flow‐subordination capture the temporal and spatial memory, respectively. Results show that the time‐subordination component of the FSF model captures a wide range of delayed flow response due to various degrees of soil heterogeneity (especially for low‐conductivity zones), while the model's flow‐subordination term accounts for the rapid flow responses along preferential flow paths. In the FSF model, parameters defining spatiotemporal memory functions may be related to soil properties, while other parameters such as scalar factors controlling the overall advection and diffusion are difficult to predict and can be estimated from subsurface stormflow hydrographs. These parameters can be constants at the hillslope scale because the spatiotemporal subordination, an upscaling technique, can capture the impact of system heterogeneity on flow dynamics, leading to a linear FSF model that might be applicable for various slopes. Valid scale, limitation and extension of the FSF model, and modification of the model for other complex hydrological dynamics are also discussed.

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Incorporating Super‐Diffusion due to Sub‐Grid Heterogeneity to Capture Non‐Fickian Transport

Cited by 16 publications

References 41 publications

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

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

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

A fully subordinated linear flow model for hillslope subsurface stormflow

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