The role of scaling laws in upscaling

Wood, Brian D.

doi:10.1016/j.advwatres.2008.08.015

Cited by 104 publications

(87 citation statements)

References 25 publications

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“…Bourgeat and Piatnitski (2004) show that, if separation of scale is possible, effective properties in random media converge as the scale of the unit cell increases, independent of its boundary conditions (periodic, Dirichlet, or Neumann). For the same reason the method of volume averaging (Bachmat and Bear 1983;Bourgeat et al 1988;Marle 1982;Whitaker 1999) also uses periodic boundary conditions to obtain transport coefficients (Wood 2009). …”

Section: Introductionmentioning

confidence: 99%

“…This article, however, does not as yet incorporate the computation of these coefficients. Another appealing aspect of homogenization (Hornung 1997;Mikelic and Rosier 2004;Sanchez-Palencia 1980) is that it does not need a closure relation for obtaining the macroscale transport equation as opposed to volume averaging (Wood 2009). However, the relevant model equations are not conventional and not easy to solve (Tardif d'Hamonville et al 2007).…”

Section: Introductionmentioning

confidence: 99%

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Computation of the Longitudinal and Transverse Dispersion Coefficient in an Adsorbing Porous Medium Using Homogenization

2011

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This article compares for the first time, local longitudinal and transverse dispersion coefficients obtained by homogenization with experimental data of dispersion coefficients in porous media, using the correct porosity dependence. It is shown that the longitudinal dispersion coefficient can be reasonably represented by a simple periodic unit cell (PUC), which consists of a single sphere in a cube. We present a slightly modified and simplified approach to derive the homogenized equations, which emphasizes physical aspects of homogenization. Subsequently, we give full dimensional expressions for the dispersion tensor based on a comparison with the convective dispersion equation used for contaminant transport, inclusive the correct dependence on porosity. For the PUC of choice, the dispersion relations are identical to the relations obtained for periodic media. We show that commercial finite element software can be readily used to compute longitudinal and transverse dispersion coefficients in 2D and 3D. The 3D results are for the first time obtained at relevant Peclet numbers. There is good agreement for longitudinal dispersion. The computed transverse dispersion coefficients for a single sphere in a cube are much too low. The effect of adsorption on the dispersion coefficient is also studied. Adsorption does not affect the transverse dispersion coefficient. However, adsorption enhances the longitudinal dispersion coefficient in agreement with an analysis of homogenization applied to Taylor dispersion discussed in the literature.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Computation of the Longitudinal and Transverse Dispersion Coefficient in an Adsorbing Porous Medium Using Homogenization

2011

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“…A different analysis, using Green's functions (cf. [52]), yields similar expressions for the perturbations. In this case, the solution is also decomposed into components corresponding to the different source terms.…”

Section: Perturbationsmentioning

confidence: 56%

Hydrodynamic dispersion within porous biofilms

et al. 2013

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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. Many microorganisms live within surface-associated consortia, termed biofilms, that can form intricate porous structures interspersed with a network of fluid channels. In such systems, transport phenomena, including flow and advection, regulate various aspects of cell behavior by controlling nutrient supply, evacuation of waste products, and permeation of antimicrobial agents. This study presents multiscale analysis of solute transport in these porous biofilms. We start our analysis with a channel-scale description of mass transport and use the method of volume averaging to derive a set of homogenized equations at the biofilm-scale in the case where the width of the channels is significantly smaller than the thickness of the biofilm. We show that solute transport may be described via two coupled partial differential equations or telegrapher's equations for the averaged concentrations. These models are particularly relevant for chemicals, such as some antimicrobial agents, that penetrate cell clusters very slowly. In most cases, especially for nutrients, solute penetration is faster, and transport can be described via an advection-dispersion equation. In this simpler case, the effective diffusion is characterized by a second-order tensor whose components depend on (1) the topology of the channels' network; (2) the solute's diffusion coefficients in the fluid and the cell clusters; (3) hydrodynamic dispersion effects; and (4) an additional dispersion term intrinsic to the two-phase configuration. Although solute transport in biofilms is commonly thought to be diffusion dominated, this analysis shows that hydrodynamic dispersion effects may significantly contribute to transport.

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“…Averaging is the formal part of the coarse-graining process while upscaling involves both mathematics and the intellectual or conceptual part of the modeling process [26]. Averaging can be done, in principle, for any heterogeneous subsurface system (such as the soil one), regardless of its complexity, or the spatial variation of its properties.…”

Section: Development Of Aggregation Methodsmentioning

confidence: 99%

Multi-scale European Soil Information System (MEUSIS): a multi-scale method to derive soil indicators

2010

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The Multi-scale Soil Information System (MEUSIS) can be a suitable framework for building a nested system of soil data that could facilitate interoperability through a common coordinate reference system, a unique grid coding database, a set of detailed and standardized metadata, and an open exchangeable format. In the context of INSPIRE Directive, MEU-SIS may be implemented as a system facilitating the update of existing soil information and accelerating the harmonization of various soil information systems. In environmental data like the soil one, it is common to generalize accurate data obtained at the field to coarser scales using either the pedotransfer rules or knowledge of experts or even some statistical solutions which combine single values of spatially distributed data. The most common statistical process for generalization is averaging the values within the study area. In this paper, we do not present a simple averaging of numerical values without any further processed information. The upscaling process is accompanied with significant statistical analysis in order to demonstrate the method suitability. The coarser resolution nested P. Panagos (B) · M. Van Liedekerke · L. Montanarella grids cells (10 × 10 km) represent broad regions where the calculated soil property (e.g., organic carbon) can be accurately upscaled. Multi-scaled approaches are urgently required to integrate different disciplines (such as Statistics) and provide a meta-model platform to improve current mechanistic modeling frameworks, request new collected data, and identify critical research questions. Past papers have described in detail the upscaling methodology while our present approach is to demonstrate an important application of this methodology accompanied with statistical evidence.

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The role of scaling laws in upscaling

Cited by 104 publications

References 25 publications

Computation of the Longitudinal and Transverse Dispersion Coefficient in an Adsorbing Porous Medium Using Homogenization

Computation of the Longitudinal and Transverse Dispersion Coefficient in an Adsorbing Porous Medium Using Homogenization

Hydrodynamic dispersion within porous biofilms

Multi-scale European Soil Information System (MEUSIS): a multi-scale method to derive soil indicators

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