Analysis and Upscaling of a Reactive Transport Model in Fractured Porous Media with Nonlinear Transmission Condition

Pop, Sorin; Bogers, J.J.P.; Kumar, Kundan

doi:10.1007/s10013-016-0198-7

Cited by 14 publications

(22 citation statements)

References 36 publications

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“…Further applications can be encountered in fluid flow through thin filters, built up by an array of obstacles, see e. g., [3]. In [17], a reactive transport model with an additional convective contribution in the thin layer and a nonlinear transmission condition of Dirichlet type at the bulk-layer interface was considered. In the latter, however, a thin homogeneous layer was considered.…”

Section: Markus Gahn Maria Neuss-radu and Peter Knabnermentioning

confidence: 99%

Effective interface conditions for processes through thin heterogeneous layers with nonlinear transmission at the microscopic bulk-layer interface

Gahn¹,

Neuss‐Radu²,

Knabner³

2018

Networks &Amp; Heterogeneous Media

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In this paper, we consider a system of reaction-diffusion equations in a domain consisting of two bulk regions separated by a thin layer with thickness of order and a periodic heterogeneous structure. The equations inside the layer depend on and the diffusivity inside the layer on an additional parameter γ ∈ [−1, 1]. On the bulk-layer interface, we assume a nonlinear Neumann-transmission condition depending on the solutions on both sides of the interface. For → 0, when the thin layer reduces to an interface Σ between two bulk domains, we rigorously derive macroscopic models with effective conditions across the interface Σ. The crucial part is to pass to the limit in the nonlinear terms, especially for the traces on the interface between the different compartments. For this purpose, we use the method of two-scale convergence for thin heterogeneous layers, and a Kolmogorov-type compactness result for Banach valued functions, applied to the unfolded sequence in the thin layer. 2010 Mathematics Subject Classification. 35B27, 35K57, 35K61, 80M40. Key words and phrases. Homogenization, nonlinear transmission conditions, reaction-diffusion systems, effective interface conditions, unfolding and averaging operator for thin heterogeneous layer, weak and strong two-scale convergence. The work of the first author was supported by the Center for Modelling and Simulation in the Biosciences (BIOMS) at the University of Heidelberg.

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Section: Markus Gahn Maria Neuss-radu and Peter Knabnermentioning

confidence: 99%

Effective interface conditions for processes through thin heterogeneous layers with nonlinear transmission at the microscopic bulk-layer interface

Gahn¹,

Neuss‐Radu²,

Knabner³

2018

Networks &Amp; Heterogeneous Media

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“…This section is concerned with the existence of a (weak) solution to Problem P ε for a fixed fracture width ε > 0. We proceed in the spirit of [50], where a linear model for reactive flow with nonlinear transmission conditions at the interfaces is considered. For the sake of readability, we drop the superscript ε since it is fixed throughout this section.…”

Section: Existencementioning

confidence: 99%

“…This corresponds to scenarios in which the inverse fracture width is an upper bound for the ratio of the fracture porosity to the matrix and the fracture width is a lower bound for the ratio of the fracture hydraulic conductivity to the matrix. We employ techniques from [18], where upscaling was considered in the context of crystal dissolution and precipitation, and [50], which is concerned with the upscaling of a reactive transport model. For a more detailed presentation of the results, we refer the reader to [39].…”

Section: Compactness Arguments Give Rise To the Following Convergent mentioning

confidence: 99%

Rigorous Upscaling of Unsaturated Flow in Fractured Porous Media

List¹,

Kumar²,

Pop³

et al. 2020

SIAM J. Math. Anal.

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In this work, we consider a mathematical model for flow in a unsaturated porous medium containing a fracture. In all subdomains (the fracture and the adjacent matrix blocks) the flow is governed by Richards' equation. The submodels are coupled by physical transmission conditions expressing the continuity of the normal fluxes and of the pressures. We start by analyzing the case of a fracture having a fixed width-length ratio, called ε > 0. Then we take the limit ε → 0 and give a rigorous proof for the convergence towards effective models. This is done in different regimes, depending on how the ratio of porosities and permeabilities in the fracture, respectively matrix scale with respect to ε, and leads to a variety of effective models. Numerical simulations confirm the theoretical upscaling results.

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“…Also, nonlinear transmission conditions may be of relevance when considering reactive flow in fractured media. Pop et al [124] employed nonlinear transmission conditions for reactive flow in fractured media. Recently, the phase field modeling of flow in fractured media has been considered by Lee et al [125] and Mikelić et al [126,127].…”

Section: Discrete Fracture Modelmentioning

confidence: 99%

Flow and Transport in Tight and Shale Formations: A Review

et al. 2017

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A review on the recent advances of the flow and transport phenomena in tight and shale formations is presented in this work. Exploration of oil and gas in resources that were once considered inaccessible opened the door to highlight interesting phenomena that require attention and understanding. The length scales associated with transport phenomena in tight and shale formations are rich. From nanoscale phenomena to field-scale applications, a unified frame that is able to encounter the varieties of phenomena associated with each scale may not be possible. Each scale has its own tools and limitations that may not, probably, be suitable at other scales. Multiscale algorithms that effectively couple simulations among various scales of porous media are therefore important. In this article, a review of the different length scales and the tools associated with each scale is introduced. Highlights on the different phenomena pertinent to each scale are summarized. Furthermore, the governing equations describing flow and transport phenomena at different scales are investigated. In addition, methods to solve these equations using numerical techniques are introduced. Cross-scale analysis and derivation of linear and nonlinear Darcy's scale laws from pore-scale governing equations are described. Phenomena occurring at molecular scales and their thermodynamics are discussed. Flow slippage at the nanosize pores and its upscaling to Darcy's scale are highlighted. Pore network models are discussed as a viable tool to estimate macroscopic parameters that are otherwise difficult to measure. Then, the environmental aspects associated with the different technologies used in stimulating the gas stored in tight and shale formations are briefly discussed.

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Analysis and Upscaling of a Reactive Transport Model in Fractured Porous Media with Nonlinear Transmission Condition

Cited by 14 publications

References 36 publications

Effective interface conditions for processes through thin heterogeneous layers with nonlinear transmission at the microscopic bulk-layer interface

Effective interface conditions for processes through thin heterogeneous layers with nonlinear transmission at the microscopic bulk-layer interface

Rigorous Upscaling of Unsaturated Flow in Fractured Porous Media

Flow and Transport in Tight and Shale Formations: A Review

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