Comparison study of phase-field and level-set method for three-phase systems including two minerals

Kelm, Mathis; Gärttner, Stephan; Bringedal, Carina; Flemisch, Bernd; Knabner, Peter; Ray, Nadja

doi:10.1007/s10596-022-10142-w

Cited by 8 publications

(2 citation statements)

References 27 publications

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“…However, the corresponding effective macroscopic problems have been derived by formal asymptotic expansion only. Numerical simulations and analytical discussion of such type of limit models can be found in [Gae+20], [Gae+22], [Kel+22].…”

Section: Introductionmentioning

confidence: 99%

Homogenisation of local colloid evolution induced by reaction and diffusion

Wiedemann¹,

Peter²

2022

Preprint

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We consider the homogenisation of a coupled reaction-diffusion process in a porous medium with evolving microstructure. A concentrationdependent reaction rate at the interface of the pores with the solid matrix induces a concentration-dependent evolution of the domain. Hence, the evolution is fully coupled with the reaction-diffusion process. In order to pass to the homogenisation limit, we employ the two-scale-transformation method. Thus, we homogenise a highly non-linear problem in a periodic and in time cylindrical domain instead. The homogenisation result is a reaction-diffusion equation, which is coupled with an internal variable, representing the local evolution of the pore structure. Contents 1. Introduction 1 2. The mathematical model 3 2.1. Weak formulation 6 2.2. Transformation of the domain 6 2.3. Transformation of the weak form 10 3. Existence and uniform a priori estimates 10 4. Derivation of the limit problem for the periodic substitute problem 20 5. Back-transformation 30 6. Acknowledgements 32 References 32 2020 Mathematics Subject Classification. 35B27, 35K57, 35R35. Key words and phrases. Homogenization; evolving microstructure; free boundary problem; two-scale convergence; porous medium; reaction-diffusion process. D.W. was partially supported by a doctoral scholarship provided by the Studienstiftung des deutschen Volkes.

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Section: Introductionmentioning

confidence: 99%

Homogenisation of local colloid evolution induced by reaction and diffusion

Wiedemann¹,

Peter²

2022

Preprint

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“…In addition, the accuracy of the approximations in this case has not been proven. Numerical methods for such two-scale (micro-macro) models are discussed, for example, in [11,12,15]. Further details on the approaches and relevant references can be found in [10,12,14].…”

Section: Introductionmentioning

confidence: 99%

Homogenized Models with Memory Effect for Heterogeneous Periodic Media

Sandrakov

Семенов

2022

JODEA

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The homogenization of initial boundary value problems for heat conduction equations with asymptotically degenerate rapidly oscillating periodic coefficients are considered. Such problems model thermal processes in heterogeneous periodic media. Homogenized problems (whose solutions determine approximate asymptotics for solutions of the original problems) are presented. Estimates for the accuracy of the asymptotics and relevant convergence theorem are discussed. The homogenized problems have the form of initial boundary value problems for integro-differential equations in convolutions. The presence of convolutions in models for media is called the memory effect. Statements about the solvability and regularity for the problems and the homogenized problems are proved. These results are optimal even in the case of zero convolutions, when the homogenized problems coincide with the classical heat conduction problems.

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Upscaling and Effective Behavior for Two-Phase Porous-Medium Flow Using a Diffuse Interface Model

Kelm,

Bringedal,

Flemisch

2024

Transp Porous Med

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We investigate two-phase flow in porous media and derive a two-scale model, which incorporates pore-scale phase distribution and surface tension into the effective behavior at the larger Darcy scale. The free-boundary problem at the pore scale is modeled using a diffuse interface approach in the form of a coupled Allen–Cahn Navier–Stokes system with an additional momentum flux due to surface tension forces. Using periodic homogenization and formal asymptotic expansions, a two-scale model with cell problems for phase evolution and velocity contributions is derived. We investigate the computed effective parameters and their relation to the saturation for different fluid distributions, in comparison to commonly used relative permeability saturation curves. The two-scale model yields non-monotone relations for relative permeability and saturation. The strong dependence on local fluid distribution and effects captured by the cell problems highlights the importance of incorporating pore-scale information into the macro-scale equations.

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Comparison study of phase-field and level-set method for three-phase systems including two minerals

Cited by 8 publications

References 27 publications

Homogenisation of local colloid evolution induced by reaction and diffusion

Homogenisation of local colloid evolution induced by reaction and diffusion

Homogenized Models with Memory Effect for Heterogeneous Periodic Media

Upscaling and Effective Behavior for Two-Phase Porous-Medium Flow Using a Diffuse Interface Model

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