A Residual a Posteriori Error Estimators for a Model for Flow in Porous Media with Fractures

Hecht, Frédéric; Mghazli, Zoubida; Naji, Ilyas; Roberts, Jean E.

doi:10.1007/s10915-018-0875-7

Cited by 6 publications

(10 citation statements)

References 25 publications

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“…The use of an iterative method to solve the system on the global interface requires an appropriate preconditioner. Possible directions for extending this work include modeling transport, and handling nonconforming meshes, either using mortar elements [56,57,16,55] or controlling the nonconformity by using a posteriori error estimates as in [39,51].…”

Section: Discussionmentioning

confidence: 99%

“…The authors of [16] note that the choice of mortar variable (pressure trace of flux) influences the analysis. In that context, our work is closer to [33] or [39]. Indeed, the mortar variable is supported by the fractures, which already have a natural mesh.…”

Section: Introductionmentioning

confidence: 90%

“…The models have been extended to deal with non-linear flow in the fractures (Forchheimer model), see [33,1] and multiphase flow, see [17,37,38,43,35,58,41]. A posteriori error estimates were studied in [51,39]. Domain decomposition (DD) methods appear naturally in the simulation of fractured media.…”

Section: Introductionmentioning

confidence: 99%

“…This is what enables a comparatively simple analysis in Sections 3 and 4. The numerical part of the work could be extended to non-matching grids along the lines of [33,32,29,51,39], and the analysis in [16] handles the non-conforming case by using mortars. The authors of [16] note that the choice of mortar variable (pressure trace of flux) influences the analysis.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, the mortar variable is supported by the fractures, which already have a natural mesh. Additionally [39] shows that the error due the nonconformity can be controlled through a posteriori error estimation.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Intersecting fractures in porous media: mathematical and numerical analysis

Amir

Kern

Mghazli³

et al. 2021

Applicable Analysis

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Intersecting fractures in porous media: mathematical and numerical analysis

Amir

Kern

Mghazli³

et al. 2021

Applicable Analysis

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Adaptive interior penalty hybridized discontinuous Galerkin methods for Darcy flow in fractured porous media

Leng

Chen

2023

IMA Journal of Numerical Analysis

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In this paper, we design and analyze an interior penalty hybridized discontinuous Galerkin (IP-HDG) method for the Darcy flow in the two- and three-dimensional fractured porous media. The discrete fracture model is used to model the fractures. The piecewise polynomials of degree $k$ are employed to approximate the pressure in the fractures and the pressure in the surrounding porous media. We prove that the IP-HDG method is well posed if the penalty parameter is large enough. Based on the discrete solutions of pressures, the discrete Darcy velocity in the matrix and the reduced fractures can be recovered, respectively, to be locally mass-conservative. A robust residual-based a posteriori error estimator is established for an energy-norm of pressure. Finally, numerical results are provided to show the efficiency of the proposed a posteriori error estimator.

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Linearization of the Travel Time Functional in Porous Media Flows

Houston¹,

Rourke²,

Zee³

2022

SIAM J. Sci. Comput.

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The travel time functional measures the time taken for a particle trajectory to travel from a given initial position to the boundary of the domain. Such evaluation is paramount in the post-closure safety assessment of deep geological storage facilities for radioactive waste where leaked, non-sorbing, solutes can be transported to the surface of the site by the surrounding groundwater. The accurate simulation of this transport can be attained using standard dual-weighted-residual techniques to derive goal-oriented a posteriori error bounds. This work provides a key aspect in obtaining a suitable error estimate for the travel time functional: the evaluation of its Gâteaux derivative. A mixed finite element method is implemented to approximate Darcy's equations and numerical experiments are presented to test the performance of the proposed error estimator. In particular, we consider a test case inspired by the Sellafield site located in Cumbria, in the UK.

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A Residual a Posteriori Error Estimators for a Model for Flow in Porous Media with Fractures

Cited by 6 publications

References 25 publications

Intersecting fractures in porous media: mathematical and numerical analysis

Intersecting fractures in porous media: mathematical and numerical analysis

Adaptive interior penalty hybridized discontinuous Galerkin methods for Darcy flow in fractured porous media

Linearization of the Travel Time Functional in Porous Media Flows

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